text: A VLA snapshot survey of Stripe 82 at 1–2 GHz 19 area radio surveys to feature in-band spectral index estimates in the catalogue without having to cross match the results with an external survey at a second frequency. Spectral measurements were derived by forming images from the entire bandwidth using the MT-MFS algorithm, identifying components in the Stokes-I images and then extracting error-weighted measurements of α and β for each com- ponent. The clean bias eﬀect remains an issue for snapshot surveys with the VLA, despite our best eﬀorts to minimise it with a con- servative approach to deconvolution. However it also appears that there are signiﬁcant biases in the spectral domain associated with the MT-MFS algorithm. The apparent trend towards ﬂatter spectra with decreasing ﬂux density is shown to be at least partially arti- ﬁcial, and dependent on the intrinsic spectral shape of the source, and the behaviour of the spectral curvature estimates is not deemed to be reliable, although this is expected for anything but the bright- ∼ est components. Above a threshold of 3 mJy we deem our median α measurements to be reliable, and in good agreement with exist- Figure 14. Euclidean-normalized diﬀerential source counts derived from ing measurements derived by cross-matching the results of pairs of the Stripe 82 survey (blue points) overlaid on other observationally-derived independent observing programs at diﬀerent frequencies. measurements by various authors, as tabulated by De Zotti et al. (2010). The use of large scale simulations to investigate the biases in the imaging process has proven to be very informative, however un- like surveys such as FIRST and NVSS it does not seem practical to derive a single correction factor for the clean bias, which appears to θmed = 2S0.3 (12) vary signiﬁcantly from pointing to pointing. A signiﬁcant contrib- uting factor is likely to be the changes in the PSF shape that arise where S is the ﬂux density in mJy. The correction factor f for due to the dynamic scheduling of the observations, which impart R the counts is then given by an undesirable level of non-uniformity to the survey. A deeper ex- amination of how the clean bias correlates with factors such as the PSF shape and the level of residual RFI will be pursued. Although 1 fR = 1 − h(θ). (13) single correction factors to the Stokes-I ﬂux density and the spectral measurements cannot be determined by running large-scale simu- lations on a per-observation basis, as was the method for FIRST In all the above calculations an area-weighted mean value of and NVSS, an estimate of the overall level of the biases can be ob- σ is used. The resulting resolution bias correction factors are unity tained, and indeed should be, in order to properly understand the for all ﬂux density bins. As expected, the counts do not need to be limitations of survey products. The simulations were computation- adjusted to compensate for this eﬀect. ally expensive and required the use of a supercomputer, although Figure 14 shows the ﬁnal values of the diﬀerential source we note that this may be a natural application for cloud computing counts, normalised to a Euclidean universe, with the visibility area platforms. and Eddington bias corrections applied. The raw and normalised Finally, the vastly increased survey potential of the now com- counts, along with the relevant correction parameters are listed in plete VLA upgrade cannot be understated. The sub-millijansky ra- Table 3. The values derived agree with existing observations. The dio sky can now be probed with a snapshot survey with very modest signiﬁcant discrepancies at the bright end are due to the relatively integration times per pointing, and the potential for the instantan- small area of the survey in terms of studying these rare and bright eous characterisation of the bulk spectral behaviour of radio sources objects. We have made no attempts to compensate for this incom- without the need for conducting two surveys (with mismatched pleteness. characteristics) is huge, provided the associated systematics, ex- 5 CONCLUSIONS amples of which are highlighted in this paper, can be understood We have used the Karl G. Jansky Very Large Array to conduct and mitigated. ∼ a snapshot survey covering 100 square degrees of the SDSS Stripe Our images and catalogues are publicly 82 region, reaching a thermal noise level of approximately 50 µJy available, and can be retrieved from the Aus- − − beam 1, but with a typical 1σ eﬀective depth of 88 µJy beam 1. The tralia Telescope National Facility’s FTP server: use of the CnB conﬁguration results in a survey with good sensit- ftp://ftp.atnf.csiro.au/pub/people/hey036/Stripe82. ivity to low surface brightness and diﬀuse emission, highly syner- ACKNOWLEDGEMENTS gistic with existing 1.4 GHz radio imaging of the ﬁeld. Our com- ponent catalogue contains 11,782 point and Gaussian components, We thank the anonymous referee and the MNRAS editorial associated with approximately 8,948 unique radio sources. Photo- staﬀ for providing very useful comments on this paper. The Na- metric and astrometric performance of the survey is excellent, as tional Radio Astronomy Observatory is a facility of the National veriﬁed by comparison with measurements from three external ra- Science Foundation operated under cooperative agreement by As- dio surveys of varying resolution and depth covering the same ﬁeld. sociated Universities, Inc. This work was supported by resources The diﬀerential source counts derived from the catalogue are in provided by the Pawsey Supercomputing Centre with funding from good agreement with previous studies. the Australian Government and the Government of Western Aus- We used the full 1–2 GHz bandwidth available with the VLA’s tralia. IH thanks the Rhodes Centre for Radio Astronomy Tech- L-band system, and in doing so have produced one of the ﬁrst large niques and Technologies (RATT) for the provision of computing MNRAS in press, 1–20 (2016) ##LTFigure## ##LTFigure## ##LTLine##

text: A

text: VLA

text: snapshot

text: survey

text: of

text: Stripe

text: at

text: GHz

text: area

text: radio

text: surveys

text: to

text: feature

text: in-band

text: spectral

text: index

text: estimates

text: in

text: the

text: catalogue

text: without

text: having

text: to

text: cross

text: match

text: the

text: results

text: with

text: an

text: external

text: survey

text: at

text: a

text: second

text: frequency.

text: Spectral

text: measurements

text: were

text: derived

text: by

text: forming

text: images

text: from

text: the

text: entire

text: bandwidth

text: using

text: the

text: MT-MFS

text: algorithm,

text: identifying

text: components

text: in

text: the

text: Stokes-I

text: images

text: and

text: then

text: extracting

text: error-weighted

text: measurements

text: of

text: α

font: EVINBB+rtxmi

text: and

text: β

text: for

text: each

text: com-

text: ponent.

text: The

text: clean

text: bias

text: eﬀect

text: remains

text: an

text: issue

text: for

text: snapshot

text: surveys

text: with

text: the

text: VLA,

text: despite

text: our

text: best

text: eﬀorts

text: to

text: minimise

text: it

text: with

text: a

text: con-

text: servative

text: approach

text: to

text: deconvolution.

text: However

text: it

text: also

text: appears

text: that

text: there

text: are

text: signiﬁcant

text: biases

text: in

text: the

text: spectral

text: domain

text: associated

text: with

text: the

text: MT-MFS

text: algorithm.

text: The

text: apparent

text: trend

text: towards

text: ﬂatter

text: spectra

text: with

text: decreasing

text: ﬂux

text: density

text: is

text: shown

text: to

text: be

text: at

text: least

text: partially

text: arti-

text: ﬁcial,

text: and

text: dependent

text: on

text: the

text: intrinsic

text: spectral

text: shape

text: of

text: the

text: source,

text: and

text: the

text: behaviour

text: of

text: the

text: spectral

text: curvature

text: estimates

text: is

text: not

text: deemed

text: to

text: be

text: reliable,

text: although

text: this

text: is

text: expected

text: for

text: anything

text: but

text: the

text: bright-

text: est

text: components.

text: Above

text: a

text: threshold

text: of

text: mJy

text: we

text: deem

text: our

text: median

text: α

text: measurements

text: to

text: be

text: reliable,

text: and

text: in

text: good

text: agreement

text: with

text: exist-

text: Figure

text: Euclidean-normalized

text: diﬀerential

text: source

text: counts

text: derived

text: from

text: ing

text: measurements

text: derived

text: by

text: cross-matching

text: the

text: results

text: of

text: pairs

text: of

text: the

text: Stripe

text: survey

text: (blue

text: points)

text: overlaid

text: on

text: other

text: observationally-derived

text: independent

text: observing

text: programs

text: at

text: diﬀerent

text: frequencies.

text: measurements

text: by

text: various

text: authors,

text: as

text: tabulated

text: by

text: De

text: Zotti

text: et

text: al.

text: The

text: use

text: of

text: large

text: scale

text: simulations

text: to

text: investigate

text: the

text: biases

text: in

text: the

text: imaging

text: process

text: has

text: proven

text: to

text: be

text: very

text: informative,

text: however

text: un-

text: like

text: surveys

text: such

text: as

text: FIRST

text: and

text: NVSS

text: it

text: does

text: not

text: seem

text: practical

text: to

text: derive

text: a

text: single

text: correction

text: factor

text: for

text: the

text: clean

text: bias,

text: which

text: appears

text: to

text: θmed

text: vary

text: signiﬁcantly

text: from

text: pointing

text: to

text: pointing.

text: A

text: signiﬁcant

text: contrib-

text: uting

text: factor

text: is

text: likely

text: to

text: be

text: the

text: changes

text: in

text: the

text: PSF

text: shape

text: that

text: arise

text: where

text: S

text: is

text: the

text: ﬂux

text: density

text: in

text: mJy.

text: The

text: correction

text: factor

text: f

text: for

text: due

text: to

text: the

text: dynamic

text: scheduling

text: of

text: the

text: observations,

text: which

text: impart

text: R

text: the

text: counts

text: is

text: then

text: given

text: by

text: an

text: undesirable

text: level

text: of

text: non-uniformity

text: to

text: the

text: survey.

text: A

text: deeper

text: ex-

text: amination

text: of

text: how

text: the

text: clean

text: bias

text: correlates

text: with

text: factors

text: such

text: as

text: the

text: PSF

text: shape

text: and

text: the

text: level

text: of

text: residual

text: RFI

text: will

text: be

text: pursued.

text: Although

text: fR

text: h(θ).

text: single

text: correction

text: factors

text: to

text: the

text: Stokes-I

text: ﬂux

text: density

text: and

text: the

text: spectral

text: measurements

text: cannot

text: be

text: determined

text: by

text: running

text: large-scale

text: simu-

text: lations

text: on

text: a

text: per-observation

text: basis,

text: as

text: was

text: the

text: method

text: for

text: FIRST

text: In

text: all

text: the

text: above

text: calculations

text: an

text: area-weighted

text: mean

text: value

text: of

text: and

text: NVSS,

text: an

text: estimate

text: of

text: the

text: overall

text: level

text: of

text: the

text: biases

text: can

text: be

text: ob-

text: σ

text: is

text: used.

