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Representing a Solid

Representing a Solid. Computer Vision #9. Classification of representation. Surface based representation represented as a collection of surfaces curvature primal sketch extended gaussian image aspect graph b-rep (winged-edge) well-tessellated surface Function based representation

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Representing a Solid

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  1. Representing a Solid Computer Vision #9

  2. Classification of representation • Surface based representation • represented as a collection of surfaces • curvature primal sketch • extended gaussian image • aspect graph • b-rep (winged-edge) • well-tessellated surface • Function based representation • represented as a function and its parameters • generalized cylinder • superquadric • symmetry seeking • spherical attribute image • Volumetric based representation • represented as a collection of primitive solids • constructive solid geometry • occupancy graph • oct-tree

  3. Represent a solid using a collection of surfaces Surface based representation • which surfaces • collection of visible surfaces -- easy to use • collection of all surfaces -- easy to maintain • what is a reasonable definition • orientation discontinuity -- if range data is available, one of the most common definition • Gaussian curvature and mean curvature Represent a solid using a collection of surfaces • color (brightness) -- view dependent only

  4. Oshima-shirai system • View dependent surface based representation • orientation discontinuity R7 AA(+) A 4(+) R6 RD R3 A8(+) A5(+) A9(+) AD(+) RC RB AB(+) R5 A1(+) R8 Model R1

  5. Model-driven vs. Data-driven

  6. Aspect representation • How many different representations are necessary? • Aspect -- topologically equivalent class of appearances C AD D ABD A ACD AB CD AC B BC A BCD B D BD The aspect graph for a tetrahedron. Nodes are of three types representing whether one, two,or three faces are visible. Arcs connect a pair of nodes when some arbitrarily small change in viewing direction suffices to change the set of visible faces from the faces visible at one node to those visible at the other node without going through any intermediate set of visible faces.

  7. Extracting possible aspects Face Label F1 F2 F3 F4 F5 F6 F7 F8 1 1 1 0 0 0 0 0 Visible Invisible

  8. Aspect generation

  9. VisionAlgorithmCompiler

  10. Spin Images • Given a set of oriented points (3D point with surface normal) Vote according to its radial and elevation coordinates 2 1 1 1

  11. Characteristics of Spin-Image Different point has different spin-image

  12. Recognition using Spin-Images

  13. body f3 f2 f1 fn <vantage> e3 e2 e1 en v3 v2 v1 vn PCW(E) PVert(E) NCCW(E) Nface(E) Pface(E) NVert(E) NCW(E) Winged edge representation • Represent topological relationship in a compact manner NVert(E) NVert(E) • VANTAGE • topological information • edge --- Winged edge representation • geometric information • face --- face equation • vertex --- vertex coordinates PCCW(E)

  14. Function-based representation represent a 3D object by a function z x y 1. easy to store 2. difficult to handle r z

  15. Function based representation 1. Classification of representation 2. Superquadric 3. Generalized cylinder 4. Symmetry seeking 5. Spherical attribute image

  16. Superquadric representation • Generalization of ellipsoid Ellipsoid SQ

  17. Shape representation using Superquadric

  18. input scene Segmentation result grasp execution grasp plan Superquadric representation of a rock

  19. Rock Sampling

  20. generalized cylinder traralelium sweep Generalized Cylinder “the sweep volume” of a two-dimensional set moved along a three-space curve <axis> 3D space curve <cross section> S A generalized cylinder and some cross-sectional coordinate systems A possibly “pathological” situation. Cross sections may be simply described as circles centered on the axis, but then their intersection makes volume calculations (for instance) less straightforward.

  21. Warped Skewed Layered GC axis Transformation rule Cross sections Cross section planes Straight Torridal Right Circular Open Polygonal Homogeneous Bilinear r b R(s) Linear s Uniform C(t) n Straight Homogeneous Generalized Cylinders 1. a(s) is a linear func 2. n-b planes are parallel 3. All cross sections have the same shape but may vary in size.

  22. Human Arm Forearm Hand Modular representation by GC This diagram illustrates the organization of shape information in a 3-Dmodel description. Each box corresponds to a 3-D model, with its model axis on the left side of the box and the arrangement of its component axes on the right. In addition, some component axes have 3-D models associated with them, as indicated by the way the boxes overlap. The relative arrangement of each model’s component axes, however, is showr improperly, since it should be in an object-centered system rather than the viewer-centered projection used here (a more correct 3-D model is given by the table shown in Figure 5-5c). The important characteristics of this type of organization are: (1) Each 3-D model is a self-centered unit of shape information and has a limited complexity; (2) information appears in shape contexts appropriate for recognition (the disposition of a finger is most stable when specified relative to the hand that contains it); and (3) the representation can be manipulated flexibly. This approach limits the representation’s scope, however, since it is only useful for shapes that have well-defined 3-D model decompositions.

