WO2001052194A1 - Utilisation de reseaux normaux dans l'imagerie 3d - Google Patents

Utilisation de reseaux normaux dans l'imagerie 3d Download PDF

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Publication number
WO2001052194A1
WO2001052194A1 PCT/US2001/001086 US0101086W WO0152194A1 WO 2001052194 A1 WO2001052194 A1 WO 2001052194A1 US 0101086 W US0101086 W US 0101086W WO 0152194 A1 WO0152194 A1 WO 0152194A1
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WO
WIPO (PCT)
Prior art keywords
normal
polyline
mesh
vertices
point
Prior art date
Application number
PCT/US2001/001086
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English (en)
Inventor
Peter Schroeder
Igor Guskov
Original Assignee
California Institute Of Technology
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Priority claimed from US09/820,383 external-priority patent/US7129947B1/en
Application filed by California Institute Of Technology filed Critical California Institute Of Technology
Priority to AU2001229416A priority Critical patent/AU2001229416A1/en
Publication of WO2001052194A1 publication Critical patent/WO2001052194A1/fr

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Classifications

    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T9/00—Image coding
    • G06T9/001—Model-based coding, e.g. wire frame

Definitions

  • Three-dimensional imaging often requires three scalar functions such as x, y, and z coordinates. These coordinates define parameters of the surface so that the surface can be visualized as a three dimensional image.
  • the present application teaches a new kind of way of describing a three dimensional surface.
  • the description is called a "normal mesh".
  • the mesh has information which defines information relative to a special tangent plane.
  • the normal mesh is defined as a normal offset from a coarser version.
  • the mesh can be stored with a single float per vertex, thus reducing the amount of information which needs to be stored.
  • Figure 1 shows how a smooth surface of three dimensions can be described in terms of single variable scalars
  • Figure 2 shows a polyline
  • Figure 3 shows construction of a normal polyline
  • Figure 4A shows a flowchart of forming a polyline
  • Figure 4B shows a flowchart of overall operation of compressing the surface
  • Figures 5A-5F show the various stages of compressing a sample surface, here a molecule
  • Figure 6 shows a based domain vertext repositioning
  • Figure 7 shows a piercing operation
  • Figure 8 shows a face splitting operation to obtain additional surface detail
  • Figure 9 shows a result of applying a naive piercing procedure.
  • Figure 1 shows how a smooth surface 100 can be locally described by single variable scalar height functions, hi, h 2 , h 3 , h over a tangent plane 110.
  • the three dimensional information for the smooth surface 100 is contained only in this single dimension h: the height over the tangent plane. In practice, this approximation only works infinitesimally. However, it may provide interesting information.
  • the normal meshes .which are described herein require only a single scalar value per vertex. This is may be done using a multiresolution and local frame. A hierarchical representation provides that all detail coefficients expressed in these frames are scalar.
  • the parameter may be a normal component, for example. In the context of compression, for example, this allows parameter information to be predicted and confines residual error to the normal direction.
  • a curve in a plane can be defined by a pair of parametric functions.
  • S(t) (x(t), y(t)) with t e [0,1] .
  • polylines may be used to approximate curves.
  • Figure 2 shows removing one point (Su+1,2t+1)) in a
  • polyline multiresolution and recording the difference with the midpoint m On the left a general polyline where the detail has both a normal and a tangential component. On the right is a normal polyline where the detail is purely normal.
  • a polyline is "normal" if a multiresolution structure exists where every removed point forms an Isosceles triangle with its neighbors. Then there is zero parameter information and the polyline can be represented with one scalar per point, namely the normal component of the associated detail.
  • Figure 3 shows construction of a normal polyline. We start with the coarsest level and each time check where the normal to the midpoint crosses the curve. For simplicity only the indices of the S j/k points are shown and only certain segments are subdivided.
  • the polyline (0,0)-(2,l)-(3,3)-(l,l) ' -(0,l) is determined by its endpoints and three scalars, the heights of the Isosceles triangles .
  • a base point and normal estimate can be produced using the well known 4 point rule. Any predictor which only depends on the coarser level is allowed. Irregular schemes described in Daubechies, I., Guskov, I., and Sweldens, W. Regularity of Irregular Subdivision. Constr. Approx. 15 (1999), 381-426. can also be used. Levels may be built by downsampling every other point, or using any other ordering. Describing this in terms of further generality, a polyline is normal if a removal order of the points exist such that each removed point lies in the normal direction from a base point, where the normal direction and base point only depend on the remaining points . Hence a normal polyline may be completely determined by a single scalar component per vertex.
  • Normal polylines are closely related to certain well known fractal curves such as the Koch Snowflake.
  • the normal coefficients can be thought of as a piecewise linear wavelet transform of the original curve. Because the tangential components are always zero, there may be half as many wavelet coefficients as the original scalar coefficients.
  • the wavelets have their usual decorrelation properties.
  • K is an abstract simplicial complex which contains all the topological, i.e., ' adjacency information.
  • the complex K is a set of subsets of ⁇ 1,...,N ⁇ . These subsets come in three types: vertices ⁇ i ⁇ , edges ⁇ i,j ⁇ , and faces ⁇ i,j,k ⁇ . Two vertices i and j are neighbors if
