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3-D Computer Vision CSc 83020

3-D Computer Vision CSc 83020. Clustering methods and boundary representations. Image Segmentation. Generate clusters (regions) of pixels that correspond to meaningful entities. Use metrics of “closeness” between values. Use algorithms for combining “close” values.

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3-D Computer Vision CSc 83020

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  1. 3-D Computer Vision CSc 83020 Clustering methods and boundary representations 3-D Computer Vision CSc 83020 – Ioannis Stamos

  2. Image Segmentation • Generate clusters (regions) of pixels that correspond to meaningful entities. • Use metrics of “closeness” between values. • Use algorithms for combining “close” values. • Apply other constraints (connectivity). 3-D Computer Vision CSc 83020 – Ioannis Stamos

  3. Example (Forsyth & Ponce) 3-D Computer Vision CSc 83020 – Ioannis Stamos

  4. Simple Clustering Methods • Divisive clustering • Everything is a big cluster at beginning • Split recursively • Agglomerative clustering • Each data (pixel) is a cluster • Merge 3-D Computer Vision CSc 83020 – Ioannis Stamos

  5. Simple clustering 3-D Computer Vision CSc 83020 – Ioannis Stamos

  6. Simple clustering • What is a good inter-cluster distance? • How many clusters are there? 3-D Computer Vision CSc 83020 – Ioannis Stamos

  7. Simple clustering 3-D Computer Vision CSc 83020 – Ioannis Stamos

  8. Clustering by K-means • Assume that the number of clusters (k) is known. • Each cluster has a center Ci (i=1..k) • Each data-point is a vector xj (j=1..Number of pixels) • Examples: xj=[x-coord, y-coord, gray-value] or xj=[gray-value] or xj=[red-value, green-value, blue-value] • Assume that elements are close to center of clusters. • Minimize: 3-D Computer Vision CSc 83020 – Ioannis Stamos

  9. Clustering by K-means • Iterative algorithm: • Allocate each point to center of closest cluster (assuming centers are known) • Calculate centers of clusters (assuming allocations are known) • How do we start? 3-D Computer Vision CSc 83020 – Ioannis Stamos

  10. Clustering by K-means 3-D Computer Vision CSc 83020 – Ioannis Stamos

  11. Example (Forsyth & Ponce) 3-D Computer Vision CSc 83020 – Ioannis Stamos

  12. Example (Forsyth & Ponce) 3-D Computer Vision CSc 83020 – Ioannis Stamos

  13. Example (Forsyth & Ponce) 3-D Computer Vision CSc 83020 – Ioannis Stamos

  14. RANSAC • RANdom SAmple Consensus • Model fitting method • Line-fitting example • Fitting a line to a set of edges with 50% outliers • Least squares would fail • Solution: M-estimator or RANSAC 3-D Computer Vision CSc 83020 – Ioannis Stamos

  15. RANSAC (line-fitting example) • Two edges (wout normal) define a line. • General idea: • Pick two points. • Write the equation of the line. • Check how many other points are “close” to line. • If number of “close” points is above threshold, done • Otherwise, pick two new points. • Questions: • Which points to pick? • Complexity in worst case? 3-D Computer Vision CSc 83020 – Ioannis Stamos

  16. RANSAC (line-fitting example) 3-D Computer Vision CSc 83020 – Ioannis Stamos

  17. RANSAC (line-fitting example) How large should k (max. number of iterations) be? 3-D Computer Vision CSc 83020 – Ioannis Stamos

  18. RANSAC (line-fitting example) How large should k (max. number of iterations) be? • Assume that w is the probability of picking a “correct” point (i.e. a point on the line). • Since we are picking n (=2 for lines) points, 1-wn is the probability of picking n “wrong” points. • If we iterate k times we want the probability of failure to be small: i.e. (1-wn)k = z => k = log(z)/log(1-wn) • If z=0.1 and w=0.5 then k=8 (n=2) • If z=0.01 and w=0.5 then k=16 (n=2) • If z=0.001 and w=0.1 then k = 687 (n=2) • How is the formula affected by n? 3-D Computer Vision CSc 83020 – Ioannis Stamos

  19. RANSAC (Conclusions) • When can this method be successful? • Can we detect circles? • In that case how many points do you need to fit a circle? • Can we detect other shapes? 3-D Computer Vision CSc 83020 – Ioannis Stamos

  20. Boundary representation of regions 3-D Computer Vision CSc 83020 – Ioannis Stamos

  21. Representation of 2-D Geometric Structures • To MATCH image boundary/region with MODEL • boundary/region, they must represented in the same • manner. • Boundary Representation • Snakes – Extraction of arbitrary contours from image. • Region Representation 3-D Computer Vision CSc 83020 – Ioannis Stamos

  22. Representation Issues • Compact: Easy to Store & Match. • Easy to manipulate & compute properties. • Captures Object/Model shape. • Computationally efficient. 3-D Computer Vision CSc 83020 – Ioannis Stamos

  23. Boundary Representation Polylines: concatenation of line segments. Breakpoint Matching on the basis of: # of line segments lengths of line segments angle between consecutive segments 3-D Computer Vision CSc 83020 – Ioannis Stamos

  24. Running Least Squares Method ei Move along the boundary At each point find line that fits previous points (Least Squares) Compute the fit error E=Sum(ei) using previous points If E exceeds threshold, declare breakpoint and start a new running line fit. 3-D Computer Vision CSc 83020 – Ioannis Stamos

