text: F. Nehlig, B. Vollmer, J. Braine : Environmental gas compression Fig. 20. Rotation curves of the four galaxies from CO(2-1) data for NGC 4501 and NGC 4567/68 and from the two-parameter ﬁt from Leroy et al. 2013 for NGC 4321. Article number, page 23 of 44 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: A&A proofs: manuscript no. 27021 Appendix A: CO(2-1) observations: rms We computed the rms of the CO(2-1) spectra for each position. The CO(2-1) emission line signal in each spec- (cid:112) trum was excluded from the rms calculation. The rms of × × the column density was deﬁned as ∆Σ = rms ∆v N − , where rms is the noise levHel2 of the CO spec- CO(2 1) − tra, ∆v the spectral velocity resolution (10.4 km s 1), and N − the number of channels containing a CO(2-1) CO(2 1) line signal. The CO(2-1) noise maps of NGC 4501 and NGC 4567/68 (Fig. A.1) do not show any systematic be- havior. Appendix B: SFR comparison Appendix B.1: Total IR The FUV heated dust emission can be traced using the total infrared emission (TIR) (see Hao et al. 2011 and Galametz et al. 2013). We tried three diﬀerent recipes to compute the TIR emission: – TIR1 based on the 3.6, 8, 24, and 70 µm from Spitzer (Eqs. B.1, B.2, and B.3). – TIR2 based on the 3.6, 24 µm from Spitzer and 100, 160 µm from Herschel PACS (Eq. B.4). – TIR3 based on the 3.6, 24, and 70 µm from Spitzer and 100, 160 µm from Herschel PACS (Eq. B.5). − I (PAH8µm) = I (8µm) 0.232 I (3.6µm) (B.1) ν ν − ν I (24µm) = I (24µm) 0.032 I (3.6µm) (B.2) ν ν ν × I(TIR1) = 0.95 νI (PAH8µm) + 1.15 ν νI (24µm) + 2.3 νI (70µm) (B.3) ν ν × I(TIR2) = 2.708 νI (24µm) + 0.734 ν νI (100µm) + 0.739 νI (160µm) (B.4) ν ν × I(TIR3) = 2.064 νI (24µm) + 0.539 ν × νI (70µm) + 0.277 ν νI (100µm) + 0.938 νI (160µm) (B.5) ν ν − Σ˙ (TIR) = 8.1 10 2I(FUV) + 0.46 I(TIR) (B.6) (cid:63) The TIR1, TIR2, and TIR3 recipes of NGC 4501, NGC 4567/68, and NGC 4321 were compared in a pixel- by-pixel analysis with the 24 µm emission (Fig. B.1). The three TIR recipes tightly correlate with the 24 µm emission (I (24µm)). The rms of these correlations are smaller than ν 0.09 dex. Appendix B.2: SFR from Hα We calculated the star formation rate based on H emis- α sion: Σ˙ (Hα + 24µm) (Eq. B.7), (cid:63) − Σ˙ (Hα + 24µm) = (634 f + 2.5 10 3 I ) cos(i) , (B.7) (cid:63) Hα 24µm − − where f is the Hα ﬂux density (ergs s 1 cm 2 Å−1) correcHtαed for Nii line contamination (NII/Hα ra- Fig. A.1. CO(2-1) rms noise maps of NGC 4501 (upper panel) tio of 0.54; see Kennicutt et al. 2008). The Hα data for and NGC 4567/68 (lower panel). The color bar is in units of NGC 4501, NGC 4567/68, and NGC 4321 are part of the − mK. The spectral velocity resolution is 10.4 km s 1. GOLD-mine database (Gavazzi et al. 2003). Σ˙ (Hα+24µm) (cid:63) and Σ˙ (FUV + 24µm) were compared pixel-by-pixel (see (cid:63) Article number, page 30 of 44 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: A&A proofs: manuscript no. 27021 Fig. B.2. Comparison of star formation rates computed with FUV+24µm and H +24µm. Residuals of the relations in dex for α NGC 4501, NGC 4567/68, and NGC 4321. The continuous line corresponds to a linear relation between the two SFR indicators, ± while the dashed and dotted lines correspond to 1σ for each galaxy. Article number, page 32 of 44 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A&A

