Lecture: Shape Analysis IntroductionSlide Number 2ShapeShape Properties: School Performance TestWhat is Shape?Example Biology Example AstronomyExample GeologyExample BiologyExample BiologyIndustrial Example: Particle Analysis, Particle Size Testing, Shape Classification Shape ClassesComputer Vision: MPEG-7BiometricsConcept of ShapeConcept of Shape?Shape: DefinitionShape DefinitionShape TransformationThe Problem of Size and ShapeLandmarksLandmarks (Dryden & Mardia)LandmarksLandmarksShape and RegistrationShape and RegistrationShape and RegistrationShape and RegistrationSize of Configuration of Landmarks?Alternative: Baseline Size A Shape Coordinate SystemBookstein CoordinatesMore General: Equivalence RelationshipsLecture:Shape AnalysisIntroductionGuido GerigCS 7960, Spring 2010References [Dryden&Mardia] Statistical Shape Analysis, Wiley, Chichester, Dryden, I.L. and Mardia, K.V. (1998).  DG Kendall (1984,Bull.Lond.Math.Soc) Bookstein (1986,Statistical Science) WS Kendall (1988,Adv.Appl.Probab.) Christopher G. Small, The Statistical Theory of Shape, Springer D’Arcy Thompson, 1917, On Growth and FormShapeThe word “shape” is very commonly used in everyday language, usually referring to the appearance of an object.Shape Properties: School Performance TestWhat is Shape?Example BiologyExample AstronomyExample GeologyExample BiologyExample BiologyThis picture clearly illustrates the typical shell shape differences between male (left) and female (right) eastern box turtles.Industrial Example: Particle Analysis, ParticleSize Testing, Shape ClassificationIn machine condition monitoring, the ability to determine the size and shape of contaminant particles is becoming a necessary, if not critical capability. Particles in an used oil sample may be due to the normal wear process, but an increase of particles, and their size and shape will assist the oil analysis laboratory in determining the source and severity of a potential malfunction.http://www.spectroinc.com/products-lasernet-fines.htmShape Classeshttp://sites.google.com/site/xiangbai/try-large.jpghttp://sites.google.com/site/xiangbai/animaldatasetComputer Vision: MPEG-7http://visionlab.uta.edu/shape_data.htmhttp://www.cis.temple.edu/~latecki/TestData/mpeg7shapeB.tar.gzVehicle shape classes: (a) Sedan, (b) Pickup, (c) Minivan, (d) SUV.Airplane shape classes: (a) Mirage, (b) Eurofighter, (c) F-14 wings closed, (d) F-14 wings opened, (e) Harrier, (f) F-22, (g) F-15.BiometricsConcept of ShapeConcept of Shape?Shape: DefinitionShape DefinitionDryden/Mardia, (Kendall 1977): Shape is all the geometrical information that remains when location, scale and rotationaleffects are filtered out from an object.Shape TransformationDog to CatSymmetry Maps and TransformsFor Perceptual Grouping and Object Recognition, Benjamin B. Kimia, BrownThe Problem of Size and ShapeDryden/Mardia(Kendall 1977): (Sometimes we are also interested in retaining scale information as well as shape):Size-and-shape is all the geometrical information that remains when location and rotational effects are filtered out from an object.LandmarksGoogle: Advanced Photoshop Pictures ContestLandmarks (Dryden & Mardia)• A landmark is a point of correspondence on each object that matches between and within populations.• An anatomical landmarkis a point assigned by an expert that corresponds between objects of study in a way meaningful in the context of the disciplinary context.• Mathematical landmarksare points located on an object according some mathematical or geometrical property of the figure.[Dryden & Mardia]LandmarksLandmarksThree landmarks along a line for simple shape comparisonCh. G. Small, The Statistical Theory of ShapeShape and RegistrationHomology: Corresponding (homologous) features in all skull images.Ch. G. Small, The Statistical Theory of ShapeShape and RegistrationShape and RegistrationShape and RegistrationSize of Configuration of Landmarks?Alternative: Baseline SizeA Shape Coordinate System==============================================• Widely used in shape analysis for planar data.• Most straightforward method for a newcomer to shape analysis.• Experienced shape analysts often use Bookstein coordinates in the first stages of an analysis.Bookstein CoordinatesMore General:Equivalence RelationshipsTwo objects are equivalent (x1 ~ x2) if they can be transformed into each other by the following transformation:define equivalence relationships: Properties:• Reflexivity: • Symmetry• TransitivitytxRx +=21*)(*ϕαmm≈mnnm≈→≈pmpnnm≈→≈∧≈tR