text: The

text: resulting

text: resolution

text: bias

text: correction

text: factors

text: are

text: unity

text: tained,

text: and

text: indeed

text: should

text: be,

text: in

text: order

text: to

text: properly

text: understand

text: the

text: for

text: all

text: ﬂux

text: density

text: bins.

text: As

text: expected,

text: the

text: counts

text: do

text: not

text: need

text: to

text: be

text: limitations

text: of

text: survey

text: products.

text: The

text: simulations

text: were

text: computation-

text: adjusted

text: to

text: compensate

text: for

text: this

text: eﬀect.

text: ally

text: expensive

text: and

text: required

text: the

text: use

text: of

text: a

text: supercomputer,

text: although

text: Figure

text: shows

text: the

text: ﬁnal

text: values

text: of

text: the

text: diﬀerential

text: source

text: we

text: note

text: that

text: this

text: may

text: be

text: a

text: natural

text: application

text: for

text: cloud

text: computing

text: counts,

text: normalised

text: to

text: a

text: Euclidean

text: universe,

text: with

text: the

text: visibility

text: area

text: platforms.

text: and

text: Eddington

text: bias

text: corrections

text: applied.

text: The

text: raw

text: and

text: normalised

text: Finally,

text: the

text: vastly

text: increased

text: survey

text: potential

text: of

text: the

text: now

text: com-

text: counts,

text: along

text: with

text: the

text: relevant

text: correction

text: parameters

text: are

text: listed

text: in

text: plete

text: VLA

text: upgrade

text: cannot

text: be

text: understated.

text: The

text: sub-millijansky

text: ra-

text: Table

text: The

text: values

text: derived

text: agree

text: with

text: existing

text: observations.

text: The

text: dio

text: sky

text: can

text: now

text: be

text: probed

text: with

text: a

text: snapshot

text: survey

text: with

text: very

text: modest

text: signiﬁcant

text: discrepancies

text: at

text: the

text: bright

text: end

text: are

text: due

text: to

text: the

text: relatively

text: integration

text: times

text: per

text: pointing,

text: and

text: the

text: potential

text: for

text: the

text: instantan-

text: small

text: area

text: of

text: the

text: survey

text: in

text: terms

text: of

text: studying

text: these

text: rare

text: and

text: bright

text: eous

text: characterisation

text: of

text: the

text: bulk

text: spectral

text: behaviour

text: of

text: radio

text: sources

text: objects.

text: We

text: have

text: made

text: no

text: attempts

text: to

text: compensate

text: for

text: this

text: incom-

text: without

text: the

text: need

text: for

text: conducting

text: two

text: surveys

text: (with

text: mismatched

text: pleteness.

text: characteristics)

text: is

text: huge,

text: provided

text: the

text: associated

text: systematics,

text: ex-

text: CONCLUSIONS

text: amples

text: of

text: which

text: are

text: highlighted

text: in

text: this

text: paper,

text: can

text: be

text: understood

text: We

text: have

text: used

text: the

text: Karl

text: G.

text: Jansky

text: Very

text: Large

text: Array

text: to

text: conduct

text: and

text: mitigated.

text: a

text: snapshot

text: survey

text: covering

text: square

text: degrees

text: of

text: the

text: SDSS

text: Stripe

text: Our

text: images

text: and

text: catalogues

text: are

text: publicly

text: region,

text: reaching

text: a

text: thermal

text: noise

text: level

text: of

text: approximately

text: µJy

text: available,

text: and

text: can

text: be

text: retrieved

text: from

text: the

text: Aus-

text: beam

text: but

text: with

text: a

text: typical

text: 1σ

text: eﬀective

text: depth

text: of

text: µJy

text: beam

text: The

text: tralia

text: Telescope

text: National

text: Facility’s

text: FTP

text: server:

text: use

text: of

text: the

text: CnB

text: conﬁguration

text: results

text: in

text: a

text: survey

text: with

text: good

text: sensit-

text: ivity

text: to

text: low

text: surface

text: brightness

text: and

text: diﬀuse

text: emission,

text: highly

text: syner-

text: ACKNOWLEDGEMENTS

text: gistic

text: with

text: existing

text: GHz

text: radio

text: imaging

text: of

text: the

text: ﬁeld.

text: Our

text: com-

text: ponent

text: catalogue

text: contains

text: point

text: and

text: Gaussian

text: components,

text: We

text: thank

text: the

text: anonymous

text: referee

text: and

text: the

text: MNRAS

text: editorial

text: associated

text: with

text: approximately

text: unique

text: radio

text: sources.

text: Photo-

text: staﬀ

text: for

text: providing

text: very

text: useful

text: comments

text: on

text: this

text: paper.

text: The

text: Na-

text: metric

text: and

text: astrometric

text: performance

text: of

text: the

text: survey

text: is

text: excellent,

text: as

text: tional

text: Radio

text: Astronomy

text: Observatory

text: is

text: a

text: facility

text: of

text: the

text: National

text: veriﬁed

text: by

text: comparison

text: with

text: measurements

text: from

text: three

text: external

text: ra-

text: Science

text: Foundation

text: operated

text: under

text: cooperative

text: agreement

text: by

text: As-

text: dio

text: surveys

text: of

text: varying

text: resolution

text: and

text: depth

text: covering

text: the

text: same

text: ﬁeld.

text: sociated

text: Universities,

text: Inc.

text: This

text: work

text: was

text: supported

text: by

text: resources

text: The

text: diﬀerential

text: source

text: counts

text: derived

text: from

text: the

text: catalogue

text: are

text: in

text: provided

text: by

text: the

text: Pawsey

text: Supercomputing

text: Centre

text: with

text: funding

text: from

text: good

text: agreement

text: with

text: previous

text: studies.

text: the

text: Australian

text: Government

text: and

text: the

text: Government

text: of

text: Western

text: Aus-

text: We

text: used

text: the

text: full

text: GHz

text: bandwidth

text: available

text: with

text: the

text: VLA’s

text: tralia.

text: IH

text: thanks

text: the

text: Rhodes

text: Centre

text: for

text: Radio

text: Astronomy

text: Tech-

text: L-band

text: system,

text: and

text: in

text: doing

text: so

text: have

text: produced

text: one

text: of

text: the

text: ﬁrst

text: large

text: niques

text: and

text: Technologies

text: (RATT)

text: for

text: the

text: provision

text: of

text: computing

text: MNRAS

text: in

text: press,

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: 4 Heywood et al. Figure 1. Coverage of the observed pointings of the survey described in this paper for the eastern (upper) and western (lower) area. The circles show the approximate size of the half-power point of the VLA primary beam at 1.5 GHz. The set of pointings contained within an individual Scheduling Block (SB) are coloured alternately with blue and pink markers. The eastern patch contains 522 pointings. The extra pointing in the western area corresponds to an image derived from the visits to the 2.9 Jy phase calibrator J0059+0006, added to the mosaic to improve the image quality around this source, bringing the total number of pointings contributing to the ﬁnal western mosaic to 505. Note the discontinuity in the western patch due to one of the Scheduling Blocks not being executed. not meet minimum signal-to-noise or number of baselines criteria beam corrected images were cut beyond the radius where the nor- when forming a solution were ﬂagged. malized, total-intensity primary beam gain as computed at the band centre exceeded 0.4. The data were then re-imaged using the FIRST-based cleaning mask. For the ﬁnal images, a second pass of deconvolution was ap- At this point the total intensity images were examined to plied with the mask removed, with deconvolution terminating after identify pointings for which the automatic calibration procedure 2000 iterations or when the peak in the residual image reached 3σ, had been inadequate. Such datasets invariably fell into one of the whichever was sooner. These ﬁnal images were restored using a 16 following categories: (i) a very strong source inside the imaged arcsec × 10 arcsec Gaussian with a position angle (PA) of 90◦ east area; (ii) a strong source outside of the imaged area, the sidelobes of north. This is slightly broader than the generally-achievable an- of which were encroaching into the main image; (iii) morphologic- gular resolution, however it imparts a desirable level of uniformity ally complex sources, or very faint extended sources, for which the to the survey products, and it encompasses the variation in the ﬁt- FIRST-based cleaning mask was inadequate; (iv) residual RFI that ted restoring beams that is caused by the dynamic scheduling of the had been missed by the automatic ﬂagging. observations. Mean values plus/minus one standard deviation of the Case (iv) was remedied by manually editing the visibility data distribution of per pointing major axes, minor axes and PAs for the and reprocessing the Measurement Set using the steps described ± ± ± survey are (13.3 2.1) arcsec, (7.5 0.3) arcsec and (88 15) deg above. Cases (ii) and (iii) were addressed by the addition of out- respectively. lier ﬁelds at the position of the confusing sources, and by set- ting manual cleaning masks. This was done on the post-NRAO The widebandpbcor task was used to correct for primary pipelined data, discarding the automatically generated phase-self- beam eﬀects in the image domain. This task works by making a calibration solutions. Having generated a model via this process spectral cube of a model of the main lobe of the VLA’s primary that was certainly more complete than the model used for the beam, in this case with one frequency plane for each of the 16 automated self-calibration procedure, we executed three iterations SPWs. This cube represents the directional sensitivity of the array of imaging and self-calibration, two passes of phase only self- as a function of frequency. The frequency cube is then transformed calibration (with 30 s and 6 s solution intervals), and one pass of into 2D images for each Taylor term t, in this case for t = [0,1,2], amplitude and phase self-calibration with a single correction ap- which are then used to correct the Taylor term images of the sky plied to the whole 2.5 minute scan. Solutions were generated in brightness distribution. The t = 0 image is practically equivalent to each of the 16 SPWs. This extended self-calibration procedure was the usual narrow-band correction method, i.e. it is divided by a total also used to improve case (i) pointings. Primary beam correction intensity beam model computed at the band centre. Examining the was then applied as described above. Of the 1,026 pointings in the diﬀerence between the widebandpbcor t = 0 image and the eﬀect- survey, 202 (20 percent) of them required manual re-processing, ive sensitivity image produced by the clean task shows deviations and thus a smaller fraction than this were subjected to amplitude of around 1–2% across the ﬁeld. Accurate primary beam-corrected self-calibration. This subset also includes ﬁelds for which the auto- ﬂux densities are recovered, aided by the spacing of the pointings and the weighting method used when forming the mosaic (see be- mated calibration procedure was suﬃcient, but had signiﬁcantly ex- tended sources in the survey which only required a deeper (interact- low). The higher-order Taylor term images should in principle now contain intrinsic spectral index and curvature estimates, with the ive) deconvolution process than our automated procedure aﬀorded. spectral corruption eﬀects of the primary beam removed. Once the Improper amplitude self-calibration can bias the ﬂux density wide band primary beam correction has been applied to the three measurements in the resulting map. This is due to the risk of the Taylor term images the spectral index (α) and spectral curvature contribution to the visibility function made by the faintest sources maps (β) are recomputed as per Equations 3 and 4. All primary which are typically absent from the sky model being subsumed into MNRAS in press, 1–20 (2016) ##LTFigure## ##LTFigure##

label: paragraph

text: Heywood

text: et

text: al.