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text: F. Nehlig, B. Vollmer, J. Braine : Environmental gas compression mainly on three key theoretical assumptions (Dickman et As expected, we do not see any clear correlation between al. 1986): (1) all GMCs have the same mean surface den- SFR and Σ for NGC 4501 and NGC 4321. Instead, for HI sity (overlapping clouds can lead to a smaller value of I , NGC 4567/68 and NGC 4321 two regimes can be seen: CO − which means that H mass can be underestimated); (2) the (1) where log (Σ ) < 0.9 (8 M(cid:12) pc 2) and (2) where 2 10 HI mean brightness temperature of the CO is well deﬁned and log (Σ ) > 0.9. The ﬁrst regime shows a shallower slope 10 HI does not vary with cloud size (CO emission is assumed to be than the second. In the central part of NGC 4501 the Hi optically thick); and (3) GMCs have a known mass spec- surface density decreases radially to the center where no trum and follow a size-linewidth relation. Moreover, the Hi is detected. Thus, the correlation between Hi and SFR conversion factor is both expected and observed to vary breaks down in the central part of NGC 4501. This behavior with metallicity (van Dishoeck & Black 1988; Wolﬁre et al. is also observed in the ﬁeld spiral galaxy NGC 628 (Leroy 2010) leading to lower value at higher metallicity. Thus, in et al. 2008); the Hi surface density also strongly decreases spiral galaxies the well-known metallicity gradient (Shaver toward the galaxy center where the gas is mostly molecular. et al. 1983, Pilkington et al. 2012 and references therein) A power law was ﬁtted to the SFR-Σ and the SFR-Σ may lead to an overestimation of Σ in the central regions relations: H2 g where the metallicity is expected tHo2be higher. Sandstrom et al. (2013) investigated the radial variations of the conver- × sion factor in a sample of 26 nearby star-forming galaxies. log (SFR) = n log (Σ ) + γ (7) These authors found a signiﬁcantly lower conversion fac- log10(SFR) = n1 × log10(ΣH)2+ γ .1 (8) tor in the nuclear regions of their sample. However, they 10 2 10 g 2 found only a weak dependency of the conversion factor on metallicity. The conversion factor is also observed to sharply We used an MCMC approach for a linear regression with increase at metallicities below one-half solar. Nevertheless, errors on both X and Y data (IDL routine LINMIX_ERR). the variation of X with metallicity is still poorly con- This Bayesian approach gives the probability density dis- CO strained, thus we prefer to use a simpler, ﬁrst-order de- tribution of both the slope and the oﬀset of the ﬁt. We also scription of XCO.×Bolatto e−t al. (2013) recommend the use computed the Spearman rank correlation ρ for the two re- of X − = 2 1020cm 2(α − = 4.4M(cid:12)(K km/s lations (Table 3). The uncertainty in the Spearman rank −CO(1 0) ± CO(1 0) pc2) 1) with an estimated 30% uncertainty. Assuming correlation ρ was estimated via the bootstrap method and a line ratio R = 0.8, this gives a value of α − = by redistributing the data within the uncertainties. The er- 21 − CO(2 1) 5.5 M(cid:12)(K km/s pc2) 1 including He. rors obtained with the two methods agree within a factor of three. To be conservative, we only present the maximum 4.1.1. Star formation eﬃciency of these error estimates. The Kennicutt-Schmidth relation (KS) (Schmidt 1959, For NGC 4501 the central area was excluded from the 1963 and Kennicutt 1989, 1998b) links the gas reservoir of ﬁt since the conversion factor is not reliable and no Hi is galaxies to the SFR using integrated values of SFR and detected in the center. For all three galaxies we found a Σreglat=ionΣ(HcIo+nsiΣdHer2i.ngKdenisnkicauvtetra(g1e9s98SbF)Rfoaunnddgaasndoenn-sliitnieeasr) pSFowRe)rinlatwheinrdanexgeno1f 0fo.9r-1t,heasmhoalsebcueleanrfoKuSndreilnatoitohner(ΣnoHn2-- × ± ΣnoSrFmRa=l stAar-foΣrmNg inwgitshpiNrals=an1.d4 sta0r.b1u5rsftrogmalaaxiseasm. pMleanoyf iinntdeerxacntingofg1a.l1ax−ies1..5FohrasthbeeΣeng-fSoFuRndr.eMlatoiroenovaerp,otwheer rlmaws 2 other works found N in the range of 1.0-2.0. Nevertheless, of the molecular and total gas KS relations are less then the use of Σ = Σ + Σ in this relation is puzzling since 20 %. Those two results suggest that gas compression, either stars form ing moleHcIular cHl2ouds and thus the information of from ram pressure or from gravitational interaction, do not the transition from H to H is hidden when using Σ . These signiﬁcantly change the KS relation. I 2 g studies consider average SFR and gas surface densities. For a direct comparison between galaxies, the SFR-Σ , Other studies have found shallower power laws (N ∼ 1) SFR-ΣHI, SFR-Σg, and Rmol-Pt∼ot relations are presentHe2d in Fig. 18. NGC 4321 shows a 0.2 dex lower molecular using only the molecular gas and SFR (Wong & Blitz 2002; K20e1n3n)i;cWutotnegt &al.B2l0it0z7(;2B0i0g2i)elceotnsailde2r00a8z;imLuetrhoyalleyt aavl.er2a0g0e8d, SPtFotE)Ho2faNnGd Ctot4a5l6S7FaEngd. Iennttihree ignanserdgisaklaocfticNdGisCks4(5a6t8htighhe molecular fraction is lower than that of the other galaxies. surface densities, while Kennicutt et al. (2007), Bigiel et al. 4.1.2. Molecular fraction (2008), and Leroy et al. (2008, 2013) have studied the KS law at kpc or sub-kpc resolution. Wong & Blitz (2002) and Blitz & Rosolowsky (2006) A weaker correlation, or even no correlation, between (BR06) investigated the ISM pressure-molecular fraction atomic gas and SFR is found by Bigiel et al. (2008) in their relation with a pixel-by-pixel analysis . They ﬁtted a rela- sample of 18 nearby galaxies. Moreover, they observe that tion between log(P /k ) and log(R ), where k is the tot B mol B the atomic gas surface density in spirals and dwarf galaxies Boltzmann constant. These authors ﬁnd that the molecu- ∼ saturates at 10 M(cid:12)pc2 so that the correlation between lar fraction is correlated with the ISM pressure mostly in atomic gas and SFR shows the saturation of the Hi line regions w(cid:18)here (cid:19)R > 1, mol (Blitz & Rosolowsky (2006), Krumholz et al. (2009)). We investigated the spatially resolved SFR-Σ , SFR- ± ΣE∼aHcIh, apnodinStFcRo-rΣreHs2porenldastiotonsaforrestohleutfioounr eglaelmaxeinets (oFfgi1g7. (cid:48)1(cid:48) 8o)r. Rmol = PPto0t 0.92 0.07 , (9) 1.5 kpc. The resulting plots for each galaxy are presented ± × − in Appendix (Fig. C.1, C.2, C.3, and C.4 for NGC 4501, with P /k = 3.5 0.6 104 cm 3K the pressure of 0 B NGC 4567, NGC 4568, and NGC 4321, respectively). the interstellar medium, where R = 1. They found that mol Article number, page 9 of 44 ##LTLine##

text: F.

label: author

text: Nehlig,

text: B.

text: Vollmer,

text: J.

text: Braine

text: Environmental

text: gas

text: compression

text: mainly

text: on

text: three

text: key

text: theoretical

text: assumptions

text: (Dickman

text: et

text: As

text: expected,

text: we

text: do

text: not

text: see

text: any

text: clear

text: correlation

text: between

text: al.

text: all

text: GMCs

text: have

text: the

text: same

text: mean

text: surface

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text: SFR

text: and

text: Σ

text: for

text: NGC

text: and

text: NGC

text: Instead,

text: for

text: HI

text: sity

text: (overlapping

text: clouds

text: can

text: lead

text: to

text: a

text: smaller

text: value

text: of

text: I

text: NGC

text: and

text: NGC

text: two

text: regimes

text: can

text: be

text: seen:

text: CO

text: which

text: means

text: that

text: H

text: mass

text: can

text: be

text: underestimated);

text: the

text: where

text: log

text: (Σ

text: pc

text: and

text: where

text: HI

text: mean

text: brightness

text: temperature

text: of

text: the

text: CO

text: is

text: well

text: deﬁned

text: and

text: log

text: (Σ

text: The

text: ﬁrst

text: regime

text: shows

text: a

text: shallower

text: slope

text: HI

text: does

text: not

text: vary

text: with

text: cloud

text: size

text: (CO

text: emission

text: is

text: assumed

text: to

text: be

text: than

text: the

text: second.

