Optical Flow I. Guido Gerig CS 6320, Spring 2015
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1 Opical Flow Guido Gerig CS 6320, Spring 2015 (credis: Marc Pollefeys UNC Chapel Hill, Comp 256 / K.H. Shafique, UCSF, CAP5415 / S. Narasimhan, CMU / Bahadir K. Gunurk, EE 7730 / Bradski&Thrun, Sanford CS223
2 Maerials Gary Bradski & Sebasian Thrun, Sanford CS223 hp://robos.sanford.edu/cs223b/inde.hml S. Narasimhan, CMU: hp:// s06/lecures/pps/lec-16.pp M. Pollefeys, ETH Zurich/UNC Chapel Hill: hp:// K.H. Shafique, UCSF: hp:// Lecure 18 (March 25, 2003), Slides: PDF/ PPT Jepson, Torono: hp:// Original paper Horn&Schunck 1981: hp:// MT A Memo Horn& Schunck 1980: hp://people.csail.mi.edu/bkph/am/am-572.pdf Bahadir K. Gunurk, EE 7730 mage Analysis Some slides and illusraions from L. Van Gool, T. Darell, B. Horn, Y. Weiss, P. Anandan, M. Black, K. Toyama
3 Opical Flow and Moion We are ineresed in finding he movemen of scene objecs from imevarying images (videos). Los of uses Moion deecion Track objecs Correc for camera jier (sabilizaion) Align images (mosaics) 3D shape reconsrucion Special effecs Games: hp:// User nerfaces: hp:// Video compression
4 Tracking Rigid Objecs
5 Tracking Non-rigid Objecs (Comaniciu e al, Siemens)
6 Tracking Non-rigid Objecs
7 7 Opical Flow: Where do piels move o?
8 Opical Flow: Where do piels move o?
9 Wha is Opical Flow (OF)? p 1 p 2 p 3 Opical Flow v 1 v 2 v 3 p 4 v 4 ( i ),{ p } ( 1) Velociy vecors { v i } Opical flow is he relaion of he moion field: he 2D projecion of he physical movemen of poins relaive o he observer o 2D displacemen of piel paches on he image plane. Common assumpion: The appearance of he image paches do no change (brighness consancy) ( p, ) ( p v, 1) i i i 9 Noe: more elaborae racking models can be adoped if more frames are process all a once
10 Opical Flow: Correspondence Basic quesion: Which Piel wen where?
11 Srucure from Moion? Known: opical flow (insananeous velociy) Moion of camera / objec?
12 Opical Flow is NOT 3D moion field Opical flow: Piel moion field as observed in image. hp://of-eval.sourceforge.ne/
13 Opical Flow is NOT 3D moion field hp://en.wikipedia.org/wiki/file:opicfloweg.png
14 Definiion of opical flow OPTCAL FLOW = apparen moion of brighness paerns deally, he opical flow is he projecion of he hree-dimensional velociy vecors on he image 14
15 Opical Flow - Agenda Brighness Consancy The Aperure problem Regularizaion Lucas-Kanade Coarse-o-fine Parameric moion models Direc deph SSD racking Robus flow Bayesian flow
16 Opical Flow - Agenda Brighness Consancy The Aperure problem Regularizaion Lucas-Kanade Coarse-o-fine Parameric moion models Direc deph SSD racking Robus flow Bayesian flow 16
17 Sar wih an Equaion: Brighness Consancy Poin moves (small), bu is brighness remains consan:,, Time: Time: + d 0
18 Mahemaical formulaion (1D) ((),) = brighness a () a ime Brighness consancy assumpion (shif of locaion bu brighness says same): d (, ) (, y, ) d Opical flow consrain equaion (chain rule): d d d d 0
19 Opical Flow: 1D Case Brighness Consancy Assumpion: ) ), ( ( ) ), ( ( ) ( d d f 0 ) ( v v 0 ) ( f Because no change in brighness wih ime Gary Bradski & Sebasian Thrun, Sanford CS223 hp://robos.sanford.edu/cs223b/inde.hml 19
20 Tracking in he 1D case: (, ) (, 1) p v? Gary Bradski & Sebasian Thrun, Sanford CS223 hp://robos.sanford.edu/cs223b/inde.hml 20
21 Tracking in he 1D case: (, ) (, 1) Temporal derivaive p v Spaial derivaive p v Assumpions: Brighness consancy Small moion 21