  23. Hierarchical Representation by GC

  24. Extracting GC • Initial light-strip image • initial axis estimation • preliminary center and axis estimation • Corn with smoothed radius function • completed analysis

  25. ACRONYM system USERS Data Structure LIBRARY HI-LEVEL MOOELER CONTEXT GRAPH modelary module (user → GC) OBJECT GRAPH GRAPHICS MODULE prediction module PREDICTOR & PLANNER OBSERVABILITY GRAPH MATCHER interpretation module INTERPRETATION GRAPH description module (image → GC) SURFACE MAPPER PICTURE GRAPH PIX PIX EDGE MAPPER

  26. Back constraints

  27. Example 7 < WING-WIDTH < 12 1000 < M < 12000 7 < WING-WIDTH < 10.5677531 2199 < M < 3322 MATCH -- L1011!!! 2356 < M < 2489

  28. Interpretation example Edge image (PIX) GC description Interpretation graph

  29. Symmetry seeking representation x s • Internal Energy • energy to maintain the axis position • energy to maintain circular shape • External Energy • gradient SSR is a shape such that it minimizes the total energy E E = internal energy + external energy

  30. Creating an image potential function. Result of applying image operations to the squash image of Fig.3.(a)-(d)σ increases progressively. Darkness indicates magnitude of local gradient of the Gaussian blurred image. Each processed image has been resealed to span the available intensity range.

  31. Reconstruction of a squash. (a) Squash image. Selected frames from an animation sequence are shown: (b) Initial state of the 3D model. (C) Intermediate shape during reconstruction. (d) Final reconstructed model. (e)-(f) Model rotating rigidly in space. The 3D model is rendered as a shaded shell.

  32. Reconstruction of a still life scene with symmetry-seeking deformable models. (a) Image of the scene. (b) Initial user-specified configurations of the 3D models. Two rendered views of the reconstructed still life: (c) Frontal view of the 3D models; (d) side view.

  33. Symmetry-Seeking Model

  34. Circular mapping (2D) 2D shape l eequal-length segments O Circular mapping

  35. Matching (2D) rotation similar shapes? contours circular representations

  36. Deformable Surface Mapping Regularization Each Mech has roughly equal area Each mesh has the same topology

  37. Semi Regular Mesh (3D) • Deform a spherical surface • internal energy • smoothness • external energy • data points • features • Regularize the deformed surface (to have roughly equal area)

  38. Spherical Attribute Image (SAI) Gaussian Curvature Deformed Surface SAI

  39. Matching Example

  40. Volume based representation • Represent a solid as a collection of primitive solids • spatial occupancy enumeration • overlapping sphere • requires many spheres or voxels to represent to represent a relatively simple smooth solid

  41. Node with descendants Node representing an empty region Node representing a full region Oct tree representation Oct tree encoding -- extension into three dimensions of quad tree encoding Quad-tree 2 1 3 4 5 7 8 9 10 6 11 12 17 14 15 16 18 19 3 13 23 25 20 21 22 24 3 1 4 5 2 9 10 6 13 16 17 23 18 7 8 11 22 19 12 14 15 24 25 20 21 6 7 2 3 7 3 2 3 5 1 Oct-tree 0 1

  42. Constructive solid geometry representation <CSG rep> :: <primitive solid> | move <CSG Rep> by <Motion params> <CSG Rep><operation><CSG Rep> leaf node: either primitive or motion params other node: Boolean operations (UNION, DIF, INT) • Input to a CAD system • boundary-evaluation function converts CSG to boundary representation such as those based on winged-edge representation

  43. G Cell decomposition representation • quasi-disjoint (do not share volume) • Representation is arbitrary

  44. 2-1/2D representation 2D image Marr’s school Object-centered representation 3D representation GC Viewer-centered representation Interpretation fusion shading Line drawing Binocular stereo motion texture 3D feature extraction (shape-from-x) bottom-up

  45. Task 3D short-term representation Deformable template 2-1/2D representation aspect 3D long term model snake motion lib stereo range finder bin stereo photometric stereo EGI shad Generalized cylinder Super quadrics 2D image Short-term representation template Task-oriented school Task → Representation → Module → Image

  46. Bin Picking Object on other objects Parellel-jaw gripper Exact model Geometric model representation Stable features Region-based segmentation

  47. rock on sand claim-shell gripper no a priori model superquadric representation imperfect features Axis Snake segmentation Rock Sampling

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