text: In

text: the

text: central

text: part

text: of

text: NGC

text: the

text: Hi

text: optically

text: thick);

text: and

text: GMCs

text: have

text: a

text: known

text: mass

text: spec-

text: surface

text: density

text: decreases

text: radially

text: to

text: the

text: center

text: where

text: no

text: trum

text: and

text: follow

text: a

text: size-linewidth

text: relation.

text: Moreover,

text: the

text: Hi

text: is

text: detected.

text: Thus,

text: the

text: correlation

text: between

text: Hi

text: and

text: SFR

text: conversion

text: factor

text: is

text: both

text: expected

text: and

text: observed

text: to

text: vary

text: breaks

text: down

text: in

text: the

text: central

text: part

text: of

text: NGC

text: This

text: behavior

text: with

text: metallicity

text: (van

text: Dishoeck

text: Black

text: Wolﬁre

text: et

text: al.

text: is

text: also

text: observed

text: in

text: the

text: ﬁeld

text: spiral

text: galaxy

text: NGC

text: (Leroy

text: leading

text: to

text: lower

text: value

text: at

text: higher

text: metallicity.

text: Thus,

text: in

text: et

text: al.

text: the

text: Hi

text: surface

text: density

text: also

text: strongly

text: decreases

text: spiral

text: galaxies

text: the

text: well-known

text: metallicity

text: gradient

text: (Shaver

text: toward

text: the

text: galaxy

text: center

text: where

text: the

text: gas

text: is

text: mostly

text: molecular.

text: et

text: al.

text: Pilkington

text: et

text: al.

text: and

text: references

text: therein)

text: A

text: power

text: law

text: was

text: ﬁtted

text: to

text: the

text: SFR-Σ

text: and

text: the

text: SFR-Σ

text: may

text: lead

text: to

text: an

text: overestimation

text: of

text: Σ

text: in

text: the

text: central

text: regions

text: relations:

text: g

text: where

text: the

text: metallicity

text: is

text: expected

text: higher.

text: Sandstrom

text: et

text: al.

text: investigated

text: the

text: radial

text: variations

text: of

text: the

text: conver-

text: sion

text: factor

text: in

text: a

text: sample

text: of

text: nearby

text: star-forming

text: galaxies.

text: log

text: (SFR)

text: n

text: log

text: (Σ

text: γ

text: These

text: authors

text: found

text: a

text: signiﬁcantly

text: lower

text: conversion

text: fac-

text: γ

text: tor

text: in

text: the

text: nuclear

text: regions

text: of

text: their

text: sample.

text: However,

text: they

text: g

text: found

text: only

text: a

text: weak

text: dependency

text: of

text: the

text: conversion

text: factor

text: on

text: metallicity.

text: The

text: conversion

text: factor

text: is

text: also

text: observed

text: to

text: sharply

text: We

text: used

text: an

text: MCMC

text: approach

text: for

text: a

text: linear

text: regression

text: with

text: increase

text: at

text: metallicities

text: below

text: one-half

text: solar.

text: Nevertheless,

text: errors

text: on

text: both

text: X

text: and

text: Y

text: data

text: (IDL

text: routine

text: the

text: variation

text: of

text: X

text: with

text: metallicity

text: is

text: still

text: poorly

text: con-

text: This

text: Bayesian

text: approach

text: gives

text: the

text: probability

text: density

text: dis-

text: CO

text: strained,

text: thus

text: we

text: prefer

text: to

text: use

text: a

text: simpler,

text: ﬁrst-order

text: de-

text: tribution

text: of

text: both

text: the

text: slope

text: and

text: the

text: oﬀset

text: of

text: the

text: ﬁt.

text: We

text: also

text: scription

text: of

text: XCO.×Bolatto

text: e−t

text: al.

text: recommend

text: the

text: use

text: computed

text: the

text: Spearman

text: rank

text: correlation

text: ρ

text: for

text: the

text: two

text: re-

text: of

text: X

text: km/s

text: lations

text: (Table

text: The

text: uncertainty

text: in

text: the

text: Spearman

text: rank

text: with

text: an

text: estimated

text: uncertainty.

text: Assuming

text: correlation

text: ρ

text: was

text: estimated

text: via

text: the

text: bootstrap

text: method

text: and

text: a

text: line

text: ratio

text: R

text: this

text: gives

text: a

text: value

text: of

text: α

text: by

text: redistributing

text: the

text: data

text: within

text: the

text: uncertainties.

text: The

text: er-

text: km/s

text: including

text: He.

text: rors

text: obtained

text: with

text: the

text: two

text: methods

text: agree

text: within

text: a

text: factor

text: of

text: three.

text: To

text: be

text: conservative,

text: we

text: only

text: present

text: the

text: maximum

text: Star

text: formation

text: eﬃciency

text: of

text: these

text: error

text: estimates.

text: The

text: Kennicutt-Schmidth

text: relation

text: (KS)

text: (Schmidt

text: For

text: NGC

text: the

text: central

text: area

text: was

text: excluded

text: from

text: the

text: and

text: Kennicutt

text: links

text: the

text: gas

text: reservoir

text: of

text: ﬁt

text: since

text: the

text: conversion

text: factor

text: is

text: not

text: reliable

text: and

text: no

text: Hi

text: is

text: galaxies

text: to

text: the

text: SFR

text: using

text: integrated

text: values

text: of

text: SFR

text: and

text: detected

text: in

text: the

text: center.

text: For

text: all

text: three

text: galaxies

text: we

text: found

text: a

text: ΣnoSrFmRa=l

text: stAar-foΣrmNg

text: pMleanoyf

text: rlmaws

text: other

text: works

text: found

text: N

text: in

text: the

text: range

text: of

text: Nevertheless,

text: of

text: the

text: molecular

text: and

text: total

text: gas

text: KS

text: relations

text: are

text: less

text: then

text: the

text: use

text: of

text: Σ

text: Σ

text: Σ

text: in

text: this

text: relation

text: is

text: puzzling

text: since

text: Those

text: two

text: results

text: suggest

text: that

text: gas

text: compression,

text: either

text: stars

text: form

text: ing

text: moleHcIular

text: and

text: thus

text: the

text: information

text: of

text: from

text: ram

text: pressure

text: or

text: from

text: gravitational

text: interaction,

text: do

text: not

text: the

text: transition

text: from

text: H

text: to

text: H

text: is

text: hidden

text: when

text: using

text: Σ

text: These

text: signiﬁcantly

text: change

text: the

text: KS

text: relation.

