Invariants for Discrete Structures – An Extension of Haar Integrals over Transformation Groups to Dirac Delta Functions Kapitel 5c Hans Burkhardt 1 , Marco Reisert 1 , and Hongdong Li 2 1 University of Freiburg, Computer Science Department, Germany 2 National ICT Australia (NICTA), Australian National University, Canberra ACT, Australia Berechnung von Invarianten für diskrete Objekte http://lmb.informatik.uni-freiburg.de/ H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 1 H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 2 Summary Introduction 1. Introduction • Increased interest in 3D models and 3D sensors 2. Invariants for continuous objects induce a growing need to support e.g. the 3. Invariants for discrete objects automatic search in such databases • Invariants for polygons • As the description of 3D objects is not canonical • 3D-meshes � use invariants for their description • Discrimination performance and completeness 4. Experiments: Object classification in a Tangram database 5. Conclusions H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 3 H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 4
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Invariant integration over Euclidean Use as kernel functions monomials of pixels of local group support and integrate over the Euclidean motion: For (cyclic) image translation and rotation: 3 1 5 2 3 ( ) f X m m m m m ( )[ , ] [ , ] 00 01 0 1 10 10 g X i j X k l cos sin k i t 0 l sin cos j t 1 all indices to be understood modulo the image dimensions. 2 N M 1 [ ]( ) ( ) A f X f g X d dt dt 1 0 2 NM t 0 t 0 0 0 1 H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 5 H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 6 Pollen examples Monomial Hasel Birke Erle Deterministic integral Monte-Carlo-Integration over the planar Euclidean motion Gräser Roggen Beifuß + 33 further species (not relevant for allergies) H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 7 H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 8
Eibe/Taxus Gänseblümchen/daisy pollen grain Integrate over Euclidean Motion H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 9 H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 10 Classification Results using 3D LSM Data The Five Platonic Solids (leave-one-out Classification) Correct Wrong classifications Octahedron Icosahedron.gif Dodecahedron Artemisia: 1 -> Compositae , 1 -> Platanus 13 Alnus: - 15 Alnus viridis: 12 - Betula: 13 2 -> Plantago Corylus: 1 -> Alnus 13 Gramineae/Poaceae: - 15 3 -> Fagus , 1 -> Tilia Secale: 11 Allergolocial irrelevant*: 2 -> Gramineae 282 2.6% Total: 97.4% Hexahedron Tetrahedron * Acer, Carpinus, Chenopodium, Compositae, Cruciferae, Fagus, Quercus, A platonic solid is a polyhedron all of whose faces are congruent regular polygons, and where Aesculus, Juglans, Fraxinus, Plantago, Platanus, Rumex, Populus, Salix, Taxus, the same number of faces meet at every vertex. The best know example is a cube (or Tilia, Ulmus, Urtica hexahedron ) whose faces are six congruent squares. H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 11 H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 12
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � �� �� � � � � � � Invariants on discrete Structures Extension of Haar-Integrals to (topologically equivalent structures) Discrete Structures Chose proper kernel functions on distributions ( ) T f f g dg X X � � G 1. Chose kernel functions which are different from zero only at the vertices and which act only on neighborhoods of degree m . 2. As each vertex can be visited in an arbitrary permutation of all points by a continuous Euclidean motion the integral is changed into a invariant sum over all vertices with Euclidean-invariant local discrete features !! 3. Use principle of rigidity to reach completeness: use a basis of features which are locally rigid and which can be pieced together in a unique way to the global object (see invariants for triangle!). Describe discrete structures with Dirac delta functions! H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 13 H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 14 Invariants for discrete objects Extension of Haar-Integrals to Discrete Structures 1. For a discrete object � and a kernel function f ( � ) it is possible to construct an invariant feature T [ f ]( � ) by integrating f ( g � ) over the transformation group g � G . 2. The kernel function is properly designed, such that it delivers a value dependent on the discrete features of a local neighborhood, when a vertex of the object moved by the continuous Euclidean motion g hits the origin and has one specific orientation. � � � � : ( ) ( , ) ( ) T f f g d dg x h g g x g x g x d x dg 3. Let us assume that our discrete object is different from zero only at its vertices. A rotation i i and translation invariant local discrete kernel function h takes care for the algebraic i G X G X relations to the neighboring vertices and we can write: h ( , x ) dg h ( , x ) i i ( , ) ( ) f h x x x i i G i i i where �� is the set of vertices and x i the vector representing vertex i . Intuitive result: get global Euclidean invariants by summation 4. In order to get finite values from the distributions it is necessary to introduce under the over discrete local Euclidean invariants h ( � , x i ) ! Haar integral another integration over the spatial domain X . 5. By choosing an arbitrary integration path in the continuous group G we can visit each vertex in an arbitrary order the integral is transformed into a sum over all local discrete Remember: The delta function has the functions allowing all possible permutations of the contributions of the vertices. f x ( ) ( x a dx ) f a ( ) following selection property: H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 15 H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 16
� � � � � � � � � � � � � � � � Choose as local Euclidean invariants distances Euclidean Invariants for Polygons We assume e.g. to have given a polygon with 10 vertices, e.g. of vertex i and its k-th righthand neighbours: d x x i k , i i k x 9 x 8 x 7 The elements: x 6 x 5 x 4 , , d d d x ,1 ,2 ,3 i i i x 3 i � 3 x 2 form a basis for a polygon, because they uniquely d define a polygon (up to a ,3 i x 0 x 1 mirror-polygon) ! d x i ,2 2 i � Principle of rigidity 0 1 1 2.5 2.5 1 1 3.5 3.5 0 x i � � 0 0 2.5 2.5 3.5 3.5 4.5 4.5 5.5 5.5 x d x i � 1 i ,1 i H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 17 H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 18 With the first two distances we get two initial configurations. Then all further vertices will be unique. The two initial configurations give two possible polygons, where one is just d ,3 i the mirror image of the other along the first edge as its axis. d i ,2 d i ,1 d 1,3 Given two edges d 1,1 and d 2,1 . Then d the third vertex is uniquely defined d d d 2,2 1,1 3,1 2,1 by the set: d 2,1 d , , d d d 1,1 ,1 ,2 ,3 i i i Because there is a unique intersection point of three circles. This means, that a whole polygon can be uniquely generated by this basis elements iteratively. H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 19 H. Burkhardt, Institut für Informatik, Universität Freiburg ME-I, Kap. 5c 20
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