text: Figure

text: Coverage

label: caption

text: of

text: the

text: observed

text: pointings

text: of

text: the

text: survey

text: described

text: in

text: this

text: paper

text: for

text: the

text: eastern

text: (upper)

text: and

text: western

text: (lower)

text: area.

text: The

text: circles

text: show

text: the

text: approximate

text: size

text: of

text: the

text: half-power

text: point

text: of

text: the

text: VLA

text: primary

text: beam

text: at

text: GHz.

text: The

text: set

text: of

text: pointings

text: contained

text: within

text: an

text: individual

text: Scheduling

text: Block

text: (SB)

text: are

text: coloured

text: alternately

text: with

text: blue

text: and

text: pink

text: markers.

text: The

text: eastern

text: patch

text: contains

text: pointings.

text: The

text: extra

text: pointing

text: in

text: the

text: western

text: area

text: corresponds

text: to

text: an

text: image

text: derived

text: from

text: the

text: visits

text: to

text: the

text: Jy

text: phase

text: calibrator

text: added

text: to

text: the

text: mosaic

text: to

text: improve

text: the

text: image

text: quality

text: around

text: this

text: source,

text: bringing

text: the

text: total

text: number

text: of

text: pointings

text: contributing

text: to

text: the

text: ﬁnal

text: western

text: mosaic

text: to

text: Note

text: the

text: discontinuity

text: in

text: the

text: western

text: patch

text: due

text: to

text: one

text: of

text: the

text: Scheduling

text: Blocks

text: not

text: being

text: executed.

text: not

text: meet

text: minimum

text: signal-to-noise

text: or

text: number

text: of

text: baselines

text: criteria

text: beam

text: corrected

text: images

text: were

text: cut

text: beyond

text: the

text: radius

text: where

text: the

text: nor-

text: when

text: forming

text: a

text: solution

text: were

text: ﬂagged.

text: malized,

text: total-intensity

text: primary

text: beam

text: gain

text: as

text: computed

text: at

text: the

text: band

text: centre

text: exceeded

text: The

text: data

text: were

text: then

text: re-imaged

text: using

text: the

text: FIRST-based

text: cleaning

text: mask.

text: For

text: the

text: ﬁnal

text: images,

text: a

text: second

text: pass

text: of

text: deconvolution

text: was

text: ap-

text: At

text: this

text: point

text: the

text: total

text: intensity

text: images

text: were

text: examined

text: to

text: plied

text: with

text: the

text: mask

text: removed,

text: with

text: deconvolution

text: terminating

text: after

text: identify

text: pointings

text: for

text: which

text: the

text: automatic

text: calibration

text: procedure

text: iterations

text: or

text: when

text: the

text: peak

text: in

text: the

text: residual

text: image

text: reached

text: had

text: been

text: inadequate.

text: Such

text: datasets

text: invariably

text: fell

text: into

text: one

text: of

text: the

text: whichever

text: was

text: sooner.

text: These

text: ﬁnal

text: images

text: were

text: restored

text: using

text: a

text: following

text: categories:

text: (i)

text: a

text: very

text: strong

text: source

text: inside

text: the

text: imaged

text: arcsec

text: arcsec

text: Gaussian

text: with

text: a

text: position

text: angle

text: (PA)

text: of

text: east

text: area;

text: (ii)

text: a

text: strong

text: source

text: outside

text: of

text: the

text: imaged

text: area,

text: the

text: sidelobes

text: of

text: north.

text: This

text: is

text: slightly

text: broader

text: than

text: the

text: generally-achievable

text: an-

text: of

text: which

text: were

text: encroaching

text: into

text: the

text: main

text: image;

text: (iii)

text: morphologic-

text: gular

text: resolution,

text: however

text: it

text: imparts

text: a

text: desirable

text: level

text: of

text: uniformity

text: ally

text: complex

text: sources,

text: or

text: very

text: faint

text: extended

text: sources,

text: for

text: which

text: the

text: to

text: the

text: survey

text: products,

text: and

text: it

text: encompasses

text: the

text: variation

text: in

text: the

text: ﬁt-

text: FIRST-based

text: cleaning

text: mask

text: was

text: inadequate;

text: (iv)

text: residual

text: RFI

text: that

text: ted

text: restoring

text: beams

text: that

text: is

text: caused

text: by

text: the

text: dynamic

text: scheduling

text: of

text: the

text: had

text: been

text: missed

text: by

text: the

text: automatic

text: ﬂagging.

text: observations.

text: Mean

text: values

text: plus/minus

text: one

text: standard

text: deviation

text: of

text: the

text: Case

text: (iv)

text: was

text: remedied

text: by

text: manually

text: editing

text: the

text: visibility

text: data

text: distribution

text: of

text: per

text: pointing

text: major

text: axes,

text: minor

text: axes

text: and

text: PAs

text: for

text: the

text: and

text: reprocessing

text: the

text: Measurement

text: Set

text: using

text: the

text: steps

text: described

text: survey

text: are

text: arcsec,

text: arcsec

text: and

text: deg

text: above.

text: Cases

text: (ii)

text: and

text: (iii)

text: were

text: addressed

text: by

text: the

text: addition

text: of

text: out-

text: respectively.

text: lier

text: ﬁelds

text: at

text: the

text: position

text: of

text: the

text: confusing

text: sources,

text: and

text: by

text: set-

text: ting

text: manual

text: cleaning

text: masks.

text: This

text: was

text: done

text: on

text: the

text: post-NRAO

text: The

text: widebandpbcor

text: task

text: was

text: used

text: to

text: correct

text: for

text: primary

text: pipelined

text: data,

text: discarding

text: the

text: automatically

text: generated

text: phase-self-

text: beam

text: eﬀects

text: in

text: the

text: image

text: domain.

text: This

text: task

text: works

text: by

text: making

text: a

text: calibration

text: solutions.

text: Having

text: generated

text: a

text: model

text: via

text: this

text: process

text: spectral

text: cube

text: of

text: a

text: model

text: of

text: the

text: main

text: lobe

text: of

text: the

text: VLA’s

text: primary

text: that

text: was

text: certainly

text: more

text: complete

text: than

text: the

text: model

text: used

text: for

text: the

text: beam,

text: in

text: this

text: case

text: with

text: one

text: frequency

text: plane

text: for

text: each

text: of

text: the

text: automated

text: self-calibration

text: procedure,

text: we

text: executed

text: three

text: iterations

text: SPWs.

text: This

text: cube

text: represents

text: the

text: directional

text: sensitivity

text: of

text: the

text: array

text: of

text: imaging

text: and

text: self-calibration,

text: two

text: passes

text: of

text: phase

text: only

text: self-

text: as

text: a

text: function

text: of

text: frequency.

text: The

text: frequency

text: cube

text: is

text: then

text: transformed

text: calibration

text: (with

text: s

text: and

text: s

text: solution

text: intervals),

text: and

text: one

text: pass

text: of

text: into

text: images

text: for

text: each

text: Taylor

text: term

text: t,

text: in

text: this

text: case

text: for

text: t

text: amplitude

text: and

text: phase

text: self-calibration

text: with

text: a

text: single

text: correction

text: ap-

text: which

text: are

text: then

text: used

text: to

text: correct

text: the

text: Taylor

text: term

text: images

text: of

text: the

text: sky

text: plied

text: to

text: the

text: whole

text: minute

text: scan.

text: Solutions

text: were

text: generated

text: in

text: brightness

text: distribution.

text: The

text: t

text: image

text: is

text: practically

text: equivalent

text: to

text: each

text: of

text: the

text: SPWs.

text: This

text: extended

text: self-calibration

text: procedure

text: was

text: the

text: usual

text: narrow-band

text: correction

text: method,

text: i.e.

text: it

text: is

text: divided

text: by

text: a

text: total

text: also

text: used

text: to

text: improve

text: case

text: (i)

text: pointings.

text: Primary

text: beam

text: correction

text: intensity

text: beam

text: model

text: computed

text: at

text: the

text: band

text: centre.

text: Examining

text: the

text: was

text: then

text: applied

text: as

text: described

text: above.

text: Of

text: the

text: pointings

text: in

text: the

text: diﬀerence

text: between

text: the

text: widebandpbcor

text: t

text: image

text: and

text: the

text: eﬀect-

text: survey,

text: percent)

text: of

text: them

text: required

text: manual

text: re-processing,

text: ive

text: sensitivity

text: image

text: produced

text: by

text: the

text: clean

text: task

text: shows

text: deviations

text: and

text: thus

text: a

text: smaller

text: fraction

text: than

text: this

text: were

text: subjected

text: to

text: amplitude

text: of

text: around

text: across

text: the

text: ﬁeld.

text: Accurate

text: primary

text: beam-corrected

text: self-calibration.

text: This

text: subset

text: also

text: includes

text: ﬁelds

text: for

text: which

text: the

text: auto-

text: ﬂux

text: densities

text: are

text: recovered,

text: aided

text: by

text: the

text: spacing

text: of

text: the

text: pointings

text: and

text: the

text: weighting

text: method

text: used

text: when

text: forming

text: the

text: mosaic

text: (see

text: be-

text: mated

text: calibration

text: procedure

text: was

text: suﬃcient,

text: but

text: had

text: signiﬁcantly

text: ex-

text: tended

text: sources

text: in

text: the

text: survey

text: which

text: only

text: required

text: a

text: deeper

text: (interact-

text: low).

