1. How to tell the differences between a Cat and a Dog Masoud Alipour (malipour@ipm.ir) Ali Farhadi (farhadi@ipm.ir) IPM – Scientific Computing Center Vision Group Institute for Studies in Theoretical Physics and Mathematics Tehran-Iran

2. Outlines - Linear Predictive Coding Coefficients - Linear Predictive Coding Coefficients - Surfaces Associated to the LPC Coefficients. - Surfaces Associated to the LPC Coefficients. - Application of Computational Geometry. - Application of Computational Geometry. - Detection via Geometric Characteristics of - Detection via Geometric Characteristics of LPC Surface. LPC Surface. - Results - Results

3. Making Surfaces – LPC Coefficients. zoom

5. LPC Surfaces : 1) By Computing the LPC coefficients for each window W of the image, we obtain a set of data in R 20. 1) By Computing the LPC coefficients for each window W of the image, we obtain a set of data in R 20. 2) Each window is identified by its upper left corner index (x,y). 2) Each window is identified by its upper left corner index (x,y). 3) Project the data for each window to R by taking the average of the marked neighbors. 3) Project the data for each window to R by taking the average of the marked neighbors.

6. Projection to R: -Take average of 6,10,11,15 coefficients. -Take average of 6,10,11,15 coefficients. Origin 10 6 11 15

7. LPC Surface (continued): 4) Denote this average by F(x,y). 4) Denote this average by F(x,y). 5) Sliding window with overlaps defines a function on a grid. 5) Sliding window with overlaps defines a function on a grid. LPC Surface = Graph of F(x,y) LPC Surface = Graph of F(x,y)

8. (i p, j p) (i p’, j p’) ZpZpZpZp Z p’ (i p, j p, z p ) (i p’,j p’,z p’ )

9. Surfaces

13. Quantifying the Oscillations where k1 and k2 are the minimum and maximum (principal) surface curvature, respectively. Strain Energy (total curvature) Bending energy function (roughness measure)

14. Quantifying Oscillations (continued): Gaussian Curvature of a surface z=F(x,y) Gaussian Curvature of a surface z=F(x,y)

15. High curvature Low curvature Images from Caltech Multires Lab

16. Discrete Curvature M a triangulated surface (not necessarily smooth) M a triangulated surface (not necessarily smooth) 0 0 0 0 0 0 0 2 0 1 Discrete Curvature at a vertex

17. Euler Number (characteristic)

18. Discrete Curvature Discrete Curvature satisfies some basic theorems of Differential Geometry cast in the discrete framework. Discrete Curvature satisfies some basic theorems of Differential Geometry cast in the discrete framework. 1. Gauss Bonnet Theorem is valid 1. Gauss Bonnet Theorem is valid 2. Every closed surface has triangulation of constant curvature. 2. Every closed surface has triangulation of constant curvature.

19. Triangulations 1)Uniform Triangulation 2)Delaunay Triangulation

20. Delaunay Triangulations Empty circle property Delaunay Triangulation and diagram Delaunay Triangulation and Voronoi diagram

21. Triangulated Surface

22. Counting the Number of Incident Edges. 1. Mapping center of gravity of each Triangle on the plate z=0. 2.Generating a uniform grid on z=o; 3. Centroid Matrix

23. Number of triangles per area Thus a triangle is assigned to a square grid if the orthogonal projection of the centroid of the triangle is located in that windows. Count the number of triangles which is assigned to the each window. Hence a matrix D is obtained.

24. 4 6 324 513 3336

25. 51716099688660942280 57355663675346123638 76927317511851651102 67954351573939923787 22302057213522531755 LPC coefficients. Centroid Matrix

26. Comparing for detection The differences between the centroid matrices for cats and dogs are obvious. The differences between the centroid matrices for cats and dogs are obvious. Simply by using the means of matrices we can differentiate between cat and dog matrices. Simply by using the means of matrices we can differentiate between cat and dog matrices. Better statistical invariants are also applicable to the matrix D. For example,  - σ 2, etc. Better statistical invariants are also applicable to the matrix D. For example,  - σ 2, etc.

27. Results 9337928676928754 20625192224871320426

28. Conclusion - This is not a image matching algorithm - This is not a shape matching algorithm - Objects are discriminated via texture analysis

29. ? Questions