text: I

text: g

text: studies

text: consider

text: average

text: SFR

text: and

text: gas

text: surface

text: densities.

text: For

text: a

text: direct

text: comparison

text: between

text: galaxies,

text: the

text: SFR-Σ

text: Other

text: studies

text: have

text: found

text: shallower

text: power

text: laws

text: (N

text: SFR-ΣHI,

text: SFR-Σg,

text: and

text: Rmol-Pt∼ot

text: relations

text: are

text: in

text: Fig.

text: NGC

text: shows

text: a

text: dex

text: lower

text: molecular

text: using

text: only

text: the

text: molecular

text: gas

text: and

text: SFR

text: (Wong

text: Blitz

text: Iennttihree

text: molecular

text: fraction

text: is

text: lower

text: than

text: that

text: of

text: the

text: other

text: galaxies.

text: surface

text: densities,

text: while

text: Kennicutt

text: et

text: al.

text: Bigiel

text: et

text: al.

text: Molecular

text: fraction

text: and

text: Leroy

text: et

text: al.

text: have

text: studied

text: the

text: KS

text: law

text: at

text: kpc

text: or

text: sub-kpc

text: resolution.

text: Wong

text: Blitz

text: and

text: Blitz

text: Rosolowsky

text: A

text: weaker

text: correlation,

text: or

text: even

text: no

text: correlation,

text: between

text: investigated

text: the

text: ISM

text: pressure-molecular

text: fraction

text: atomic

text: gas

text: and

text: SFR

text: is

text: found

text: by

text: Bigiel

text: et

text: al.

text: in

text: their

text: relation

text: with

text: a

text: pixel-by-pixel

text: analysis

text: They

text: ﬁtted

text: a

text: rela-

text: sample

text: of

text: nearby

text: galaxies.

text: Moreover,

text: they

text: observe

text: that

text: tion

text: between

text: log(P

text: /k

text: and

text: log(R

text: where

text: k

text: is

text: the

text: tot

text: B

text: mol

text: B

text: the

text: atomic

text: gas

text: surface

text: density

text: in

text: spirals

text: and

text: dwarf

text: galaxies

text: Boltzmann

text: constant.

text: These

text: authors

text: ﬁnd

text: that

text: the

text: molecu-

text: saturates

text: at

text: so

text: that

text: the

text: correlation

text: between

text: lar

text: fraction

text: is

text: correlated

text: with

text: the

text: ISM

text: pressure

text: mostly

text: in

text: atomic

text: gas

text: and

text: SFR

text: shows

text: the

text: saturation

text: of

text: the

text: Hi

text: line

text: regions

text: mol

text: (Blitz

text: Rosolowsky

text: Krumholz

text: et

text: al.

text: We

text: investigated

text: the

text: spatially

text: resolved

text: SFR-Σ

text: SFR-

text: ΣE∼aHcIh,

text: eglaelmaxeinets

text: Rmol

text: kpc.

text: The

text: resulting

text: plots

text: for

text: each

text: galaxy

text: are

text: presented

text: in

text: Appendix

text: (Fig.

text: and

text: for

text: NGC

text: with

text: P

text: /k

text: cm

text: the

text: pressure

text: of

text: B

text: NGC

text: NGC

text: and

text: NGC

text: respectively).

text: the

text: interstellar

text: medium,

text: where

text: R

text: They

text: found

text: that

text: mol

text: Article

text: number,

text: page

text: of

text: ##LTLine##


text: SUBMITTED TO IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 1 A Closed-Form Solution to Tensor Voting: Theory and Applications Tai-Pang Wu, Sai-Kit Yeung, Jiaya Jia, Chi-Keung Tang, Ge´rard Medioni Abstract— We prove a closed-form solution to tensor voting (CFTV): given a point set in any dimensions, our closed-form solution provides an exact, continuous and efﬁcient algorithm for computing a structure-aware tensor that simultaneously achieves salient structure detection and outlier attenuation. Using CFTV, we prove the convergence of tensor voting on a Markov random ﬁeld (MRF), thus termed as MRFTV, where the structure-aware tensor at each input site reaches a stationary state upon convergence in structure propagation. We then embed structure-aware tensor into expectation maximization (EM) for optimizing a single linear structure to achieve efﬁcient and robust parameter estimation. Speciﬁcally, our EMTV algorithm optimizes both the tensor and ﬁtting parameters and does not require random sampling consensus typically used in existing robust statistical techniques. We performed quantitative evaluation on its accuracy and robustness, showing that EMTV performs better than the original TV and other state-of-the-art techniques in fundamental matrix estimation for multiview stereo matching. The extensions of CFTV and EMTV for extracting multiple and nonlinear structures are underway. An addendum is included in this arXiv version. Index Terms—Tensor voting, closed-form solution, structure inference, parameter estimation, multiview stereo. (cid:70) 1 INTRODUCTION as well as general theory of tensor voting. This paper T HIS paper reinvents tensor voting [19] for robust focuses on the special theory, where the above data computer vision, by proving a closed-form solution communication is data driven without using constraints to computing an exact structure-aware tensor after data other than proximity and continuity. The special theory, communication in a feature space of any dimensions, sometimes coined as “ﬁrst voting pass,” is applied to process raw input data to detect structures and outliers. where the goal is salient structure inference from noisy In addition to structure detection and outlier attenuation, and corrupted data. To infer structures from noisy data corrupted by in the general theory of tensor voting, tensor votes are outliers, in tensor voting, input points communicate propagated along preferred directions to achieve data among themselves subject to proximity and continuity communication when such directions are available, typ- constraints. Consequently, each point is aware of its ically after the ﬁrst pass, such that useful tensor votes structure saliency via a structure-aware tensor. Structure are reinforced whereas irrelevant ones are suppressed. refers to surfaces, curves, or junctions if the feature space Expressing tensor voting in a single and compact is three dimensional where a structure-aware tensor can equation, or a closed-form solution, offers many ad- be visualized as an ellipsoid: if a point belongs to a vantages: not only an exact and efﬁcient solution can smooth surface, the resulting ellipsoid after data com- be achieved with less implementation effort for salient munication resembles a stick pointing along the surface structure detection and outlier attenuation, formal and normal; if a point lies on a curve the tensor resembles useful mathematical operations such as differential cal- a plate, where the curve tangent is perpendicular to culus can be applied which is otherwise impossible the plate tensor; if it is a point junction where surfaces using the original tensor voting procedure. Notably, we intersect, the tensor will be like a ball. An outlier is can prove the convergence of tensor voting on Markov characterized by a set of inconsistent votes it receives random ﬁelds (MRFTV) where a structure-aware tensor after data communication. at each input site achieves a stationary state upon con- We develop in this paper a closed-form solution to vergence. tensor voting (CFTV), which is applicable to the special Using CFTV, we contribute a mathematical derivation based on expectation maximization (EM) that applies the • exact tensor solution for extracting the most salient linear T.-P. Wu is with the Enterprise and Consumer Electronics Group, Hong Kong Applied Science and Technology Research Institute (ASTRI), Hong structure, despite that the input data is highly corrupted. • Kong. E-mail: tpwu@astri.org. Our algorithm is called EMTV, which optimizes both S.-K. Yeung and C.-K. Tang are with the Department of Computer Science the tensor and ﬁtting parameters upon convergence and and Engineering, Hong Kong University of Science and Technology, Clear { } Water Bay, Hong Kong. E-mail: saikit,cktang @cse.ust.hk. does not require random sampling consensus typical • J. Jia is with the Department of Computer Science and Engineering, of existing robust statistical techniques. The extension Chinese University of Hong Kong, Shatin, Hong Kong. E-mail: leo- to extract salient multiple and nonlinear structures is jia@cse.cuhk.edu.hk • G. Medioni is with the Department of Computer Science, University of underway. Southern California, USA. E-mail: medioni@usc.edu. While the mathematical derivation may seem in- ##LTLine## ##LTLine## ##LTLine##