text: The

text: higher-order

text: Taylor

text: term

text: images

text: should

text: in

text: principle

text: now

text: contain

text: intrinsic

text: spectral

text: index

text: and

text: curvature

text: estimates,

text: with

text: the

text: ive)

text: deconvolution

text: process

text: than

text: our

text: automated

text: procedure

text: aﬀorded.

text: spectral

text: corruption

text: eﬀects

text: of

text: the

text: primary

text: beam

text: removed.

text: Once

text: the

text: Improper

text: amplitude

text: self-calibration

text: can

text: bias

text: the

text: ﬂux

text: density

text: wide

text: band

text: primary

text: beam

text: correction

text: has

text: been

text: applied

text: to

text: the

text: three

text: measurements

text: in

text: the

text: resulting

text: map.

text: This

text: is

text: due

text: to

text: the

text: risk

text: of

text: the

text: Taylor

text: term

text: images

text: the

text: spectral

text: index

text: (α)

text: and

text: spectral

text: curvature

text: contribution

text: to

text: the

text: visibility

text: function

text: made

text: by

text: the

text: faintest

text: sources

text: maps

text: (β)

text: are

text: recomputed

text: as

text: per

text: Equations

text: and

text: All

text: primary

text: which

text: are

text: typically

text: absent

text: from

text: the

text: sky

text: model

text: being

text: subsumed

text: into

text: MNRAS

text: in

text: press,

text: ##LTFigure##

text: ##LTFigure##


text: A VLA snapshot survey of Stripe 82 at 1–2 GHz 5 the gain solutions. The end result of this is suppression of the un- to them. An attempt is made to measure the intrinsic shape of each modelled sources, as well as more subtle, counterintuitive eﬀects, component by deconvolving the restoring beam from the compon- e.g. Grobler et al. (2014), see also Franzen et al. (2015) and Hey- ent, and then ﬁtting a Gaussian component to the result. If no satis- wood et al. (2016). We do not expect to suﬀer signiﬁcantly from factory ﬁts are obtained then the component is deemed to be indis- such self-cal biases, as (i) the model should be largely complete, tinguishable from being point-like. being formed via interactive, constrained deconvolution which was PyBDSM also attempts to identify individual sources within terminated when the residuals contained only noise, or noise plus each island via a pairwise examination of the ﬁtted components. artefacts; (ii) our solution interval was conservative, with only a The criteria which must be satisﬁed for components to be part of single amplitude and phase correction being applied for the entire the same source are (i) the pixels along the line joining the two com- scan; and (iii) the VLA is robust against such biases due to the self ponent centres must all be above the 3σ island threshold, and (ii) calibration problem being well constrained, with 27 complex gain the length of this line is less than the sum of the full width half max- terms being derived from 351 equations. ima of the two components. Generating a source catalogue from a Pointings that were processed manually were re-imaged using radio survey in most cases involves a trade-oﬀ between maximising their individual clean masks and primary beam corrected following the catalogue completeness and minimising the risk of including the procedures used in the au(cid:107)tomated case. Once all the ﬁnal images of spurious features. Although setting a cutoﬀ at 5σ will naturally were in place, the Montage software was used to construct linear result in the exclusion of numerous faint sources, it has long been mosaics in total intensity, α, β, and α-error. For all mosaics, each established that below this threshold Eddington bias (Eddington, pointing is weighted by the assumed variance in the total intensity 1913) will preclude sensible interpretation of the results due to the map (i.e. the square of the assumed Stokes-I primary beam). inclusion of very high numbers of spurious noise features (Condon 3 DATA PRODUCTS 1974; Hogg 2001). Around strong sources in particular, incomplete deconvolution The principal products from this snapshot survey are wide area and residual calibration errors can leave spurious PSF-like features. mosaics for the east and west regions, formed directly (Stokes-I, α, The Y-shape of the VLA results in linear structure with six-fold β, α-error) or derivatively (model, residual and sensitivity) from the symmetry in the PSF for any observation that is not suﬃciently long individual pointing images as described in Section , and the corres- enough to achieve complete Fourier plane coverage, and for snap- ponding lists of components derived from them using automatic shot surveys such as this one these features are very pronounced. source-ﬁnding software. As mentioned in Section , even with tapered weighting of the vis- 3.1 Radio mosaics ibilities the PSF exhibits sidelobes at 10–20 percent of the peak Stokes-I mosaics of the east and west regions are shown in level. This exacerbates the error patterns around sources that do not Figure 2. As each mosaic contains approximately a third of a billion deconvolve properly, and in numerous parts of the mosaic compon- pixels only the brightest sources are obvious in the upper panels ents that do not represent genuine astronomical signals will ﬁnd showing the full area. The discontinuity in the western ﬁeld due to their way into the source catalogue. The position of every entry the unobserved SBs is apparent as the horizontal gap spanning the in the component list with a peak ﬂux density exceeding 100 mJy ◦ × ◦ − width of the ﬁeld. Representative zooms of a 1 1 region in the beam 1 was therefore examined and any visually identiﬁed arte- ◦ × ◦ eastern ﬁeld and a 0.5 0.5 region in the western ﬁeld can be facts were removed. Pointings with known calibration deﬁciencies − seen in the lower panels. The pixel brightness scale runs from 1 (typically around the phase calibrator J0059+0006) were also ex- − to 3 mJy beam 1 for all components of Figure 2, as indicated by the amined for spurious features. In general the use of a local RMS colour bar at the top. The image products are examined further in estimate as opposed to ﬁxed threshold results in a catalogue that is Section . not ﬂux limited but does have high reliability. The spectral indices 3.2 Component catalogue presented in the catalogue are discussed in detail in Section . The ﬁnal eastern catalogue contains 5,674 individual point The source catalogue for this survey was generated by run- and Gaussian components, ﬁtted to 4,602 islands and grouped into ning the Python Blob Detection and Source Management (PyBDSM) 4,354 sources. The western catalogue contains 6,094 components, source ﬁnder (Mohan & Raﬀerty 2015) on each of the wide area 4,869 islands and 4,594 sources. radio images presented in Section . A brief overview of the source Table 1 lists the ﬁrst ten components drawn from the eastern detection algorithm is as follows. The spatial variation of the image catalogue in order to illustrate the structure of the component table. noise was estimated by moving a box across the image in overlap- The columns are deﬁned as follows: ping steps, calculating the root mean square (RMS) of the pixels (1) Identiﬁer for the component formed from its HHMMSS.SS+/- within the box and interpolating the values measured from each × DDMMSS.SS right ascension and declination position in J2000 step. The box size was 250 250 pixels, stepped across the im- coordinates. age in units of 50 pixels. This procedure provided the source ﬁnder (2) Right ascension of the component in degrees. with an estimate of the spatial variation of the image noise for de- (3) Declination of the component in degrees. tection thresholding purposes. This variation was written to a FITS (4) The 1σ uncertainty in right ascension in arcseconds. image in order to provide a map of the sensitivity of the eastern and (5) The 1σ uncertainty in declination in arcseconds. western ﬁelds, and these images are presented in Figure 3. (6) Integrated ﬂux density of the component in mJy. Image peaks that exceeded 5σ where σ is the local value of (7) The 1σ uncertainty in the integrated ﬂux density of the the RMS noise are identiﬁed. These detections were then grown component in mJy. into islands, deﬁned as regions where there is contiguous emission − (8) Peak intensity of the component in mJy beam 1. above a secondary threshold, in this case 3σ. The islands were then (9) The 1σ uncertainty in the peak intensity of the component in decomposed into a component list by ﬁtting Gaussian components − mJy beam 1. (10) Estimate of the local RMS noise at the position of the (cid:107) − http://montage.ipac.caltech.edu/ component in µJy beam 1. MNRAS in press, 1–20 (2016)

text: A

text: VLA

text: snapshot

text: survey

text: of

text: Stripe

text: at

text: GHz

text: the

text: gain

text: solutions.

text: The

text: end

text: result

text: of

text: this

text: is

text: suppression

text: of

text: the

text: un-

text: to

text: them.

text: An

text: attempt

text: is

text: made

text: to

text: measure

text: the

text: intrinsic

text: shape

text: of

text: each

text: modelled

text: sources,

text: as

text: well

text: as

text: more

text: subtle,

text: counterintuitive

text: eﬀects,

text: component

text: by

text: deconvolving

text: the

text: restoring

text: beam

text: from

text: the

text: compon-

text: e.g.

text: Grobler

text: et

text: al.

text: see

text: also

text: Franzen

text: et

text: al.

text: and

text: Hey-

text: ent,

text: and

text: then

text: ﬁtting

text: a

text: Gaussian

text: component

text: to

text: the

text: result.

text: If

text: no

text: satis-

text: wood

text: et

text: al.

text: We

text: do

text: not

text: expect

text: to

text: suﬀer

text: signiﬁcantly

text: from

text: factory

text: ﬁts

text: are

text: obtained

text: then

text: the

text: component

text: is

text: deemed

text: to

text: be

text: indis-

text: such

text: self-cal

text: biases,

text: as

text: (i)

text: the

text: model

text: should

text: be

text: largely

text: complete,

text: tinguishable

text: from

text: being

text: point-like.

text: being

text: formed

text: via

text: interactive,

text: constrained

text: deconvolution

text: which

text: was

text: PyBDSM

text: also

text: attempts

text: to

text: identify

text: individual

text: sources

text: within

text: terminated

text: when

text: the

text: residuals

text: contained

text: only

text: noise,

text: or

text: noise

text: plus

text: each

text: island

text: via

text: a

text: pairwise

text: examination

text: of

text: the

text: ﬁtted

text: components.

text: artefacts;

text: (ii)

text: our

text: solution

text: interval

text: was

text: conservative,

text: with

text: only

text: a

text: The

text: criteria

text: which

text: must

text: be

text: satisﬁed

text: for

text: components

text: to

text: be

text: part

text: of

text: single

text: amplitude

text: and

text: phase

text: correction

text: being

text: applied

text: for

text: the

text: entire

text: the

text: same

text: source

text: are

text: (i)

text: the

text: pixels

text: along

text: the

text: line

text: joining

text: the

text: two

text: com-

text: scan;

text: and

text: (iii)

text: the

text: VLA

text: is

text: robust

text: against

text: such

text: biases

text: due

text: to

text: the

text: self

text: ponent

text: centres

text: must

text: all

text: be

text: above

text: the

text: 3σ

text: island

text: threshold,

text: and

text: (ii)

text: calibration

text: problem

text: being

text: well

text: constrained,

text: with

text: complex

text: gain

text: the

text: length

text: of

text: this

text: line

text: is

text: less

text: than

text: the

text: sum

text: of

text: the

text: full

text: width

text: half

text: max-

text: terms

text: being

text: derived

text: from

text: equations.

