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1 Yago Diez and J. Antoni Sellarès Universitat de Girona Colored Point Set Matching under rigid motion in 3D

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2 Overview - Introduction - Matching Algorithm - Enumeration - Testing - Lossless Filtering Preprocessing Step - Conclusions and Future Work

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3 Overview - Introduction - Matching Algorithm - Enumeration - Testing - Lossless Filtering Preprocessing Step - Conclusions and Future Work

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4 Substructure search in proteins Motivation Given a Protein, find all the occurrences of a secondary structure: Proteins are formed by amino acid molecules Only twenty types of standard amino acids. Every amino acid relates to a unique alpha-carbon.

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5 Problem formulation : Given two sets of colored points in space A and B, |A| < |B|, find all the subsets B of B that can be expressed as rigid motions of A. Introduction We want: the points to approximately match (fuzzy nature of real data). the colors to coincide. (*) Rigid motion: composition of a translation and a rotation.

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6 A B Given two colored points sets A and B, n=|A|, m=|B|, n < m, and a real positive number ε, determine all the rigid motions τ for which there exists a subset B of B, |B|=|A|, such that: d b (τ(A),B) ε (Bottleneck distance) Colored Point Set Matching Problem Definition

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7 Overview - Introduction - Matching Algorithm - Enumeration - Testing - Lossless Filtering Preprocessing Step - Conclusions and Future Work

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8 Matching Algorithm -Two main parts: Enumeration More difficult than in 2D Numerical problems Testing Bipartite matching algorithm The data structures used in 2D can be adapted. Noisy Matching Algorithm

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9 Matching Algorithm IDEA: Generate all possible motions τ that may bring set A near some B Enumeration: (2D) LEMMA: If there exist a matching of A to B then there is one that maps two points b i, b j to the circles of center a k, a l and radius ε (boundary of noise regions).

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10 Matching Algorithm Enumeration: 2D We rule out all those pairs of point whose colors do not coincide. 1.- For every quadruple, generate all possible rigid motions Partition of [0,2 π [ 2.- For every representative rigid motion, execute Testing algorithm.

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11 Matching Algorithm IDEA: Generate all possible motions τ that may bring set A near some B Enumeration: 3D LEMMA: If there exist a matching of A to B then there is one that maps three points b i, b j b k to the spheres of center a p a q, a r and radius ε (boundary of noise regions).

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12 Matching Algorithm Enumeration: 3D Generate: - Volume described by the fourth vertex of a teterahedron when the other three vertices move on the surfaces of three spheres of the same radius. - Algebraic volume. - Numerical problems.

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16 Matching Algorithm Enumeration: 3D In order to know when two points may be matched, intesect the resulting surfaces with spheres.

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18 Matching Algorithm Enumeration: 3D We obtain cells in the space of parameters (cube of side 2π )

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21 Overview - Introduction - Matching Algorithm - Enumeration - Testing - Lossless Filtering Preprocessing Step - Conclusions and Future Work

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22 Matching Algorithm For every motion τ representative of an equivalence class, find a matching of cardinality n between τ(A) and S. Testing (bi`partite matching algorithm) A set of calls to Neighbor operation corresponds to one range search operation in a skip octree Neighbor ( D(T), q ) Delete ( D(T), s ) Corresponds to a deletion operation in a skip octree. Amortized cost of Neighbor, Delete: t(n) Under adequate assumptions t(n) = log n

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23 Matching Algorithm Arrangement size: O( n 5 m 5 λ q (nm) ) Total cost (testing every cell): O( n 6 m 5 λ q (nm) t(n,n) ) Computational Cost

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24 Overview - Introduction - Matching Algorithm - Enumeration - Testing - Lossless Filtering Preprocessing Step - Conclusions and Future Work

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25 Our Approach The main goal is to transform the problem into a series of smaller instances. We will use a conservative strategy to discard, cheaply and at an early stage, those subsets of B where no match may happen. Then we will execute the matching algorithm that we have already described

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26 Candidate Zone Determination What geometric parameters, invariant for rigid motion, do we consider ? - number of Colored Points, - histogram of colors, - maximum and minimum distance between disks of the same color, - … There cannot be any subset B of B that approximately matches A fully contained in the four top-left squares, because A contains twelve colored Points and the squares only six.

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27 Candidate Zone Determination (3D)

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28 Candidate Zone Determination (2D) Candidate Zone Search Algorithm a b b c Three search functions needed for every type of zone according to the current node: -Search type a zones. -Search type b zones. -Search type c zones.

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29 Candidate Zone Determination (3D) Candidate Zone Search Algorithm Four search functions needed for every type of zone according to the current node: - Type a zones -> Inside the node. - Type b zones -> Overlapping two nodes. - Type a zones -> Overlapping four nodes. - Type a zones -> Overlapping eight nodes. The search begins at the root and continues until nodes of size s are reached. Early discards will rule out of the search bigger subsets of B than later ones.