text: SUBMITTED

font: XDRGIF+NimbusSanL-Regu

text: TO

text: IEEE

text: TRANSACTIONS

text: ON

text: PATTERN

text: ANALYSIS

text: AND

text: MACHINE

text: INTELLIGENCE

text: A

label: title

text: Closed-Form

text: Solution

text: to

text: Tensor

text: Voting:

text: Theory

text: and

text: Applications

text: Tai-Pang

text: Wu,

text: Sai-Kit

text: Yeung,

text: Jiaya

text: Jia,

text: Chi-Keung

text: Tang,

text: Ge´rard

text: Medioni

text: Abstract—

font: QSPFRV+NimbusSanL-Bold

text: We

label: abstract

text: prove

text: a

text: closed-form

text: solution

text: to

text: tensor

text: voting

text: (CFTV):

font: IAEGZA+URWPalladioL-Bold

text: given

text: a

text: point

text: set

text: in

text: any

text: dimensions,

text: our

text: closed-form

text: solution

text: provides

text: an

text: exact,

text: continuous

text: and

text: efﬁcient

text: algorithm

text: for

text: computing

text: a

text: structure-aware

text: tensor

text: that

text: simultaneously

text: achieves

text: salient

text: structure

text: detection

text: and

text: outlier

text: attenuation.

text: Using

text: CFTV,

text: we

text: prove

text: the

text: convergence

text: of

text: tensor

text: voting

text: on

text: a

text: Markov

text: random

text: ﬁeld

text: (MRF),

text: thus

text: termed

text: as

text: MRFTV,

text: where

text: the

text: structure-aware

text: tensor

text: at

text: each

text: input

text: site

text: reaches

text: a

text: stationary

text: state

text: upon

text: convergence

text: in

text: structure

text: propagation.

text: We

text: then

text: embed

text: structure-aware

text: tensor

text: into

text: expectation

text: maximization

text: (EM)

text: for

text: optimizing

text: a

text: single

text: linear

text: structure

text: to

text: achieve

text: efﬁcient

text: and

text: robust

text: parameter

text: estimation.

text: Speciﬁcally,

text: our

text: EMTV

text: algorithm

text: optimizes

text: both

text: the

text: tensor

text: and

text: ﬁtting

text: parameters

text: and

text: does

text: not

text: require

text: random

text: sampling

text: consensus

text: typically

text: used

text: in

text: existing

text: robust

text: statistical

text: techniques.

text: We

text: performed

text: quantitative

text: evaluation

text: on

text: its

text: accuracy

text: and

text: robustness,

text: showing

text: that

text: EMTV

text: performs

text: better

text: than

text: the

text: original

text: TV

text: and

text: other

text: state-of-the-art

text: techniques

text: in

text: fundamental

text: matrix

text: estimation

text: for

text: multiview

text: stereo

text: matching.

text: The

text: extensions

text: of

text: CFTV

text: and

text: EMTV

text: for

text: extracting

text: multiple

text: and

text: nonlinear

text: structures

text: are

text: underway.

text: An

text: addendum

text: is

text: included

text: in

text: this

text: arXiv

text: version.

text: Index

text: Terms—Tensor

text: voting,

text: closed-form

text: solution,

text: structure

text: inference,

text: parameter

text: estimation,

text: multiview

text: stereo.

text: INTRODUCTION

text: as

text: well

text: as

text: general

text: theory

text: of

text: tensor

text: voting.

text: This

text: paper

text: T

text: HIS

text: paper

text: reinvents

text: tensor

text: voting

text: for

text: robust

text: focuses

text: on

text: the

text: special

text: theory,

text: where

text: the

text: above

text: data

text: computer

text: vision,

text: by

text: proving

text: a

text: closed-form

text: solution

text: communication

text: is

text: data

text: driven

text: without

text: using

text: constraints

text: to

text: computing

text: an

text: exact

text: structure-aware

text: tensor

text: after

text: data

text: other

text: than

text: proximity

text: and

text: continuity.

text: The

text: special

text: theory,

text: communication

text: in

text: a

text: feature

text: space

text: of

text: any

text: dimensions,

text: sometimes

text: coined

text: as

text: “ﬁrst

text: voting

text: pass,”

text: is

text: applied

text: to

text: process

text: raw

text: input

text: data

text: to

text: detect

text: structures

text: and

text: outliers.

text: where

text: the

text: goal

text: is

text: salient

text: structure

text: inference

text: from

text: noisy

text: In

text: addition

text: to

text: structure

text: detection

text: and

text: outlier

text: attenuation,

text: and

text: corrupted

text: data.

text: To

text: infer

text: structures

text: from

text: noisy

text: data

text: corrupted

text: by

text: in

text: the

text: general

text: theory

text: of

text: tensor

text: voting,

text: tensor

text: votes

text: are

text: outliers,

text: in

text: tensor

text: voting,

text: input

text: points

text: communicate

text: propagated

text: along

text: preferred

text: directions

text: to

text: achieve

text: data

text: among

text: themselves

text: subject

text: to

text: proximity

text: and

text: continuity

text: communication

text: when

text: such

text: directions

text: are

text: available,

text: typ-

text: constraints.