text: ima

text: of

text: the

text: two

text: components.

text: Generating

text: a

text: source

text: catalogue

text: from

text: a

text: Pointings

text: that

text: were

text: processed

text: manually

text: were

text: re-imaged

text: using

text: radio

text: survey

text: in

text: most

text: cases

text: involves

text: a

text: trade-oﬀ

text: between

text: maximising

text: their

text: individual

text: clean

text: masks

text: and

text: primary

text: beam

text: corrected

text: following

text: the

text: catalogue

text: completeness

text: and

text: minimising

text: the

text: risk

text: of

text: including

text: the

text: procedures

text: used

text: in

text: the

text: case.

text: Once

text: all

text: the

text: ﬁnal

text: images

text: of

text: spurious

text: features.

text: Although

text: setting

text: a

text: cutoﬀ

text: at

text: 5σ

text: will

text: naturally

text: were

text: in

text: place,

text: the

text: Montage

text: software

text: was

text: used

text: to

text: construct

text: linear

text: result

text: in

text: the

text: exclusion

text: of

text: numerous

text: faint

text: sources,

text: it

text: has

text: long

text: been

text: mosaics

text: in

text: total

text: intensity,

text: α,

text: β,

text: and

text: α-error.

text: For

text: all

text: mosaics,

text: each

text: established

text: that

text: below

text: this

text: threshold

text: Eddington

text: bias

text: (Eddington,

text: pointing

text: is

text: weighted

text: by

text: the

text: assumed

text: variance

text: in

text: the

text: total

text: intensity

text: will

text: preclude

text: sensible

text: interpretation

text: of

text: the

text: results

text: due

text: to

text: the

text: map

text: (i.e.

text: the

text: square

text: of

text: the

text: assumed

text: Stokes-I

text: primary

text: beam).

text: inclusion

text: of

text: very

text: high

text: numbers

text: of

text: spurious

text: noise

text: features

text: (Condon

text: DATA

text: PRODUCTS

text: Hogg

text: Around

text: strong

text: sources

text: in

text: particular,

text: incomplete

text: deconvolution

text: The

text: principal

text: products

text: from

text: this

text: snapshot

text: survey

text: are

text: wide

text: area

text: and

text: residual

text: calibration

text: errors

text: can

text: leave

text: spurious

text: PSF-like

text: features.

text: mosaics

text: for

text: the

text: east

text: and

text: west

text: regions,

text: formed

text: directly

text: (Stokes-I,

text: α,

text: The

text: Y-shape

text: of

text: the

text: VLA

text: results

text: in

text: linear

text: structure

text: with

text: six-fold

text: β,

text: α-error)

text: or

text: derivatively

text: (model,

text: residual

text: and

text: sensitivity)

text: from

text: the

text: symmetry

text: in

text: the

text: PSF

text: for

text: any

text: observation

text: that

text: is

text: not

text: suﬃciently

text: long

text: individual

text: pointing

text: images

text: as

text: described

text: in

text: Section

text: and

text: the

text: corres-

text: enough

text: to

text: achieve

text: complete

text: Fourier

text: plane

text: coverage,

text: and

text: for

text: snap-

text: ponding

text: lists

text: of

text: components

text: derived

text: from

text: them

text: using

text: automatic

text: shot

text: surveys

text: such

text: as

text: this

text: one

text: these

text: features

text: are

text: very

text: pronounced.

text: source-ﬁnding

text: software.

text: As

text: mentioned

text: in

text: Section

text: even

text: with

text: tapered

text: weighting

text: of

text: the

text: vis-

text: Radio

text: mosaics

text: ibilities

text: the

text: PSF

text: exhibits

text: sidelobes

text: at

text: percent

text: of

text: the

text: peak

text: Stokes-I

text: mosaics

text: of

text: the

text: east

text: and

text: west

text: regions

text: are

text: shown

text: in

text: level.

text: This

text: exacerbates

text: the

text: error

text: patterns

text: around

text: sources

text: that

text: do

text: not

text: Figure

text: As

text: each

text: mosaic

text: contains

text: approximately

text: a

text: third

text: of

text: a

text: billion

text: deconvolve

text: properly,

text: and

text: in

text: numerous

text: parts

text: of

text: the

text: mosaic

text: compon-

text: pixels

text: only

text: the

text: brightest

text: sources

text: are

text: obvious

text: in

text: the

text: upper

text: panels

text: ents

text: that

text: do

text: not

text: represent

text: genuine

text: astronomical

text: signals

text: will

text: ﬁnd

text: showing

text: the

text: full

text: area.

text: The

text: discontinuity

text: in

text: the

text: western

text: ﬁeld

text: due

text: to

text: their

text: way

text: into

text: the

text: source

text: catalogue.

text: The

text: position

text: of

text: every

text: entry

text: the

text: unobserved

text: SBs

text: is

text: apparent

text: as

text: the

text: horizontal

text: gap

text: spanning

text: the

text: in

text: the

text: component

text: list

text: with

text: a

text: peak

text: ﬂux

text: density

text: exceeding

text: mJy

text: width

text: of

text: the

text: ﬁeld.

text: Representative

text: zooms

text: of

text: a

text: region

text: in

text: the

text: beam

text: was

text: therefore

text: examined

text: and

text: any

text: visually

text: identiﬁed

text: arte-

text: eastern

text: ﬁeld

text: and

text: a

text: region

text: in

text: the

text: western

text: ﬁeld

text: can

text: be

text: facts

text: were

text: removed.

text: Pointings

text: with

text: known

text: calibration

text: deﬁciencies

text: seen

text: in

text: the

text: lower

text: panels.

text: The

text: pixel

text: brightness

text: scale

text: runs

text: from

text: (typically

text: around

text: the

text: phase

text: calibrator

text: were

text: also

text: ex-

text: to

text: mJy

text: beam

text: for

text: all

text: components

text: of

text: Figure

text: as

text: indicated

text: by

text: the

text: amined

text: for

text: spurious

text: features.

text: In

text: general

text: the

text: use

text: of

text: a

text: local

text: RMS

text: colour

text: bar

text: at

text: the

text: top.

text: The

text: image

text: products

text: are

text: examined

text: further

text: in

text: estimate

text: as

text: opposed

text: to

text: ﬁxed

text: threshold

text: results

text: in

text: a

text: catalogue

text: that

text: is

text: Section

text: not

text: ﬂux

text: limited

text: but

text: does

text: have

text: high

text: reliability.

text: The

text: spectral

text: indices

text: Component

text: catalogue

text: presented

text: in

text: the

text: catalogue

text: are

text: discussed

text: in

text: detail

text: in

text: Section

text: The

text: ﬁnal

text: eastern

text: catalogue

text: contains

text: individual

text: point

text: The

text: source

text: catalogue

text: for

text: this

text: survey

text: was

text: generated

text: by

text: run-

text: and

text: Gaussian

text: components,

text: ﬁtted

text: to

text: islands

text: and

text: grouped

text: into

text: ning

text: the

text: Python

text: Blob

text: Detection

text: and

text: Source

text: Management

text: (PyBDSM)

text: sources.

text: The

text: western

text: catalogue

text: contains

text: components,

text: source

text: ﬁnder

text: (Mohan

text: Raﬀerty

text: on

text: each

text: of

text: the

text: wide

text: area

text: islands

text: and

text: sources.

text: radio

text: images

text: presented

text: in

text: Section

text: A

text: brief

text: overview

text: of

text: the

text: source

text: Table

text: lists

text: the

text: ﬁrst

text: ten

text: components

text: drawn

text: from

text: the

text: eastern

text: detection

text: algorithm

text: is

text: as

text: follows.

text: The

text: spatial

text: variation

text: of

text: the

text: image

text: catalogue

text: in

text: order

text: to

text: illustrate

text: the

text: structure

text: of

text: the

text: component

text: table.

text: noise

text: was

text: estimated

text: by

text: moving

text: a

text: box

text: across

text: the

text: image

text: in

text: overlap-

text: The

text: columns

text: are

text: deﬁned

text: as

text: follows:

text: ping

text: steps,

text: calculating

text: the

text: root

text: mean

text: square

text: (RMS)

text: of

text: the

text: pixels

text: Identiﬁer

text: for

text: the

text: component

text: formed

text: from

text: its

text: HHMMSS.SS+/-

text: within

text: the

text: box

text: and

text: interpolating

text: the

text: values

text: measured

text: from

text: each

text: DDMMSS.SS

text: right

text: ascension

text: and

text: declination

text: position

text: in

text: step.

text: The

text: box

text: size

text: was

text: pixels,

text: stepped

text: across

text: the

text: im-

text: coordinates.

text: age

text: in

text: units

text: of

text: pixels.

text: This

text: procedure

text: provided

text: the

text: source

text: ﬁnder

text: Right

text: ascension

text: of

text: the

text: component

text: in

text: degrees.

text: with

text: an

text: estimate

text: of

text: the

text: spatial

text: variation

text: of

text: the

text: image

text: noise

text: for

text: de-

text: Declination

text: of

text: the

text: component

text: in

text: degrees.

text: tection

text: thresholding

text: purposes.

text: This

text: variation

text: was

text: written

text: to

text: a

text: FITS

text: The

text: 1σ

text: uncertainty

text: in

text: right

text: ascension

text: in

text: arcseconds.

text: image

text: in

text: order

text: to

text: provide

text: a

text: map

text: of

text: the

text: sensitivity

text: of

text: the

text: eastern

text: and

text: The

text: 1σ

text: uncertainty

text: in

text: declination

text: in

text: arcseconds.

text: western

text: ﬁelds,

text: and

text: these

text: images

text: are

text: presented

text: in

text: Figure

text: Integrated

text: ﬂux

text: density

text: of

text: the

text: component

text: in

text: mJy.

text: Image

text: peaks

text: that

text: exceeded

text: 5σ

text: where

text: σ

text: is

text: the

text: local

text: value

text: of

text: The

text: 1σ

text: uncertainty

text: in

text: the

text: integrated

text: ﬂux

text: density

text: of

text: the

text: the

text: RMS

text: noise

text: are

text: identiﬁed.

