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A Multi-Resolution Technique for Comparing Images Using the Hausdorff Distance Daniel P. Huttenlocher and William J. Rucklidge Nov 7 2000 Presented by Chang Ha Lee

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Introduction Registration methods Correlation and sequential methods Fourier methods Point mapping Elastic model-based matching

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The Hausdorff Distance Given two finite point sets A = {a 1,…,a p } and B = {b 1,…,b q } H(A, B) = max(h(A, B), h(B, A)) where

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With transformations A: a set of image points B: a set of model points transformations: translations, scales t = (t x, t y, s x, s y ) w = (w x, w y ) => t(w) = (s x w x +t x, s y w y +t y ) f(t) = H(A, t(B))

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With transformations Forward distance f B (t) = h(t(B), A) a hypothesize reverse distance f A (t) = h(A, t(B)) a test method

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Computing Hausdorff distance Voronoi surface d(x) = {(x, d(x))|x R 2 }

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Computing Hausdorff distance The forward distance for a transformation t = (t x, t y, s x, s y )

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Comparing Portion of Shapes Partial distance The partial bidirectional Hausdorff distance H LK (A,t(B))=max(h L (A,t(B)), h K (t(B),A))

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A Multi-Resolution Approach Claim 1. 0 b x x max, 0 b y y max for all b=(b x,b y ) B, t = (t x, t y, s x, s y ),

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A Multi-Resolution Approach If H LK (A, t(B)) = >, then any transformation t with (t,t) < - cannot have H LK (A, t(B)) A rectilinear region(cell) The center of this cell If then no transformation t R have H LK (A, t(B))

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A Multi-Resolution Approach 1. Start with a rectilinear region(cell) which contains all transformations 2. For each cell, If Mark this cell as interesting 3. Make a new list of cells that cover all interesting cells. 4. Repeat 2,3 until the cell size becomes small enough

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The Hausdorff distance for grid points A = {a 1,…,a p } and B = {b 1,…,b q } where each point a A, b B has integer coordinates The set A is represented by a binary array A[k,l] where the k,l-th entry is nonzero when the point (k,l) A The distance transform of the image set A D[x,y] is zero when A[k,l] is nonzero, Other locations of D[x,y] specify the distance to the nearest nonzero point

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Rasterizing transformation space Translations An accuracy of one pixel Scales b=(b x,b y ) B, 0 b x x max, 0 b y y max x-scale: an accuracy of 1/x max y-scale: an accuracy of 1/y max Lower limits: s xmin, s ymin Ratio limits: s x /s y > a max or s y /s x > a max Transformations can be represented by ( i x, i y, j x, j y ) represents the transformation (i x, i y, j x /x max, j y /y max )

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Rasterizing transformation space Restrictions where A[k,l], 0 k m a, 0 l n a

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Rasterizing transformation space Forward distance h K (t(B),A) Bidirectional distance

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Reverse distances in cluttered images When the model is considerably smaller than the image Compute a partial reverse distance only for image points near the current position of the model

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Increasing the efficiency of the computation Early rejection K = f 1 q where q = |B| If the number of probe values greater than the threshold exceeds q-K, then stop for the current cell Early Acceptance Accept false positive and no false negative A fraction s, 0~~
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~~

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Increasing the efficiency of the computation Skipping forward Relies on the order of cell scanning Assume that cells are scanned in the x-scale order. D +x [x,y]: the distance in the increasing x direction to the nearest location where D[x,y] Computing D +x [x,y]: Scan right-to-left along each row of D[x,y] F B [i x,i y,j x,j y ],…,FB[i x,i y,j x + -1,j y ] must be greater than

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Examples Camera images Edge detection

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Examples Engineering drawing Find circles

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Conclusion A multi-resolution method for searching possible transformations of a model with respect to an image Problem domain Two dimensional images which are taken of an object in the 3-dimensional world Engineering drawings

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