Computer Vision Seminar Shape Representation Yaron Berlinsky 21092021

Computer Vision Seminar Shape Representation Yaron Berlinsky 21/09/2021

Shape Representation - what for? 4 To reason about an entity we must first represent the entity. 21/09/2021

Shape Representation - what for? • Real world Image processing • Shape representation • Image understanding shape 21/09/2021 recognition 0110010001000000100111 0010100111101

Topics 4 Definitions 4 Attributes 4 Popular Stratagies 21/09/2021

Definitions 4 Object 4 Abstraction 4 Representation 21/09/2021

Object 4 An object is something that can be seen or touched, material thing (Oxford dictionary) 21/09/2021

Abstraction 4 Idea of quality separate from actual examples (Oxford dictionary) 21/09/2021 • ball • sphere

Representation 4 A way of symbolizing an object 21/09/2021

Attributes Of A Good Representation 4 sufficient 4 wide domain 4 unique 4 unambiguous 4 generative 4 stable 4 convenient 21/09/2021

Sufficient Is this representation sufficient enough? Depends on the application. . . 21/09/2021

Wide Domain 4 Able to represent many different classes of entities 4 E. g. numbers for elements in a queue 3 2 21/09/2021 1

Unique 4 every distinct member of its domain has a single distinct representation. Not unique : dog unique : collie pitbull 21/09/2021 dog cocker-spaniel

Unambiguous 4 An entity may have different representations but no two distinct entities may have a common representation. שלוש 3 III 21/09/2021 שנים 2 II

Generative 4 Capable of directly generating (recovering) the represented entity 1 8 2 Chain-code : 436476872832 21/09/2021 7 3 6 4 5

Stable 4 small perturbations ( )הפרעות do not induce large changes in the representation of the entity. 21/09/2021

Convenient 4 A representation may exhibit all the characteristics that we have discussed so far and yet not be convenient for a task. e. g. an assembly line robot who’s task is to filter out rectangels form other shapes might use chaine code representation. Yet for a police computerized camera that is supposed to compare faces in the crowd to a DB of suspects this is not enough. 21/09/2021

Popular Strategies 4 Volume based : – Describe the object volumetrically – Combination of primitive volumes commonly used, eg. cubes, tetrahedra, and discs – Provide more access to global relationships – Do not provide direct surface information about the object – Can represent only closed (boundaryless) surfaces e. g. Symmetric Axis Transform 21/09/2021

Popular Strategies 4 Surface based : – Surface of object represented by a single closed parametric grid. – Better suited for partial surfaces. E. g. Parametric Bicubic Patches Gaussian- Image Representations 21/09/2021

Popular Strategies 4 Others: – Chain code. – Distance vs. Angle. – Fourier Transform Moment – Generalized Cylinders – Visual Potential 21/09/2021

Chain Code 21/09/2021

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Attributes 4 wide domain 4 unique 4 unambiguous 2 generative - 2 D only stable - depends on tolerance 21/09/2021

Distance vs. Angle 21/09/2021

Distance vs. Angle 4 Find point of balance. 4 Measure distance to edges. 4 Plot the graph of distance vs. angle. Taking the dist. vs. angle one step further we get. . . 21/09/2021

Fourier Descriptor 21/09/2021

Attributes 4 wide domain 4 unique 4 unambiguous 4 generative stable - depends upon tolerance 21/09/2021

Moment 4 find center of mass 4 the moment: 21/09/2021

Moment 21/09/2021

Moments 4 since we get invariant values the moments are not affected by transition or rotation. 4 By normalizing the values, we can make the moment be indifferent to scaling 21/09/2021

Attributes 4 wide domain 8 unique - no 8 unambiguous - no 8 generative - no 4 stable - no 21/09/2021

Parametric Bicubic Patches 4 Split the object to simpler bicubic patches that can be represented by relatively simple mathematical equations. 21/09/2021

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Attributes 4 generative 4 stable 4 popular in computer graphics 4 Possibility of several equally acceptable bicubic approximations to any given surface makes it inappropriate for surface matching 21/09/2021

Symmetric Axis Transform • A. k. a Blum Transform, Medial Axis Transform. • Formally : the object is the logical union of all its maximal discs. 21/09/2021

Symmetric Axis Transform(cont. ) 4 Description : – Locus of centers a. k. a symmetric axis. – Radius at each point 4 May be thought of as grass fire spreading from the border inwards. 21/09/2021

Attributes 4 wide domain 4 unique 4 unambiguous 4 generative 8 not stable - small changes affect dramatically 4 Primarily used in biological applications 21/09/2021

Generalized Cylinders 4 Binford 1971 4 an extension of SAT. 4 Shape represented by an ordinary cylinder sweeping the cross-section along an arbitrary space curve (axis/spine) 21/09/2021

Generalized Cylinders 4 Decomposition of 3 D shape-description problems into lower-order problems 21/09/2021

Generalized Cylinders 4 A generalized cylinder is thus defined by 2 a cross section, 2 an axis 2 a sweeping rule. 21/09/2021

Generalized Cylinders 21/09/2021

Attributes 4 wide domain - depends on implementation (redundancy vs. wide domain ) 4 unique 4 generative 4 stability is doubtful 4 Decomposition of complex structures is difficult. 4 Used mainly for object recognition 4 been used in working systems. 21/09/2021

Gaussian Image 4 Surface normal vector information for any object can be mapped onto a unit sphere, called the Gaussian sphere. 4 Mapping is called the Gaussian image of the object. 21/09/2021

Extended Gaussian Image 4 The mapping is: Surface normals for each point of the object are placed so that their 4 tails lie at the center of the Gaussian sphere 4 heads lie on a point on the sphere appropriate to the particular surface orientation 21/09/2021

Extended Gaussian Image We can extend this process so that 4 a weight is assigned to each point on the Gaussian sphere equal to the area of the surface having the given normal 4 This mapping is called the extended Gaussian image (EGI). 4 Weights are represented by vectors parallel to the surface normals, with length equal to the weight 21/09/2021

EGI 21/09/2021

Attributes 4 wide domain 8 not unique 8 not unambiguous 8 not generative 4 stable - with limitations 4 invariant to translation and scaling 4 position independence 21/09/2021

Visual Potential 4 Aspect - topological structure of the singularities in a single view. 4 A graph in which each node represents an aspect of the object, and each edge the possibility of transiting from one aspect to another under motion of the observer [Turner 74]. 4 Represents in a concise way any visual experience an observer can obtain by looking at the object when traversing any orbit through space. 21/09/2021

Visual Potential 21/09/2021

Attributes 4 wide domain 4 unique 4 unambiguous 4 generative 7 not stable 21/09/2021

Summary 4 The above methods are deterministic; in practice, uncertainties in shape that result from the noise in measurement have to be considered [Ayache 88, Ikeuchi 88] 4 The above methods are based on geometric models, they are more appropriate for describing specific objects, particularly artificial ones with regular structures 21/09/2021

Summary 4 Symbolic models are more appropriate for natural objects that are better defined in terms of generic characteristics (e. g. . small, red, rough) than precise shape; useful for matching 21/09/2021

Questions? 21/09/2021

References 4 Books : – Nalwa “a guided tour of computer vision” – Gonzaless, “Image analysis and computer vision” – Jain 4 Web: – http: //web. mit. edu/manoli/ecimorph/www/code/MMorph. html – http: //www. cs. ubc. ca/~clee/cg/proj/index. html – http: //www. dai. ed. ac. uk/CVonline/LOCAL_COPIES/MARSHAL L/node 53. html#figuresame_EGI – http: //www. cs. uwa. edu. au/~cheng/index. html 21/09/2021