text: These

text: detections

text: were

text: then

text: grown

text: component

text: in

text: mJy.

text: into

text: islands,

text: deﬁned

text: as

text: regions

text: where

text: there

text: is

text: contiguous

text: emission

text: Peak

text: intensity

text: of

text: the

text: component

text: in

text: mJy

text: beam

text: above

text: a

text: secondary

text: threshold,

text: in

text: this

text: case

text: The

text: islands

text: were

text: then

text: The

text: 1σ

text: uncertainty

text: in

text: the

text: peak

text: intensity

text: of

text: the

text: component

text: in

text: decomposed

text: into

text: a

text: component

text: list

text: by

text: ﬁtting

text: Gaussian

text: components

text: mJy

text: beam

text: Estimate

text: of

text: the

text: local

text: RMS

text: noise

text: at

text: the

text: position

text: of

text: the

text: component

text: in

text: µJy

text: beam

text: MNRAS

text: in

text: press,


text: 10 Heywood et al. Figure 6. Normalized histograms of the RMS values from Figure 3 as de- termined by the PyBDSM source ﬁnder and those measured via the formation of Stokes-V images of the individual survey pointings. The Stokes-V meas- − urements peak at 52 µJy beam 1, however the eﬀective survey “noise” as estimated by the source ﬁnder is higher, with the distribution peaking at − 88 µJy beam 1. See text in Sections and for further details. discussed in Section ). Component lists from each of the external surveys are then cross referenced with the ﬁltered CnB sample in order to ﬁnd the nearest associated component, although the cross- match is rejected if the nearest component is separated by more than 5 arcseconds to exclude spurious detections and components that are not detected in both surveys. This results in 2,626 and 2,773 cross matches for the external A- and B-array surveys respectively. The oﬀsets in right ascension and declination are determ- ined for matched components, and the distribution of these oﬀ- × sets is shown as a two-dimensional histogram with 51 51 bins in Figure 7, with the A- and B-array comparisons in the up- ± per and lower panels respectively. The mean ( 1 standard devi- ation) values of the distributions in (RA, Dec) in units of arc- − ± − ± − ± seconds are ( 0.018 1.171, 0.023 1.018) and ( 0.03 1.186, − ± 0.069 1.094) for the A and B comparisons respectively. The cyan ellipses on each panel are centred on the mean oﬀset position, and the extent of the major and minor axes are one standard devi- ation of the distribution of oﬀsets in RA and Dec. The distributions also conﬁrm that the <5 arcsec separation criterion is not biasing the results. For the A-array and B-array comparisons respectively, the 1σ scatter values are 0.8 arcsec and 0.89 arcsec in right ascension and 0.5 arcsec and 0.6 arcsec in declination. The primary source of the increased scatter in the CnB measurements is likely to be the an- Figure 7. Measured positional diﬀerences between catalogued sources from gular resolution of the survey, noting the increased right ascension the survey presented in this paper, and those detected by Hodge et al. (2011; oﬀset is consistent with the PSF being elongated in that direction. upper panel) and the FIRST survey (Becker, White & Helfand., 1995; lower Scaling the RMS oﬀsets reported by Hodge et al. (2011) by the ra- panel). The points on the above ﬁgure relate only to those entries in the tios of the angular resolutions of the A and CnB surveys results in component catalogue that are isolated, i.e. entries where the island identi- higher scatter than is reported here, however Hodge et al. (2011) ﬁer belongs only to one component. The images show the two dimensional × reason that bandwidth smearing contributes signiﬁcantly to their histograms (51 51 bins) of the oﬀsets in right ascension and declination. uncertainties. This does not signiﬁcantly aﬀect our measurements The ellipses are centred on the mean oﬀset positions and their extents show due to our choice of imaging method. Measuring the mean value of the corresponding 1σ scatter in the points in right ascension and declination the oﬀsets as a function of ﬂux density cut oﬀ reveals no system- between the CnB survey and the corresponding external survey. The extents of the panels above are one half of the minor axis of the restoring beam used atic bias. Note that the positional uncertainties in the component in the CnB survey. catalogue are derived from the uncertainties in the ﬁt performed by PyBDSM, and are therefore purely statistical. 4.5 Photometric accuracy The calibration of the absolute ﬂux density scale against the MNRAS in press, 1–20 (2016) ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Heywood

text: et

text: al.

text: Figure

text: Normalized

text: histograms

text: of

text: the

text: RMS

text: values

text: from

text: Figure

text: as

text: de-

text: termined

text: by

text: the

text: PyBDSM

text: source

text: ﬁnder

text: and

text: those

text: measured

text: via

text: the

text: formation

text: of

text: Stokes-V

text: images

text: of

text: the

text: individual

text: survey

text: pointings.

text: The

text: Stokes-V

text: meas-

font: QMCBKB+txsy

text: urements

text: peak

text: at

text: µJy

text: beam

text: however

text: the

text: eﬀective

text: survey

text: “noise”

text: as

text: estimated

text: by

text: the

text: source

text: ﬁnder

text: is

text: higher,

text: with

text: the

text: distribution

text: peaking

text: at

text: µJy

text: beam

text: See

text: text

text: in

text: Sections

text: and

text: for

text: further

text: details.

text: discussed

text: in

text: Section

text: Component

text: lists

text: from

text: each

text: of

text: the

text: external

text: surveys

text: are

text: then

text: cross

text: referenced

text: with

text: the

text: ﬁltered

text: CnB

text: sample

text: in

text: order

text: to

text: ﬁnd

text: the

text: nearest

text: associated

text: component,

text: although

text: the

text: cross-

text: match

text: is

text: rejected

text: if

text: the

text: nearest

text: component

text: is

text: separated

text: by

text: more

text: than

text: arcseconds

text: to

text: exclude

text: spurious

text: detections

text: and

text: components

text: that

text: are

text: not

text: detected

text: in

text: both

text: surveys.

text: This

text: results

text: in

text: and

text: cross

text: matches

text: for

text: the

text: external

text: A-

text: and

text: B-array

text: surveys

text: respectively.

text: The

text: oﬀsets

text: in

text: right

text: ascension

text: and

text: declination

text: are

text: determ-

text: ined

text: for

text: matched

text: components,

text: and

text: the

text: distribution

text: of

text: these

text: oﬀ-

text: sets

text: is

text: shown

text: as

text: a

text: two-dimensional

text: histogram

text: with

text: bins

text: in

text: Figure

text: with

text: the

text: A-

text: and

text: B-array

text: comparisons

text: in

text: the

text: up-

text: per

text: and

text: lower

text: panels

text: respectively.

text: The

text: mean

text: standard

text: devi-

text: ation)

text: values

text: of

text: the

text: distributions

text: in

text: (RA,

text: Dec)

text: in

text: units

text: of

text: arc-

text: seconds

text: are

text: and

text: for

text: the

text: A

text: and

text: B

text: comparisons

text: respectively.

text: The

text: cyan

text: ellipses

text: on

text: each

text: panel

text: are

text: centred

text: on

text: the

text: mean

text: oﬀset

text: position,

text: and

text: the

text: extent

text: of

text: the

text: major

text: and

text: minor

text: axes

text: are

text: one

text: standard

text: devi-

text: ation

text: of

text: the

text: distribution

text: of

text: oﬀsets

text: in

text: RA

text: and

text: Dec.

text: The

text: distributions

text: also

text: conﬁrm

text: that

text: the

text: arcsec

text: separation

text: criterion

text: is

text: not

text: biasing

text: the

text: results.

text: For

text: the

text: A-array

text: and

text: B-array

text: comparisons

text: respectively,

text: the

text: 1σ

text: scatter

text: values

text: are

text: arcsec

text: and

text: arcsec

text: in

text: right

text: ascension

text: and

text: arcsec

text: and

text: arcsec

text: in

text: declination.

text: The

text: primary

text: source

text: of

text: the

text: increased

text: scatter

text: in

text: the

text: CnB

text: measurements

text: is

text: likely

text: to

text: be

text: the

text: an-

text: Figure

text: Measured

text: positional

text: diﬀerences

text: between

text: catalogued

text: sources

text: from

text: gular

text: resolution

text: of

text: the

text: survey,

text: noting

text: the

text: increased

text: right

text: ascension

text: the

text: survey

text: presented

text: in

text: this

text: paper,

text: and

text: those

text: detected

text: by

text: Hodge

text: et

text: al.

text: oﬀset

text: is

text: consistent

text: with

text: the

text: PSF

text: being

text: elongated

text: in

text: that

text: direction.

text: upper

text: panel)

text: and

text: the

text: FIRST

text: survey

text: (Becker,

text: White

text: Helfand.,

text: lower

text: Scaling

text: the

text: RMS

text: oﬀsets

text: reported

text: by

text: Hodge

text: et

text: al.

text: by

text: the

text: ra-

text: panel).

text: The

text: points

text: on

text: the

text: above

text: ﬁgure

text: relate

text: only

text: to

text: those

text: entries

text: in

text: the

text: tios

text: of

text: the

text: angular

text: resolutions

text: of

text: the

text: A

text: and

text: CnB

text: surveys

text: results

text: in

text: component

text: catalogue

text: that

text: are

text: isolated,

text: i.e.

text: entries

text: where

text: the

text: island

text: identi-

text: higher

text: scatter

text: than

text: is

text: reported

text: here,

text: however

text: Hodge

text: et

text: al.

text: ﬁer

text: belongs

text: only

text: to

text: one

text: component.

text: The

text: images

text: show

text: the

text: two

text: dimensional

text: reason

text: that

text: bandwidth

text: smearing

text: contributes

text: signiﬁcantly

text: to

text: their

text: histograms

text: bins)

text: of

text: the

text: oﬀsets

text: in

text: right

text: ascension

text: and

text: declination.

text: uncertainties.

text: This

text: does

text: not

text: signiﬁcantly

text: aﬀect

text: our

text: measurements

text: The

text: ellipses

text: are

text: centred

text: on

text: the

text: mean

text: oﬀset

text: positions

text: and

text: their

text: extents

text: show

text: due

text: to

text: our

text: choice

text: of

text: imaging

text: method.