text: Consequently,

text: each

text: point

text: is

text: aware

text: of

text: its

text: ically

text: after

text: the

text: ﬁrst

text: pass,

text: such

text: that

text: useful

text: tensor

text: votes

text: structure

text: saliency

text: via

text: a

text: structure-aware

text: tensor.

text: Structure

text: are

text: reinforced

text: whereas

text: irrelevant

text: ones

text: are

text: suppressed.

text: refers

text: to

text: surfaces,

text: curves,

text: or

text: junctions

text: if

text: the

text: feature

text: space

text: Expressing

text: tensor

text: voting

text: in

text: a

text: single

text: and

text: compact

text: is

text: three

text: dimensional

text: where

text: a

text: structure-aware

text: tensor

text: can

text: equation,

text: or

text: a

text: closed-form

text: solution,

text: offers

text: many

text: ad-

text: be

text: visualized

text: as

text: an

text: ellipsoid:

text: if

text: a

text: point

text: belongs

text: to

text: a

text: vantages:

text: not

text: only

text: an

text: exact

text: and

text: efﬁcient

text: solution

text: can

text: smooth

text: surface,

text: the

text: resulting

text: ellipsoid

text: after

text: data

text: com-

text: be

text: achieved

text: with

text: less

text: implementation

text: effort

text: for

text: salient

text: munication

text: resembles

text: a

text: stick

text: pointing

text: along

text: the

text: surface

text: structure

text: detection

text: and

text: outlier

text: attenuation,

text: formal

text: and

text: normal;

text: if

text: a

text: point

text: lies

text: on

text: a

text: curve

text: the

text: tensor

text: resembles

text: useful

text: mathematical

text: operations

text: such

text: as

text: differential

text: cal-

text: a

text: plate,

text: where

text: the

text: curve

text: tangent

text: is

text: perpendicular

text: to

text: culus

text: can

text: be

text: applied

text: which

text: is

text: otherwise

text: impossible

text: the

text: plate

text: tensor;

text: if

text: it

text: is

text: a

text: point

text: junction

text: where

text: surfaces

text: using

text: the

text: original

text: tensor

text: voting

text: procedure.

text: Notably,

text: we

text: intersect,

text: the

text: tensor

text: will

text: be

text: like

text: a

text: ball.

text: An

text: outlier

text: is

text: can

text: prove

text: the

text: convergence

text: of

text: tensor

text: voting

text: on

text: Markov

text: characterized

text: by

text: a

text: set

text: of

text: inconsistent

text: votes

text: it

text: receives

text: random

text: ﬁelds

text: (MRFTV)

text: where

text: a

text: structure-aware

text: tensor

text: after

text: data

text: communication.

text: at

text: each

text: input

text: site

text: achieves

text: a

text: stationary

text: state

text: upon

text: con-

text: We

text: develop

text: in

text: this

text: paper

text: a

text: closed-form

text: solution

text: to

text: vergence.

text: tensor

text: voting

text: (CFTV),

text: which

text: is

text: applicable

text: to

text: the

text: special

text: Using

text: CFTV,

text: we

text: contribute

text: a

text: mathematical

text: derivation

text: based

text: on

text: expectation

text: maximization

text: (EM)

text: that

text: applies

text: the

text: exact

text: tensor

text: solution

text: for

text: extracting

text: the

text: most

text: salient

text: linear

text: T.-P.

text: Wu

text: is

text: with

text: the

text: Enterprise

text: and

text: Consumer

text: Electronics

text: Group,

text: Hong

text: Kong

text: Applied

text: Science

text: and

text: Technology

text: Research

text: Institute

text: (ASTRI),

text: Hong

text: structure,

text: despite

text: that

text: the

text: input

text: data

text: is

text: highly

text: corrupted.

text: Kong.

text: E-mail:

text: tpwu@astri.org.

text: Our

text: algorithm

text: is

text: called

text: EMTV,

text: which

text: optimizes

text: both

text: S.-K.

text: Yeung

text: and

text: C.-K.

text: Tang

text: are

text: with

text: the

text: Department

text: of

text: Computer

text: Science

text: the

text: tensor

text: and

text: ﬁtting

text: parameters

text: upon

text: convergence

text: and

text: and

text: Engineering,

text: Hong

text: Kong

text: University

text: of

text: Science

text: and

text: Technology,

text: Clear

text: Water

text: Bay,

text: Hong

text: Kong.

text: E-mail:

text: saikit,cktang

text: @cse.ust.hk.

text: does

text: not

text: require

text: random

text: sampling

text: consensus

text: typical

text: J.

text: Jia

text: is

text: with

text: the

text: Department

text: of

text: Computer

text: Science

text: and

text: Engineering,

text: of

text: existing

text: robust

text: statistical

text: techniques.

text: The

text: extension

text: Chinese

text: University

text: of

text: Hong

text: Kong,

text: Shatin,

text: Hong

text: Kong.

text: E-mail:

text: leo-

text: to

text: extract

text: salient

text: multiple

text: and

text: nonlinear

text: structures

text: is

text: jia@cse.cuhk.edu.hk

text: G.

text: Medioni

text: is

text: with

text: the

text: Department

text: of

text: Computer

text: Science,

text: University

text: of

text: underway.

text: Southern

text: California,

text: USA.

text: E-mail:

text: medioni@usc.edu.

