Early Work by D Arcy Thompson

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Percival Singleton
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Potetial Projects Recogitio ad classificatio Face recogitio: uder age variatio Image-based disease diagosis Shoe image classificatio Scee uderstadig CIS 5543 Computer Visio Shape Aalysis Matchig &

1 Potetial Projects Recogitio ad classificatio Face recogitio: uder age variatio Image-based disease diagosis Shoe image classificatio Scee uderstadig CIS 5543 Computer Visio Shape Aalysis Matchig & registratio TPS-RPM registratio for medical structures (shapes) Image stitchig Video aalysis Visual trackig (through occlusio) Actio recogitio Detectio Ladmark poit detectio Licese plate detectio Blurred object detectio Others Haibi Lig May slides revised from D Jacobs Shape Aalysis Topics Shape similarity Eample: kow poit correspodeces, determie similarity Early Work by D Arcy Thompso Shape morphig (warpig) Eample: kow poit correspodeces, determie warpig fuctio Shape matchig Eample: determie poit correspodeces Combied tasks D Arcy Thompso, 97 Key Poits Math is helpful for morphology Homologous structures ecessary: correspodece Give these, compute trasformatios of plae Uses: Nature of trasformatio gives clues to forces of growth Shapes related by simple trasformatio -> species are related May compellig eamples Morph betwee species, predict itermediate species Ca predict missig parts of skeleto Homologies Had a log traditio Aristotle: Save oly for a differece i the way of ecess or defect, the parts are idetical i the case of such aimals as are of oe ad the same geus I biology, study of homologous structures i species preceded provided backgroud for Darwi Homologous structures eplaied by God creatig differet species accordig to a commo pla Otogey provided clues to homology

Trasformatios Give matchig poits i two images, we fid a

udercostraied problem Implicitly, seeks simple trasformatio Not

Ituitively pretty clear i eamples cosidered Cao-boe of o,

Logarithmically varyig: eg, tapir s toes Descriptios of shape:

relevat eve without compariso Fourier descriptors Shape cotet

2 Trasformatios Give matchig poits i two images, we fid a trasformatio of plae Homeomorphism (cotiuous, oe-to-oe) This is udercostraied problem Implicitly, seeks simple trasformatio Not well defied here, will be subject of much future research Ituitively pretty clear i eamples cosidered Cao-boe of o, sheep, giraffe Simplest, subset of affie Piecewise affie Logarithmically varyig: eg, tapir s toes Descriptios of shape: Clues to Growth Somewhat differet topic, shape descriptios relevat eve without compariso Fourier descriptors Shape cotet Equal growth i all directios leads to circle (or sphere) Smooth: amphipods (a kid of crustacea)

3 No growth i oe directio (as i a leaf o a stem), growth icreases i directios away from this so r = si( Asymmetric amouts of growth o two sides Related Species Ivetio of Morphig? Give trasformatio betwee species, liearly iterpolate itermediate trasformatios Itermediate morphs predict itermediate species Pages 7-7 Figure 537 3

4 Coclusios Stress o homologies Shape compariso through o-trivial trasformatios Simplicity of trasformatio -> similarity of shape What is the simplest trasformatio? How do we fid it? Trasformatio may leave some deviatios, how are these hadled? Shape Spaces Procrustes Aalysis Matchig Sets of Poit Features Fid best trasformatio Similarity trasformatio, thi-plate splies Measure how good it is Chamfer distace, Haussdorf distace, Euclidea distace, procrustea distace, deformatio eergy Assumptios Two sets of D poits Mostly we assume there eists a correct oe-to-oe correspodece Ad this correspodece is give This is very atural i morphometrics, where poits are measured ad labeled I visio we must solve for correspodece Shape Space What is shape? all the geometrical iformatio that remais whe locatio, scale ad rotatioal effects are filtered out from a object D G Kedall (984) So describe poits idepedet of similarity trasformatio Remove traslatio Simplest way: traslate so poit is at origi, the remove it More elegat, traslate ceter of mass to origi, remove a poit Remove scale Scale so that sum X i ^ = Resultig set of poits is called pre-shape Pre because we have t removed rotatio yet Pre-shape Notatio: U ad X deote sets of ormalized poits Poits called X i ad U i, with coordiates ( i,y i ), (u i, v i ) If we started with poits, we ow have - so that: sum i=- i^ + y i^ = So we ca thik of these coordiates as lyig o a uit hypersphere i (-)-dimesioal space 4

5 Shape If we cosider all possible rotatios of a set of ormalized poits, these trace out a closed, D curve i pre-shape space Maifold Distaces betwee shapes ca be thought of as distaces betwee these curves Note: to compute distace, without loss of geerality, we ca assume that oe set of poits (U) does ot rotate, sice rotatig both poit sets by the same amout does t chage distaces Procrustes Distaces Full Procrustes Distace D F mi (s,) U sxr Fid a scalig ad rotatio of X that miimizes the Euclidea distace to U R() meas rotate by Partial Procrustes Distace D P mi U XR Rotate X to miimize the Euclidea distace to U Procrustes Distace Rotate X to miimize the geodesic distace o the sphere from X to U Liear Pose Solvig Liear Pose: D rotatio, traslatio & scale We ca liearly fid optimal similarity trasformatio that matches X to U (ie, miimize sum AX i -U i ^, where A is a similarity trasformatio This is asymmetric betwee ee X ad U I same way we ca liearly compute Full Procrustes Distace This is symmetric Leads immediately to other procrustes distaces u u u cos si t s y y v v v si cos t y a b t y y b a t y with a s cos, b ssi Notice a ad b ca take o ay values Equatios liear i a, b, traslatio s a b Solve eactly with poits, or overcostraied system with more y y cos a s Similarity Matchig Give poit sets X ad U, compare by fidig similarity trasformatio A that miimizes AX-U Note that we ow also kow how to calculate the Full Procrustes Distace This is just a leastsquares solutio to the over-costraied problem: X = poits X,, X U = poits U,, U Fid A to miimize sum AX i U i ^ A straightforward, liear problem Takig derivatives with respect to four ukows of A gives four liear equatios i four ukows u v u v u cos si s v si cos y a b b a y y y y y 5