  25. Approximating Curves with Polylines • Draw Straight line between end-points of curve • For every curve point find distance to line. • If distance is less than tolerance level for all points, Exit • Else, pick point that is farthest away and use as breakpoint. • Introduce new segments. • Recursively apply algorithm to new segments. 3-D Computer Vision CSc 83020 – Ioannis Stamos

  26. Ψ-s Curve y Ψ 2π 6 5 π 4 3 ψ s 1 2 x s y 4 5 3 6 1 2 x 3-D Computer Vision CSc 83020 – Ioannis Stamos

  27. Ψ-s Curve y Ψ 2π 6 5 π 4 3 ψ s 1 2 x s y 4 Ψ-s is periodic (2π) Horizontal section in Ψ-s curve => straight line in the Image Non horizontal section in Ψ-s => arc in Image 5 3 6 1 2 x 3-D Computer Vision CSc 83020 – Ioannis Stamos

  28. Slope-Density Function y H(Ψ) HISTOGRAM Lines ψ s Ψ x π 2π Arcs 3-D Computer Vision CSc 83020 – Ioannis Stamos

  29. Slope-Density Function y H(Ψ) HISTOGRAM Lines ψ s Ψ x π 2π Arcs H(Ψ) shifts as objects rotates. H(Ψ) wraps around 3-D Computer Vision CSc 83020 – Ioannis Stamos

  30. y Fourier Descriptors Find: Ψ(s) Define: Φ(s)= Ψ(s)-(2πs)/P P: Perimeter. 2π: Period of Φ(s) ψ s x Φ(s) is a Continuous, Periodic function. Fourier Series for Periodic Functions: Fourier Coefficients: Φk’s capture shape information Match shapes by matching Φk’s Use finite number of Φk’s

  31. Fourier Descriptors Input Shape Reconstruction Power Spectrum # of coefficients Φk Note: Reconstructed shapes are often not closed since only a finite # of Φk’s are used.

  32. B-Splines • Piecewise continuous polynomials used to INTERPOLATE • between Data Points. • Smooth, Flexible, Accurate. x2 Spline X(s) x1 s=2 x0 s=1 Data Point xi s s=0 x N BASIS FUNCTIONS We want to find X(s) from points xi Cubic Polynomials are popular: COEFFICIENTS

  33. B-Splines Bi(s) has limited support (4 spans) i+2 s: i-2 i-1 i i+1 Each span (i->i+1) has only 4 non-zero Basis Functions: Bi-1(s), Bi(s), Bi+1(s), Bi+2(s) Bi+1(s) Bi(s) Bi+2(s) Bi-1(s) s: i i+1 t=0 t t=1

  34. B-Splines 4/6 Bi(s) Bi+1(s) Bi-1(s) Bi+2(s) 1/6 i i+1 t=0 t t=1 Cubic Polynomials 3-D Computer Vision CSc 83020 – Ioannis Stamos

  35. B-Splines 4/6 Bi(s) Bi+1(s) Bi-1(s) Bi+2(s) 1/6 i i+1 t=0 t t=1 Cubic Polynomials 3-D Computer Vision CSc 83020 – Ioannis Stamos

  36. B-Splines If we compute v0, … vN => continuous representation for the curve. 3-D Computer Vision CSc 83020 – Ioannis Stamos

  37. B-Splines We have our N+1 data points: And in matrix form: We can solve for the vi’s from the xi’s Boundary condition for closed curves: v0=vN, v1=vN+1.

  38. B-Splines So, for any i we can find: xi(t) t xi xi+1 t=0 t=1 Note: Local support. Spline passes through all data points xi. B-Spline demo: http://www.doc.ic.ac.uk/~dfg/AndysSplineTutorial/BSplines.html

  39. Snakes Elastic band of arbitrary shape. Located near the image contour. 3-D Computer Vision CSc 83020 – Ioannis Stamos

  40. Snakes – Implementation Discrete version: sum over the points. How can we minimize e? Greedy minimization: For each point compute the best move over a small area. 3-D Computer Vision CSc 83020 – Ioannis Stamos

  41. Snakes – Implementation Discrete version: sum over the points. How can we minimize e? Greedy minimization: For each point compute the best move over a small area. Set βi=0 for points of high curvature (corners) 3-D Computer Vision CSc 83020 – Ioannis Stamos

  42. Snakes – Implementation Discrete version: sum over the points. How can we minimize e? Greedy minimization: For each point compute the best move over a small area. Set βi=0 for points of high curvature (corners) Stop when a user-specified fraction of points does not move. 3-D Computer Vision CSc 83020 – Ioannis Stamos

  43. Synthetic Results Real Experiment Snakes demo: http://www.markschulze.net/snakes/ 3-D Computer Vision CSc 83020 – Ioannis Stamos

  44. Region Representation Spatial Occupancy Array 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 • Easy to implement. • Large Storage area. • Can apply set operations (unite/intersect). • Expensive for matching! 3-D Computer Vision CSc 83020 – Ioannis Stamos

  45. Quad Trees Efficient encoding of Spatial Occupancy Array using Resolution Pyramids. Black: Fully Occupied. White: Empty. Gray: Partially Occupied. 3-D Computer Vision CSc 83020 – Ioannis Stamos

  46. Quad Tree Level 0 Level 1 Level 2 Level 3 Quad Tree Generation: Start with level 0. If Black or White, Terminate. Else declare Gray node & expand with four sons. For each son repeat above step. NW NE SW SE 3-D Computer Vision CSc 83020 – Ioannis Stamos

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