text: While

text: the

text: mathematical

text: derivation

text: may

text: seem

text: in-

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: SUBMITTED TO IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 2 Fig. 1. Inlier/outlier and tensor inverse illustration. (a) The normal votes received at a surface point cast by points in x’s neighborhood. Three salient outliers are present. (b) For a non-surface point, there is no preference to any normals. (c) The ≥ structure-aware tensor induced by the normal observations in (a), which is represented by a d-D ellipsoid, where d 2. The orange curve (dashed curve) represents the variance produced along all possible directions. (d) The structure-aware tensor collected after collecting the received votes in (b). (e) and (f) correspond to the inverse of (c) and (d), respectively. volved, our main results for CFTV, MRFTV and EMTV, sian distribution to model inliers, the authors of [28] that is, Eqns (11), (12), (18), (19), (26), and (30). have proposed to use a two-step scale estimator (TSSE) to rigorous mathematical foundations, are applicable to any reﬁne the model scale: ﬁrst, a non-Gaussian distribution dimensions, produce more robust and accurate results is used to model inliers where local peaks of density are as demonstrated in our qualitative and quantitative found by mean shift [5]; second, the scale parameter is evaluation using challenging synthetic and real data, estimated by a median scale estimator with the estimated but on the other hand are easier to implement. The peaks and valleys. On the other hand, the projection- source codes accompanying this paper are available in based M-estimator (pbM) [3], an improvement made on the supplemental material. the M-estimator, uses a Parzen window for scale estima- 2 RELATED WORK tion, so the scale parameter is automatically found by searching for the normal direction (projection direction) While this paper is mainly concerned with tensor that maximizes the sharpest peak of the density. This voting, we provide a concise review on robust estimation does not require an input scale from the user. While these and expectation maximization. recent methods can tolerate more outliers, most of them Robust estimators. Robust techniques are widely used still rely on or are based on RANSAC and a number of and an excellent review of the theoretical foundations of random sampling trials is required to achieve the desired robust methods in the context of computer vision can be robustness. found in [20]. To reject outliers, a multi-pass method using L∞- The Hough transform [13] is a robust voting-based norms was proposed to successively detect outliers technique operating in a parameter space capable of which are characterized by maximum errors [24]. extracting multiple models from noisy data. Statistical Expectation Maximization. EM has been used in han- Hough transform [6] can be used for high dimensional dling missing data and identifying outliers in robust spaces with sparse observations. Mean shift [5] has been computer vision [8], and its convergence properties were widely used since its introduction to computer vision studied [18]. In essence, EM consists of two steps [2], for robust feature space analysis. The Adaptive mean [18]: shift [11] with variable bandwidth in high dimensions was introduced in texture classiﬁcation and has since 1) E-Step. Computing an expected value for the com- been applied to other vision tasks. Another popular plete data set using incomplete data and the cur- robust method in computer vision is in the class of rent estimates of the parameters. random sampling consensus (RANSAC) procedures [7] 2) M-Step. Maximizing the complete data log- which have spawned a lot of follow-up work (e.g., likelihood using the expected value computed in optimal randomized RANSAC [4]). the E-step. Like RANSAC [7], robust estimators including the LMedS [22] and the M-estimator [14] adopted a statistical EM is a powerful inference algorithm, but it is also approach. The LMedS, RANSAC and the Hough trans- well-known from [8] that: 1) initialization is an issue form can be expressed as M-estimators with auxiliary because EM can get stuck in poor local minima, and 2) scale [20]. The choice of scales and parameters related treatment of data points with small expected weights to the noise level are major issues. Existing works on requires great care. They should not be regarded as neg- robust scale estimation use random sampling [26], or ligible, as their aggregate effect can be quite signiﬁcant. operate on different assumptions (e.g., more than 50% In this paper we initialize EMTV using structure-aware of the data should be inliers [23]; inliers have a Gaus- tensors obtained by CFTV. As we will demonstrate, sian distribution [15]). Among them, the Adaptive Scale such initialization not only allows the EMTV algorithm Sample Consensus (ASSC) estimator [28] has shown the to converge quickly (typically within 20 iterations) but best performance where the estimation process requires also produces accurate and robust solution in parameter no free parameter as input. Rather than using a Gaus- estimation and outlier rejection. ##LTFigure##

text: SUBMITTED

text: TO

text: IEEE

text: TRANSACTIONS

text: ON

text: PATTERN

text: ANALYSIS

text: AND

text: MACHINE

text: INTELLIGENCE

text: Fig.

text: Inlier/outlier

text: and

text: tensor

text: inverse

text: illustration.

text: (a)

text: The

text: normal

text: votes

text: received

text: at

text: a

text: surface

text: point

text: cast

text: by

text: points

text: in

text: x’s

text: neighborhood.

text: Three

text: salient

text: outliers

text: are

text: present.

text: (b)

text: For

text: a

text: non-surface

text: point,

text: there

text: is

text: no

text: preference

text: to

text: any

text: normals.

text: (c)

text: The

text: structure-aware

text: tensor

text: induced

text: by

text: the

text: normal

text: observations

text: in

text: (a),

text: which

text: is

text: represented

text: by

text: a

text: d-D

text: ellipsoid,

text: where

text: d

text: The

text: orange

text: curve

text: (dashed

text: curve)

text: represents

text: the

text: variance

text: produced

text: along

text: all

text: possible

text: directions.

text: (d)

text: The

text: structure-aware

text: tensor

text: collected

text: after

text: collecting

text: the

text: received

text: votes

text: in

text: (b).

text: (e)

text: and

text: (f)

text: correspond

text: to

text: the

text: inverse

text: of

text: (c)

text: and

text: (d),

text: respectively.

text: volved,

text: our

text: main

text: results

text: for

text: CFTV,

text: MRFTV

text: and

text: EMTV,

text: sian

text: distribution

text: to

text: model

text: inliers,

text: the

text: authors

text: of

text: that

text: is,

text: Eqns

text: and

text: have

text: proposed

text: to

text: use

text: a

text: two-step

text: scale

text: estimator

text: (TSSE)

text: to

text: rigorous

text: mathematical

text: foundations,

text: are

text: applicable

text: to

text: any

text: reﬁne

text: the

text: model

text: scale:

text: ﬁrst,

text: a

text: non-Gaussian

text: distribution

text: dimensions,

text: produce

text: more

text: robust

text: and

text: accurate

text: results

text: is

text: used

text: to

text: model

text: inliers

text: where

text: local

text: peaks

text: of

text: density

text: are

text: as

text: demonstrated

text: in

text: our

text: qualitative

text: and

text: quantitative

text: found

text: by

text: mean

text: shift

text: second,

text: the

text: scale

text: parameter

text: is

text: evaluation

text: using

text: challenging

text: synthetic

text: and

text: real

text: data,

text: estimated

text: by

text: a

text: median

text: scale

text: estimator

text: with

text: the

text: estimated

text: but

text: on

text: the

text: other

text: hand

text: are

text: easier

text: to

text: implement.

text: The

text: peaks

text: and

text: valleys.

text: On

text: the

text: other

text: hand,

text: the

text: projection-

text: source

text: codes

text: accompanying

text: this

text: paper

text: are

text: available

text: in

text: based

text: M-estimator

text: (pbM)

text: an

text: improvement

text: made

text: on

text: the

text: supplemental

text: material.