22 Tracking in he 1D case: eraing helps refining he velociy vecor (, ) (, 1) Temporal derivaive a 2 nd ieraion p Can keep he same esimae for spaial derivaive v v previous Converges in abou 5 ieraions 22
23 From 1D o 2D racking 0 ) ( 1D: 0 ) ( y y 2D: 0 ) ( v y u Shoo! One equaion, wo velociy (u,v) unknowns 23
24 The aperure problem u d d v dy d, y y u y v 0 1 equaion in 2 unknowns Horn and Schunck opical flow equaion
25 Opical Flow Brighness Consancy The Aperure problem Regularizaion Lucas-Kanade Coarse-o-fine Parameric moion models Direc deph SSD racking Robus flow Bayesian flow 26
26 How does his show up visually? Known as he Aperure Problem Gary Bradski & Sebasian Thrun, Sanford CS223 hp://robos.sanford.edu/cs223b/inde.hml
27 Aperure Problem Eposed Moion along jus an edge is ambiguous Gary Bradski & Sebasian Thrun, Sanford CS223 hp://robos.sanford.edu/cs223b/inde.hml
28 How does his show up visually? Known as he Aperure Problem Gary Bradski & Sebasian Thrun, Sanford CS223 hp://robos.sanford.edu/cs223b/inde.hml
29 How does his show up visually? Known as he Aperure Problem Gary Bradski & Sebasian Thrun, Sanford CS223 hp://robos.sanford.edu/cs223b/inde.hml
30 How does his show up visually? Known as he Aperure Problem Gary Bradski & Sebasian Thrun, Sanford CS223 hp://robos.sanford.edu/cs223b/inde.hml
31 Opical Flow vs. Moion: Aperure Problem Barber shop pole: hp://
32 Normal Flow Wha we can ge!! We ge a mos Normal Flow wih one poin we can only deec movemen perpendicular o he brighness gradien. Soluion is o ake a pach of piels around he piel of ineres.
33 Recall: Aperure Problem
34 Recall: Aperure Problem
35 Aperure Problem and Normal Flow v d p (u,v ) le (u, v ) be rue flow rue flow has wo componens: u Normal flow: d Parallel flow: p normal flow can be compued parallel flow canno
36 37
37 Compuing True Flow Schunck Horn & Schunck Lukas and Kanade
38 Possible Soluion: Neighbors Two adjacen piels which are par of he same rigid objec: we can calculae normal flows v n1 and v n2 Two OF equaions for 2 parameers of flow:. 0. 0
39 Schunck: Considering Neighbor Piels
40 Schunck: Considering Neighbor Piels Cluser cener provides velociy vecor common for all piels in pach. Jepson, Torono: hp://
41 Opical Flow Brighness Consancy The Aperure problem Regularizaion: Horn & Schunck Lucas-Kanade Coarse-o-fine Parameric moion models Direc deph SSD racking Robus flow Bayesian flow 42
42 43 Horn & Schunck algorihm
43 Horn & Schunck algorihm Addiional smoohness consrain (usually moion field varies smoohly in he image penalize deparure from smoohness) : e s (( u 2 u 2 y ) ( v 2 v 2 y )) ddy, OF consrain equaion erm (formulae error in opical flow consrain) : e c ( u minimize es+ec y v ) 2 ddy, 44
44 Horn & Schunck algorihm Variaional calculus: Pair of second order differenial equaions ha can be solved ieraively. 45
45 Horn & Schunck algorihm Δ0 Δ0 Approimae Laplacian by weigh averaged compued in a neighborhood around he piel (,y): Δ,,, Δ,,, Rearranging erms: equaions in 2 unknowns, wrie v in erms of u and plug i in he oher equaion
46 Horn & Schunck algorihm 2 equaions in 2 unknowns, wrie v in erms of u and plug i in he oher equaion 47
47 48 The Euler-Lagrange equaions : 0 0 y y v v v u u u F y F F F y F F n our case,, ) ( ) ( ) ( y y y v u v v u u F so he Euler-Lagrange equaions are, ) (, ) ( y y y v u v v u u y is he Laplacian operaor Horn & Schunck algorihm
48 49 Remarks : 1. Coupled PDEs solved using ieraive mehods and finie differences 2. More han wo frames allow a beer esimaion of 3. nformaion spreads from corner-ype paerns, ) (, ) ( y y y v u v v v u u u Horn & Schunck algorihm