6 Give two poits o the hypersphere, we ca draw the plae cotaiig these poits ad the origi Why Procrustes Distace? D P D F Procrustes Distaces is D P = si ( /) D F = si These are all mootoic i So the same choice of rotatio miimizes all three D F is easy to compute, others are easy to compute from D F Procrustes distace is most atural Ituitio: give two objects, we ca produce a sequece of itermediate objects o a straight lie betwee them, so the distace betwee the two objects is the sum of the distaces betwee ee itermediate objects This requires a geodesic Taget Space Ca compute a hyperplae taget to the hypersphere at a poit i preshape space Project all poits oto that plae All distaces Euclidea Average shape easy to fid This is reasoable whe all shapes similar I this case, all distaces are similar too Note that whe is small,, si( /), si() are all similar Warpig Thi-Plate Splies A fuctio, f, R -> R is a thi-plate splie if: Costrait: Give correspodig poits: X X ad U U, f(xi)=ui Eergy: f miimizes the followig bedig eergy: R f f y f y ddy Thi-Plate Splies Solutio: The fuctio f ca be computed usig straightforward liear algebra See Pricipal Warps: Thi-Plate Splies ad the Decompositio of Deformatios, Bookstei Statistical Shape Aalysis, Dryde ad Mardia Etesio: Ca pealize mismatch of poits (usig fuctio of Ui f(xi) ) Results: Much like D Arcy Thompso If we thik of this as the amout of bedig produced by f Allows arbitrary affie trasformatio 6

7 Chamfer Matchig Chamfer Matchig p i d i For every edge poit p i i the trasformed object, compute the distace to the earest image edge poit Sum distaces i mi( pi, q, pi, q, pi, q m ) Chamfer Matchig Mai Feature Every model poit matches a image poit A image poit ca match,, or more model poits Chamfer Matchig The, miimize this distace over pose Eample: miimum Chamfer distace over all traslatios t mi mi( pi t, q, pi t, q, pi t, q t i m ) Variatios Sum a differet distace f(d) = d or Mahatta distace f(d) = if d < threshold, otherwise This is called bouded error Use maimum distace istead of sum This is called: directed Hausdorff distace Use media distace Use other features Corers Lies The positio ad agles of lies must be similar Model lie may be subset of image lie Shape Cotet S Belogie, J Malik ad J Puzicha PAMI Slides revised from J Malik 7

Matchig Framework Comparig Poitsets model target Fid correspodeces betwee poits o shape

umber of poits iside each bi, eg: Cout = 4 Cout = F Compact represetatio of distributio

ivariat to rotatio by usig local taget orietatio frame Tolerat to small affie distortio

image registratio Comparig Shape Cotets Compute matchig costs usig Chi Squared distace:

8 Matchig Framework Comparig Poitsets model target Fid correspodeces betwee poits o shape Fast pruig Estimate trasformatio & measure similarity Shape Cotet Shape Cotet Cout the umber of poits iside each bi, eg: Cout = 4 Cout = F Compact represetatio of distributio of poits relative to each poit Shape Cotets Ivariat uder traslatio ad scale Ca be made ivariat to rotatio by usig local taget orietatio frame Tolerat to small affie distortio Log-polar bis make spatial blur proportioal to r Cf Spi Images (Johso & Hebert) - rage image registratio Comparig Shape Cotets Compute matchig costs usig Chi Squared distace: Recover correspodeces by solvig liear assigmet problem with costs C ij [Joker & Volgeat 987] 8

Articulatio Ivariat Shape Matchig Ier-Distace, Articulatio Isesitivity PAMI 7, Lig & Jacobs Ier-distace Legth of

is up bouded by a small value Computatio Shortest path Fast marchig Computatio: Step : Build graph Poits Nodes

C 5 9 5 3 3 θ 3 p q 4 5 θ: ier-agle Shape Cotet (SC) [Belogie et al ] Ier-Distace Shape Cotet (IDSC) Three objects

9 Articulatio Ivariat Shape Matchig Ier-Distace, Articulatio Isesitivity PAMI 7, Lig & Jacobs Ier-distace Legth of the shortest path betwee ladmark poits Articulatio Isesitivity Theorem: The chage of ierdistaces durig articulatio is up bouded by a small value Computatio Shortest path Fast marchig Computatio: Step : Build graph Poits Nodes Visible ibl poit pairs Edges Step : Shortest path Eg, Bellma-Ford Ier-Distace Shape Cotet (IDSC) IDSC Eamples A B C θ 3 p q 4 5 θ: ier-agle Shape Cotet (SC) [Belogie et al ] Ier-Distace Shape Cotet (IDSC) Three objects from the MPEG7 database Two poits p, q o each shape ad the shape cotet (SC) ad ierdistace shape cotet (IDSC) at p ad q 9

Image Spaces. What might an image space be

Image Spaces. What might an image space be Image Spaces What might a image space be Map each image to a poit i a space Defie a distace betwee two poits i that space Mabe also a shortest path (morph) We have alread see a simple versio of this, i