text: the

text: M-estimator,

text: uses

text: a

text: Parzen

text: window

text: for

text: scale

text: estima-

text: RELATED

text: WORK

text: tion,

text: so

text: the

text: scale

text: parameter

text: is

text: automatically

text: found

text: by

text: searching

text: for

text: the

text: normal

text: direction

text: (projection

text: direction)

text: While

text: this

text: paper

text: is

text: mainly

text: concerned

text: with

text: tensor

text: that

text: maximizes

text: the

text: sharpest

text: peak

text: of

text: the

text: density.

text: This

text: voting,

text: we

text: provide

text: a

text: concise

text: review

text: on

text: robust

text: estimation

text: does

text: not

text: require

text: an

text: input

text: scale

text: from

text: the

text: user.

text: While

text: these

text: and

text: expectation

text: maximization.

text: recent

text: methods

text: can

text: tolerate

text: more

text: outliers,

text: most

text: of

text: them

text: Robust

text: estimators.

text: Robust

text: techniques

text: are

text: widely

text: used

text: still

text: rely

text: on

text: or

text: are

text: based

text: on

text: RANSAC

text: and

text: a

text: number

text: of

text: and

text: an

text: excellent

text: review

text: of

text: the

text: theoretical

text: foundations

text: of

text: random

text: sampling

text: trials

text: is

text: required

text: to

text: achieve

text: the

text: desired

text: robust

text: methods

text: in

text: the

text: context

text: of

text: computer

text: vision

text: can

text: be

text: robustness.

text: found

text: in

text: To

text: reject

text: outliers,

text: a

text: multi-pass

text: method

text: using

text: L∞-

text: The

text: Hough

text: transform

text: is

text: a

text: robust

text: voting-based

text: norms

text: was

text: proposed

text: to

text: successively

text: detect

text: outliers

text: technique

text: operating

text: in

text: a

text: parameter

text: space

text: capable

text: of

text: which

text: are

text: characterized

text: by

text: maximum

text: errors

text: extracting

text: multiple

text: models

text: from

text: noisy

text: data.

text: Statistical

text: Expectation

text: Maximization.

text: EM

text: has

text: been

text: used

text: in

text: han-

text: Hough

text: transform

text: can

text: be

text: used

text: for

text: high

text: dimensional

text: dling

text: missing

text: data

text: and

text: identifying

text: outliers

text: in

text: robust

text: spaces

text: with

text: sparse

text: observations.

text: Mean

text: shift

text: has

text: been

text: computer

text: vision

text: and

text: its

text: convergence

text: properties

text: were

text: widely

text: used

text: since

text: its

text: introduction

text: to

text: computer

text: vision

text: studied

text: In

text: essence,

text: EM

text: consists

text: of

text: two

text: steps

text: for

text: robust

text: feature

text: space

text: analysis.

text: The

text: Adaptive

text: mean

text: shift

text: with

text: variable

text: bandwidth

text: in

text: high

text: dimensions

text: was

text: introduced

text: in

text: texture

text: classiﬁcation

text: and

text: has

text: since

text: E-Step.

text: Computing

text: an

text: expected

text: value

text: for

text: the

text: com-

text: been

text: applied

text: to

text: other

text: vision

text: tasks.

text: Another

text: popular

text: plete

text: data

text: set

text: using

text: incomplete

text: data

text: and

text: the

text: cur-

text: robust

text: method

text: in

text: computer

text: vision

text: is

text: in

text: the

text: class

text: of

text: rent

text: estimates

text: of

text: the

text: parameters.

text: random

text: sampling

text: consensus

text: (RANSAC)

text: procedures

text: M-Step.

text: Maximizing

text: the

text: complete

text: data

text: log-

text: which

text: have

text: spawned

text: a

text: lot

text: of

text: follow-up

text: work

text: (e.g.,

text: likelihood

text: using

text: the

text: expected

text: value

text: computed

text: in

text: optimal

text: randomized

text: RANSAC

text: the

text: E-step.

text: Like

text: RANSAC

text: robust

text: estimators

text: including

text: the

text: LMedS

text: and

text: the

text: M-estimator

text: adopted

text: a

text: statistical

text: EM

text: is

text: a

text: powerful

text: inference

text: algorithm,

text: but

text: it

text: is

text: also

text: approach.

text: The

text: LMedS,

text: RANSAC

text: and

text: the

text: Hough

text: trans-

text: well-known

text: from

text: that:

text: initialization

text: is

text: an

text: issue

text: form

text: can

text: be

text: expressed

text: as

text: M-estimators

text: with

text: auxiliary

text: because

text: EM

text: can

text: get

text: stuck

text: in

text: poor

text: local

text: minima,

text: and

text: scale

text: The

text: choice

text: of

text: scales

text: and

text: parameters

text: related

text: treatment

text: of

text: data

text: points

text: with

text: small

text: expected

text: weights

text: to

text: the

text: noise

text: level

text: are

text: major

text: issues.

text: Existing

text: works

text: on

text: requires

text: great

text: care.

text: They

text: should

text: not

text: be

text: regarded

text: as

text: neg-

text: robust

text: scale

text: estimation

text: use

text: random

text: sampling

text: or

text: ligible,

text: as

text: their

text: aggregate

text: effect

text: can

text: be

text: quite

text: signiﬁcant.

text: operate

text: on

text: different

text: assumptions

text: (e.g.,

text: more

text: than

text: In

text: this

text: paper

text: we

text: initialize

text: EMTV

text: using

text: structure-aware

text: of

text: the

text: data

text: should

text: be

text: inliers

text: inliers

text: have

text: a

text: Gaus-

text: tensors

text: obtained

text: by

text: CFTV.

text: As

text: we

text: will

text: demonstrate,

text: sian

text: distribution

text: Among

text: them,

text: the

text: Adaptive

text: Scale

text: such

text: initialization

text: not

text: only

text: allows

text: the

text: EMTV

text: algorithm

text: Sample

text: Consensus

text: (ASSC)

text: estimator

text: has

text: shown

text: the

text: to

text: converge

text: quickly

text: (typically

text: within

text: iterations)

text: but

text: best

text: performance

text: where

text: the

text: estimation

text: process

text: requires

text: also

text: produces

text: accurate

text: and

text: robust

text: solution

text: in

text: parameter

text: no

text: free

text: parameter

text: as

text: input.

text: Rather

text: than

text: using

text: a

text: Gaus-

text: estimation

text: and

text: outlier

text: rejection.

text: ##LTFigure##