49 Discree Opical Flow Algorihm Consider image piel Deparure from Smoohness Consrain: s ij 1 4 [( u ( v i1, j i1, j ( i, j) u v i, j i, j ) ) 2 2 ( u ( v i, j1 i, j1 u v i, j i, j ) ) 2 2 ] Error in Opical Flow consrain equaion: c ij ( ij u ij 2 ) We seek he se ha minimize: ij y { u ij } & { vij} v ij ij e ( s c ) ij ij i j { u ij } & { vij} NOTE: show up in more han one erm
50 Discree Opical Flow Algorihm e Differeniaing w.r. and seing o zero: e ukl e v kl kl kl kl kl 2 ( u u ) 2 ( u v ) kl kl are averages of around piel kl u kl kl kl kl kl 2 ( v v ) 2 ( u v ) kl kl v & kl u kl v & ( u, v) ( k, l) kl kl y y kl kl y 0 0 Updae Rule: u n1 kl u n kl kl u n kl 1 [( kl kl y ) 2 v n kl ( kl y kl ) 2 ] kl v n1 kl v n kl kl u n kl 1 [( kl kl y ) 2 v n kl ( kl y kl ) 2 ] kl y
51 Horn-Schunck Algorihm : Discree Case Derivaives (and error funcionals) are approimaed by difference operaors Leads o an ieraive soluion: u v n1 ij n1 ij u, v are u v n ij n ij y he averages u n ij 1 ( y 2 v n ij 2 y of values of neighbors )
52 nuiion of he eraive Scheme v (E,E y ) ( u, v) Consrain line (u,v) u The new value of (u,v) a a poin is equal o he average of surrounding values minus an adjusmen in he direcion of he brighness gradien
53 Horn - Schunck Algorihm
54 Eample hp://of-eval.sourceforge.ne/
55 Resuls
56 Resuls
57 Opical Flow Resul
58 Horn & Schunck, remarks 1. Errors a boundaries (smooh over) 2. Eample of regularizaion (selecion principle for he soluion of ill-posed problems) 60
59 Resuls of an enhanced sysem
60 Resuls hp://www-suden.informaik.uni-bonn.de/~gerdes/opicalflow/inde.hml
61 Resuls hp://
62 Opical Flow Brighness Consancy The Aperure problem Regularizaion Lucas-Kanade Coarse-o-fine Parameric moion models Direc deph SSD racking Robus flow Bayesian flow 64
63 Lucas & Kanade Assume single velociy for all piels wihin a pach. negrae over a pach.
64 Lucas & Kanade
65 Lucas & Kanade
66 ), ( 0 2 ), ( y y y v u dv v u de v u du v u de
67 69
68 Discussion Horn-Schunck: Add smoohness consrain. Lucas-Kanade: Provide consrain by minimizing over local neighborhood: Horn-Schunck and Lucas-Kanade opical mehods work only for small moion. f objec moves faser, he brighness changes rapidly, derivaive masks fail o esimae spaioemporal derivaives. Pyramids can be used o compue large opical flow vecors.
69 eraive Refinemen (eraive Lucas-Kanade) Esimae velociy a each piel using one ieraion of Lucas and Kanade esimaion Warp one image oward he oher using he esimaed flow field (easier said han done) Refine esimae by repeaing he process
70 Reduce he Resoluion!
71 Opical Flow Brighness Consancy The Aperure problem Regularizaion Lucas-Kanade Coarse-o-fine Parameric moion models Direc deph SSD racking Robus flow Bayesian flow 73
72
73 Revisiing he Small Moion Assumpion s his moion small enough? Probably no i s much larger han one piel (2 nd order erms dominae) How migh we solve his problem?
74 Coarse-o-fine Opical Flow Esimaion u=1.25 piels u=2.5 piels u=5 piels image H u=10 piels image Gaussian pyramid of image H Gaussian pyramid of image
75 Coarse-o-fine Opical Flow Esimaion run ieraive OF upsample run ieraive OF... image HJ image Gaussian pyramid of image H Gaussian pyramid of image
76 78
77 Video Segmenaion
78 Ne: Moion Field Srucure from Moion
79 Moion Field mage velociy of a poin moving in he scene v o v i r i f ' Y Perspecive projecion: Moion field d v i r d i f Ẑ ' X 1 r f ' i ro r Zˆ o r o ro Zv o vo r Z o Scene poin velociy: mage velociy: ro Z Z r o f ' 2 Z r vo r Z 2 o v o i v o dr i d Z d o r d
80
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