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30 Overview - Introduction - Matching Algorithm - Enumeration - Testing - Lossless Filtering Preprocessing Step - Conclusions and Future Work

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31 Our Approach: Conclusions CONCLUSIONS - Uses a conservative pruning technique. - Is implementable. - There exist the possibility of adapting it to specific problems. -Is parallelizable. -FUTURE WORK - Approximate algorithm -

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32 QUESTIONS?

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33 Yago Diez and J. Antoni Sellarès Universitat de Girona Colored Point Set Matching under rigid motion in 3D

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34 Matching Algorithm Exact Enumeration Approximate

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35 Candidate Zone Determination Initial step 1. Determine an adequate square bounding box of A. 2 s (size s) 2. Calculate associated geometric information.

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36 Candidate Zone Determination

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37 Candidate Zone Determination Candidate zone determination algorithm complexity: O(max{m log m, c(t - s)}) - c: number of candidate zones - t (root quadtree size) and s depend on the size of the bounding boxes of A and B. - In practice we expect t-s to be close to constant and c in O(m).

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38 Constellation recognition problem Motivation Given a constellation of stars: determine whether it exists in the night sky (fixed magnification): Stars can be seen as disks with radii determined by their brightness (discrete range).

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39 Candidate Zone Determination Calculate quadtree of B with geometric parameters

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40 Candidate Zone Determination - Searchs first step: 3 – Candidate Zone Search Algorithm Sites = 550 Sites = 173 Sites = 113 Sites = 131 Sites = Target number of sites = 25 - Launch search1? yes (in four sons) - Launch search2? yes (all possible couples) - Launch search3? yes (possible quartet)

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41 Candidate Zone Determination - Searchs first step: 3 – Candidate Zone Search Algorithm Sites = 550 Sites = 173 Sites = 113 Sites = 131 Sites = Target number of sites = 25 - Launch search1? yes (in three sons) - Launch search2? yes (all possible couples) - Launch search3? yes (possible quartet)

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42 Example Consider: A B Find:

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43 Candidate Zone Determination

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44 Candidate Zone Determination Execution Tests \A\\B\NUMBER OF CANDIDATE ZONES TOTAL TIME (seconds)

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45 Given two ball sets A and B, n=|A|, m=|B|, n < m, and a real number ε, determine all the isometric transformations τ for which there exists a subset S of B such that: d b (τ(A),S ) ε Noisy Ball Matching Problem Definition

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46 Efrat et al. Matching Algorithm All endpoints of these angle intervals partition the interval [0, 2 [ in O(n 2 ) subintervals, so that for all angles in one interval the same points of B are mapped into the same neighborhoods of points of A. All these relationships are represented as edges in a bipartite graph whose two sides of nodes are A and B. If there is an angle for which the corresponding graph has a perfect matching, then the associated motion approximately matches A and B.

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47 Efrat et al. Matching Algorithm To find all perfect matchings, traverse the subintervals from left to right: -For angle = 0, find the maximum matching. If it is a perfect matching, keep the associated motion. -For each one of the remaining O(n 2 ) subintervals, check if its corresponding graph together with the previous matching contains an augmenting path. If we obtain a perfect matching, keep the associated motion. -Finding an augmenting path, using a layered graph, takes O(n log n) time.

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48 Efrat et al. Matching Algorithm For a query point q, returns a point in D(B) whose distance to q is at most ε, if it exists. Neighbor(D(B),q) Delete(D(B),s) Deletes the point s from D(B). Cost of Neighbor and Delete: O(log m), m=|B|. Difficult to implement. To build the layered graph, a data structure D(B) with two main operations is used:

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49 Efrat et al. Matching Algorithm For each one of the O(n 4 ) 4-tuples: a k, a l, b i, b j there are O(n 2 ) subintervals where to find a perfect matching in O(n log n) time. Total cost: O(n 4 ) O(n 2 ) O(n log n) = O(n 7 log n). Computational Cost

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50 Candidate Zone Determination - Searchs first step: 3 – Candidate Zone Search Algorithm Sites = 550 Sites = 173 Sites = 113 Sites = 131 Sites = Target number of sites = 25 - Launch search1? yes (in two sons) - Launch search2? yes (three possible couples) - Launch search3? yes (possible quartet)

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51 Candidate Zone Determination Sites = 550 Sites = 173 Sites = 113 Sites = 131Sites = Example with geometric parameter: number of disks

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52 Candidate Zone Determination Sites = 550 Sites = 173 Sites = 113 Sites = 131Sites = Example with geometric parameter: number of disks

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53 Introduction Let A, B be two disk sets of the same cardinality. A radius preserving bijective mapping f : A B maps each disk D(a, r) to a distinct and unique disk f(D(a,r))= D(b,s) so that r = s. Let F be the set of all radius preserving bijective mappings between A and B. The Bottleneck Distance between A and B is is defined as: d b (A, B) = min f F max D(a,r) A d(D(a,r), f(D(a,r))). |F| may severely diminish when the disks in A and B have a high number of different radii.

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