text: Measuring

text: the

text: mean

text: value

text: of

text: the

text: corresponding

text: 1σ

text: scatter

text: in

text: the

text: points

text: in

text: right

text: ascension

text: and

text: declination

text: the

text: oﬀsets

text: as

text: a

text: function

text: of

text: ﬂux

text: density

text: cut

text: oﬀ

text: reveals

text: no

text: system-

text: between

text: the

text: CnB

text: survey

text: and

text: the

text: corresponding

text: external

text: survey.

text: The

text: extents

text: of

text: the

text: panels

text: above

text: are

text: one

text: half

text: of

text: the

text: minor

text: axis

text: of

text: the

text: restoring

text: beam

text: used

text: atic

text: bias.

text: Note

text: that

text: the

text: positional

text: uncertainties

text: in

text: the

text: component

text: in

text: the

text: CnB

text: survey.

text: catalogue

text: are

text: derived

text: from

text: the

text: uncertainties

text: in

text: the

text: ﬁt

text: performed

text: by

text: PyBDSM,

text: and

text: are

text: therefore

text: purely

text: statistical.

text: Photometric

text: accuracy

text: The

text: calibration

text: of

text: the

text: absolute

text: ﬂux

text: density

text: scale

text: against

text: the

text: MNRAS

text: in

text: press,

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: Understanding Convolutional Neural Networks Jayanth Koushik Language Technologies Institute Carnegie Mellon University Pittsburgh, PA 15213 jkoushik@cs.cmu.edu Abstract Convoulutional Neural Networks (CNNs) exhibit extraordinary performance on a variety of machine learning tasks. However, their mathematical properties and behavior are quite poorly understood. There is some work, in the form of a framework, for analyzing the operations that they perform. The goal of this project is to present key results from this theory, and provide intuition for why CNNs work. 1 Introduction 1.1 The supervised learning problem We begin by formalizing the supervised learning problem which CNNs are designed to solve. We will consider both regression and classiﬁcation, but restrict the label (dependent variable) to be univariate. ∈ X ⊂ R ∈ Y ⊂ R Let X d and Y be two random variables. We typically have Y = f(X) for some { } unknown f. Given a sample (x , y ) drawn from the joint distribution of X and Y , the i i i=1,...,n X → Y goal of supervised learning is to learn a mapping fˆ : which minimizes the expected loss, as Y ×Y → R deﬁned by a suitable loss function L : . However, minimizing over the set of all functions X Y F from to is ill-posed, so we restrict the space of hypotheses to some set , and deﬁne fˆ = arg min E[L(Y, f(X))] (1) ∈F f 1.2 Linearization A common strategy for learning classiﬁers, and the one employed by kernel methods, is to linearize the variations in f with a feature representation. A feature representation is any transformation of the input variable X; a change of variable. Let this transformation be given by Φ(X). Note that the transformed variable need not have a lower dimension than X. We would like to construct a feature representation such that f is linearly separable in the transformed space i.e. (cid:104) (cid:105) f(X) = Φ(X), w (2) for regression, or (cid:104) (cid:105) f(X) = sign( Φ(X), w ) (3) 29th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain. ##LTLine##

text: Understanding

label: title

text: Convolutional

text: Neural

text: Networks

text: Jayanth

label: author

text: Koushik

text: Language

text: Technologies

text: Institute

text: Carnegie

text: Mellon

text: University

text: Pittsburgh,

text: PA

text: jkoushik@cs.cmu.edu

text: Abstract

text: Convoulutional

label: abstract

text: Neural

text: Networks

text: (CNNs)

text: exhibit

text: extraordinary

text: performance

text: on

text: a

text: variety

text: of

text: machine

text: learning

text: tasks.

text: However,

text: their

text: mathematical

text: properties

text: and

text: behavior

text: are

text: quite

text: poorly

text: understood.

text: There

text: is

text: some

text: work,

text: in

text: the

text: form

text: of

text: a

text: framework,

text: for

text: analyzing

text: the

text: operations

text: that

text: they

text: perform.

text: The

text: goal

text: of

text: this

text: project

text: is

text: to

text: present

text: key

text: results

text: from

text: this

text: theory,

text: and

text: provide

text: intuition

text: for

text: why

text: CNNs

text: work.

label: section

text: Introduction

text: The

text: supervised

text: learning

text: problem

text: We

text: begin

text: by

text: formalizing

text: the

text: supervised

text: learning

text: problem

text: which

text: CNNs

text: are

text: designed

text: to

text: solve.

text: We

text: will

text: consider

text: both

text: regression

text: and

text: classiﬁcation,

text: but

text: restrict

text: the

text: label

text: (dependent

text: variable)

text: to

text: be

text: univariate.

text: X

text: R

text: Y

text: R

text: Let

text: X

text: d

text: and

text: Y

text: be

text: two

text: random

text: variables.

text: We

text: typically

text: have

text: Y

text: f(X)

text: for

text: some

text: unknown

text: f.

text: Given

text: a

text: sample

text: (x

text: y

text: drawn

text: from

text: the

text: joint

text: distribution

text: of

text: X

text: and

text: Y

text: the

text: i

text: i

text: X

text: Y

text: goal

text: of

text: supervised

text: learning

text: is

text: to

text: learn

text: a

text: mapping

text: fˆ

text: which

text: minimizes

text: the

text: expected

text: loss,

text: as

text: Y

text: ×Y

text: R

text: deﬁned

text: by

text: a

text: suitable

text: loss

text: function

text: L

text: However,

text: minimizing

text: over

text: the

text: set

text: of

text: all

text: functions

text: X

text: Y

text: F

text: from

text: to

text: is

text: ill-posed,

text: so

text: we

text: restrict

text: the

text: space

text: of

text: hypotheses

text: to

text: some

text: set

text: and

text: deﬁne

text: fˆ

text: arg

text: min

text: E[L(Y,

text: f(X))]

text: ∈F

text: f

text: Linearization

text: A

text: common

text: strategy

text: for

text: learning

text: classiﬁers,

text: and

text: the

text: one

text: employed

text: by

text: kernel

text: methods,

text: is

text: to

text: linearize

text: the

text: variations

text: in

text: f

text: with

text: a

text: feature

text: representation.

text: A

text: feature

text: representation

text: is

text: any

text: transformation

text: of

text: the

text: input

text: variable

text: X;

text: a

text: change

text: of

text: variable.

text: Let

text: this

text: transformation

text: be

text: given

text: by

text: Φ(X).

text: Note

text: that

text: the

text: transformed

text: variable

text: need

text: not

text: have

text: a

text: lower

text: dimension

text: than

text: X.

text: We

text: would

text: like

text: to

text: construct

text: a

text: feature

text: representation

text: such

text: that

text: f

text: is

text: linearly

text: separable

text: in

text: the

text: transformed

text: space

text: i.e.

text: f(X)

text: Φ(X),

text: w

text: for

text: regression,

text: or

text: f(X)

text: sign(

text: Φ(X),

text: w

text: Conference

text: on

text: Neural

text: Information

text: Processing

text: Systems

text: (NIPS

text: Barcelona,

text: Spain.

text: ##LTLine##


text: This produces a hierarchical transformation with a tree structure rather than a full network. It is possible to show that the above transformation has an equivalent representation through wavelet ≡ ﬁlters i.e. there exists a sequence p (λ , . . . , λ ) such that 1 m ≡ ∗ ≡ ∗ ∗ ∗ ∗ x (u, k ) = S [p]x(u) (U[p]x φ )(u) (ρ(ρ(. . . ρ(x ψ ) . . . ) ψ ) φ )(u) (16) J J J J λ1 λm J where the ψ s are suitably choses wavelet ﬁlters and φ is the averaging ﬁlter deﬁned in Theorem 2. This is thλeiwavelet scattering transfom; its structureJis similar to that of a convolutional neural network as shown in ﬁgure 2, but its ﬁlters are deﬁned by ﬁxed wavelet functions instead of being learned from the data. Further, we have the following theorem about the scattering transform. Theorem 3 Let S [p] be the scattering transform as deﬁned by equation (16). Then there exists J R C > 0 such that for all diffeomorphisms g, and all L2( n) signals x, (cid:107) − (cid:107) ≤ | |(cid:107) (cid:107) S [p]g.x S [p]x Cm g x (17) J J | | with the diffeomorphism norm g given by equation (13). Theorem 3 shows that the scattering transform is Lipschitz continuous to the action of diffemorphisms. So the action of small deformations is linearized over scattering coefﬁcients. Further, because of its structure, it is naturally locally invariant to translations. It has several other desirable properties [4], and can be used to achieve state of the art classiﬁcation errors on the MNIST digits dataset [2]. 5 General Convolutional Neural Network Architectures The scattering transform described in the previous section provides a simple view of a general convolutional neural netowrk. While it provides intuition behind the working of CNNs, the transfor- mation suffers from high variance and loss of information because we only consider single channel convolutions. To analyze the properties of general CNN architectures, we must allow for channel combinations. Mallat [10] extends previously introduced tools to develop a mathematical framework for this analysis. The theory is, however, out of the scope of this paper. At a high level, the extension is achieved by replacing the requirement of contractions and invariants to translations by contractions along adaptive groups of local symmetries. Further, the wavelets are replaced by adapted ﬁlter weights similar to deep learning models. 6 Conclusion In this paper, we tried to analyze the properties of convolutional neural networks. A simpliﬁed model, the scattering transform was introduced as a ﬁrst step towards understanding CNN operations. We saw that the feature transformation is built on top of wavelet transforms which separate variations at different scales using a wavelet transform. The analysis of general CNN architectures was not considered in this paper, but even this analysis is only a ﬁrst step towards a full mathematical understanding of convolutional neural networks. References [1] Joakim Andén and Stéphane Mallat. Deep scattering spectrum. Signal Processing, IEEE Transactions on, 62(16):4114–4128, 2014. [2] Joan Bruna and Stéphane Mallat. Invariant scattering convolution networks. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 35(8):1872–1886, 2013. [3] Corinna Cortes and Vladimir Vapnik. Support-vector networks. Machine learning, 20(3): 273–297, 1995. 6

text: This

text: produces

text: a

text: hierarchical

text: transformation

text: with

text: a

text: tree

text: structure

text: rather

text: than

text: a

text: full

text: network.

text: It

text: is

text: possible

text: to

text: show

text: that

text: the

text: above

text: transformation

text: has

text: an

text: equivalent

text: representation

text: through

text: wavelet

text: ﬁlters

text: i.e.

text: there

text: exists

text: a

text: sequence

text: p

text: (λ

text: λ

text: such

text: that

text: m

label: equation

text: x

text: (u,

text: k

text: S

text: [p]x(u)

text: (U[p]x

text: φ

text: )(u)

text: (ρ(ρ(.

text: ρ(x

text: ψ

text: ψ

text: φ

text: )(u)

text: J

text: J

text: J

text: J

text: λ1

text: λm

text: J

text: where

text: the

text: ψ

text: s

text: are

text: suitably

text: choses

text: wavelet

text: ﬁlters

text: and

text: φ

text: is

text: the

text: averaging

text: ﬁlter

text: deﬁned

text: in

text: Theorem

text: This

text: is

text: thλeiwavelet

text: scattering

text: transfom;

text: its

text: structureJis

text: similar

text: to

text: that

text: of

text: a

text: convolutional

text: neural

text: network

text: as

text: shown

text: in

text: ﬁgure

text: but

text: its

text: ﬁlters

text: are

text: deﬁned

text: by

text: ﬁxed

text: wavelet

text: functions

text: instead

text: of

text: being

text: learned

text: from

text: the

text: data.

text: Further,

text: we

text: have

text: the

text: following

text: theorem

text: about

text: the

text: scattering

text: transform.

text: Theorem

text: Let

text: S

text: [p]

text: be

text: the

text: scattering

text: transform

text: as

text: deﬁned

text: by

text: equation

text: Then

text: there

text: exists

text: J

text: R

text: C

text: such

text: that

text: for

text: all

text: diffeomorphisms

text: g,

text: and

text: all

text: n)

text: signals

text: x,

text: S

text: [p]g.x

text: S

text: [p]x

text: Cm

text: g

text: x

text: J

text: J

text: with

text: the

text: diffeomorphism

text: norm

text: g

text: given

text: by

text: equation

text: Theorem

text: shows

text: that

text: the

text: scattering

text: transform

text: is

text: Lipschitz

text: continuous

text: to

text: the

text: action

text: of

text: diffemorphisms.

text: So

text: the

text: action

text: of

text: small

text: deformations

text: is

text: linearized

text: over

text: scattering

text: coefﬁcients.

text: Further,

text: because

text: of

text: its

text: structure,

text: it

text: is

text: naturally

text: locally

text: invariant

text: to

text: translations.

text: It

text: has

text: several

text: other

text: desirable

text: properties

text: and

text: can

text: be

text: used

text: to

text: achieve

text: state

text: of

text: the

text: art

text: classiﬁcation

text: errors

text: on

text: the

text: MNIST

text: digits

text: dataset

text: General

text: Convolutional

text: Neural

text: Network

text: Architectures

text: The

text: scattering

text: transform

text: described

text: in

text: the

text: previous

text: section

text: provides

text: a

text: simple

text: view

text: of

text: a

text: general

text: convolutional

text: neural

text: netowrk.

text: While

text: it

text: provides

text: intuition

text: behind

text: the

text: working

text: of

text: CNNs,

text: the

text: transfor-

text: mation

text: suffers

text: from

text: high

text: variance

text: and

text: loss

text: of

text: information

text: because

text: we

text: only

text: consider

text: single

text: channel

text: convolutions.

text: To

text: analyze

text: the

text: properties

text: of

text: general

text: CNN

text: architectures,

text: we

text: must

text: allow

text: for

text: channel

text: combinations.

text: Mallat

text: extends

text: previously

text: introduced

text: tools

text: to

text: develop

text: a

text: mathematical

text: framework

text: for

text: this

text: analysis.

text: The

text: theory

text: is,

text: however,

text: out

text: of

text: the

text: scope

text: of

text: this

text: paper.

text: At

text: a

text: high

text: level,

text: the

text: extension

text: is

text: achieved

text: by

text: replacing

text: the

text: requirement

text: of

text: contractions

text: and

text: invariants

text: to

text: translations

text: by

text: contractions

text: along

text: adaptive

text: groups

text: of

text: local

text: symmetries.

text: Further,

text: the

text: wavelets

text: are

text: replaced

text: by

text: adapted

text: ﬁlter

text: weights

text: similar

text: to

text: deep

text: learning

text: models.

text: Conclusion

text: In

text: this

text: paper,

text: we

text: tried

text: to

text: analyze

text: the

text: properties

text: of

text: convolutional

text: neural

text: networks.

text: A

text: simpliﬁed

text: model,

text: the

text: scattering

text: transform

text: was

text: introduced

text: as

text: a

text: ﬁrst

text: step

text: towards

text: understanding

text: CNN

text: operations.

text: We

text: saw

text: that

text: the

text: feature

text: transformation

text: is

text: built

text: on

text: top

text: of

text: wavelet

text: transforms

text: which

text: separate

text: variations

text: at

text: different

text: scales

text: using

text: a

text: wavelet

text: transform.

text: The

text: analysis

text: of

text: general

text: CNN

text: architectures

text: was

text: not

text: considered

text: in

text: this

text: paper,

text: but

text: even

text: this

text: analysis

text: is

text: only

text: a

text: ﬁrst

text: step

text: towards

text: a

text: full

text: mathematical

text: understanding

text: of

text: convolutional

text: neural

text: networks.

text: References

text: Joakim

text: Andén

text: and

text: Stéphane

text: Mallat.

text: Deep

text: scattering

text: spectrum.

text: Signal

text: Processing,

text: IEEE

text: Transactions

text: on,

text: Joan

text: Bruna

text: and

text: Stéphane

text: Mallat.

text: Invariant

text: scattering

text: convolution

text: networks.

text: Pattern

text: Analysis

text: and

text: Machine

text: Intelligence,

text: IEEE

text: Transactions

text: on,

text: Corinna

text: Cortes

text: and

text: Vladimir

text: Vapnik.

text: Support-vector

text: networks.

text: Machine

text: learning,


text: It is not easy to solve the problem in Eq. (16) directly. After some deductions, it is equal to (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:104) (cid:105) (cid:104) (cid:105) − 2 min V(s), Φ(Z(s) ) + V¯ , Φ(Z¯ ) Y V(s),V¯ ,w1,w2 T1+1 T1+1 T1+1 (17) 1 (cid:107) (cid:107) 1 (cid:107) (cid:107) ≥ ≥ +γ( × V(s) 2 + × V¯ 2), with w + w = 1, w 0, w 0. 1 2 1 2 c w (d(a) + c) w 1 2 The optimization problem in Eq. (17) seems very similar to traditional regularized square loss re- gression. Nevertheless, they are diﬀerent since the regularization parameter is ﬁxed in traditional method, while it is also an optimization parameter in our setting. There are two groups of optimization variables, i.e., the coeﬃcients for classiﬁcation and the balance parameters. It is diﬃcult to optimize them together and we optimize them alternatively. When V(s) and V¯ are ﬁxed, the optimal balance parameters w and w can be computed by the 1 2 following proposition directly. ≥ ≥ Proposition 3 When w + w = 1, w 0, w 0, 1 2 1 2 (cid:112) 1 (cid:107) (cid:107) 1 (cid:107) (cid:107) √1 (cid:107) (cid:107) 1 (cid:107) (cid:107) min ( × V(s) 2 + × V¯ 2) = ( V(s) + V¯ )2. (18) w1,w2 w1 c w2 (d(a) + c) c d(a) + c The optimal solution is (cid:112) √(cid:112) (cid:112) (cid:107) (cid:107) (cid:107) (cid:107) ∗ √ V(s) / c ∗ √V¯ / d(a) + c w1 = (cid:107)V(s)(cid:107)/ c + (cid:107)V¯ (cid:107)/ d(a) + c, w2 = (cid:107)V(s)(cid:107)/ c + (cid:107)V¯ (cid:107)/ d(a) + c. (19) Proof: The optimization problem is 1 (cid:107) (cid:107) 1 (cid:107) (cid:107) min ( × V(s) 2 + × V¯ 2) , w1,w2 w1 c w2 (d(a) + c) (20) ≥ ≥ s.t. w + w = 1, w 0, w 0. 1 2 1 2 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: It

text: is

text: not

text: easy

text: to

text: solve

text: the

text: problem

text: in

text: Eq.

text: directly.

text: After

text: some

text: deductions,

text: it

text: is

text: equal

text: to

text: min

text: V(s),

text: Φ(Z(s)

text: V¯

text: Φ(Z¯

text: Y

text: V(s),V¯

text: +γ(

text: V(s)

text: V¯

text: with

text: w

text: w

text: w

text: w

text: c

text: w

text: (d(a)

text: c)

text: w

text: The

text: optimization

text: problem

text: in

text: Eq.

text: seems

text: very

text: similar

text: to

text: traditional

text: regularized

text: square

text: loss

text: re-

text: gression.

text: Nevertheless,

text: they

text: are

text: diﬀerent

text: since

text: the

text: regularization

text: parameter

text: is

text: ﬁxed

text: in

text: traditional

text: method,

text: while

text: it

text: is

text: also

text: an

text: optimization

text: parameter

text: in

text: our

text: setting.

text: There

text: are

text: two

text: groups

text: of

text: optimization

text: variables,

text: i.e.,

text: the

text: coeﬃcients

text: for

text: classiﬁcation

text: and

text: the

text: balance

text: parameters.

text: It

text: is

text: diﬃcult

text: to

text: optimize

text: them

text: together

text: and

text: we

text: optimize

text: them

text: alternatively.

text: When

text: V(s)

text: and

text: V¯

text: are

text: ﬁxed,

text: the

text: optimal

text: balance

text: parameters

text: w

text: and

text: w

text: can

text: be

text: computed

text: by

text: the

text: following

text: proposition

text: directly.

text: Proposition

text: When

text: w

text: w

text: w

text: w

text: √1

text: min

text: V(s)

text: V¯

text: V(s)

text: V¯

text: c

text: (d(a)

text: c)

text: c

text: d(a)

text: c

text: The

text: optimal

text: solution

text: is

text: V(s)

text: c

text: √V¯

text: d(a)

text: c

text: c

text: d(a)

text: c,

text: c

text: d(a)

text: c.

text: Proof:

text: The

text: optimization

text: problem

text: is

text: min

text: V(s)

text: V¯

text: c

text: (d(a)

text: c)

text: s.t.

text: w

text: w

text: w

text: w

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