  • V(i) ⁇ jlmid ⁇ i,j ⁇ ⁇ E ⁇ .
  • a mesh M is normal in case a sequence of vertex removals exists so that each removed vertex lies on a line defined by a base point and normal direction which only depends on the remaining vertices.
  • a normal mesh can be described by a small base domain and one scalar coefficient per vertex.
  • a mesh in general is not normal, just as a curve is in general not normal.
  • the present application therefore uses a special kind of mesh, called a semi-regular mesh.
  • the semi-regular mesh has a connectivity which is formed by successive quadrasection of coarse base domain faces.
  • the operation is shown in Figure 4 at 440, the operation begins with a coarsest level or base domain. If there are no new vertices, the operation is complete at 410. For each new vertex determined at 405, a new base point is computed and a normal direction are found at 415. A determination is made of where the line defined by the base point and normal intersects the surface 420. 425 determines how many intersection points exist. If only one point exists, it is accepted at 430. In the surface situation, there might be no intersection point or many intersection points, not all of which are correct.
  • V(i) is the 1-ring neighborhood of the vertex i
  • an initial set of curves is defined, to connect the vertices of the base domain with a net of non intersecting curves on the different levels of the mesh simplification hierarchy. This can be -done using the MAPS parameterization.
  • MAPS uses polar maps to build a bijection between a 1-ring and its retriangulation after the center vertex is removed. The concatenation of these maps is a bijective mapping between different levels ( j, j) in the hierarchy.
  • the desired curves include the image of the base domain edges under this mapping. Because of the bijection, no intersection can occur. Note that the curves start and finish at a vertex of the base domain.
  • the repositioning is typically done on some intermediate level j .
  • Boundary conditions are assigned using arc length parameterization.
  • Parameter coordinates are iteratively computed for each level j vertex inside the shaded region.
  • the point qi may be replaced with any level point from P j in the shaded region.
  • the new q ⁇ ⁇ may be the point of Pj that in the parameter domain is closest to the center of the disk.
  • the curves can be redrawn by taking the inverse mapping of straight lines from the new point in the parameter plane. This procedure may be iterated. It may alternatively suffice to cycle once through all base domain vertices.
  • User controlled repositioning may allow the user to replace the center vertex with any P j point in the shaded region. Parameterization may be used to recompute the curves from that point.
  • Figure 5C shows the repositioned vertices. Notice how some of them, like the rightmost ones have moved considerably.
  • Figure 6 shows base domain vertex repositioning with the left showing original patches around qi, middle: parameter domain, right: repositioned q ⁇ and new patch boundaries. This is replaced with the vertex whose parameter coordinate are the closest to the center. The inverse mapping (right) is used to find the new position qi' and the new curves. 4. Fixing the global edges: The image of the global edges on the finest level will later be the patch boundaries of the normal mesh. For this reason, the smoothness of the associated curves be improved at the finest level. 465 defines fixing global edges using a procedure similar to Eck, M., DeRose, T., Duchamp, T., Hoppe, H., Lounsbery, M. , and Stuetzle, W.
  • FIG. 7 shows the canonical step for a new vertex of the semi-regular mesh to find its position on the original mesh.
  • every edge of level j generates a new vertex on level j+1.
  • compute a base point is computed using interpolating Butterfly subdivision as well as an approximation of the normal. This defines a straight line. This line may have multiple or no intersection points with the original surface.
  • the new vertex q may lie halfway along the edge ⁇ a, c ⁇ with incident triangles ⁇ a, c, b ⁇ and ⁇ c, a, d ⁇ , see Figure 7, Let the two incident patches form the region R.
  • Figure 8 shows a Face split: Quadrisection in the parameter plane (left) leads to three new curves within the triangular surface region (right) .
  • the aperture parameter K of the piercing procedure provides control
  • Figure 9 shows 4 levels of naive piercing for the torus starting from a 102 vertex base mesh. Clearly, there are several regions with flipped and self- intersecting triangles. The error is about 20 times larger than the true normal mesh.
  • Normal meshes have numerous applications. The following are examples. Compression Usually a wavelet transform of a standard mesh has three components which need to be quantized and encoded. Information theory tells us that the more non uniform the distribution of the coefficients the lower the first order entropy. Having 2/3 of the coefficients exactly zero will further reduce the bit budget. From an implementation viewpoint, the normal mesh coefficients may be connected to the best known scalar wavelet image compression code.
  • Texturing Normal semi-regular meshes are very smooth inside patches, across global edges, and around global vertices even when the base domain is exceedingly coarse, cf. the skull model.
  • the implied parameterizations are highly suitable for all types of mapping applications.

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  • Engineering & Computer Science (AREA)
  • Multimedia (AREA)
  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Theoretical Computer Science (AREA)
  • Image Generation (AREA)
  • Processing Or Creating Images (AREA)

Abstract

L'invention concerne un ensemble spécial de réseaux normaux (100) dans lequel les erreurs (Fig. 1) et les résidus vont également se trouver dans une direction qui minimise les erreurs de codage. Ces réseaux normaux (100) peuvent être utilisés pour créer une représentation d'une surface 3D.
PCT/US2001/001086 2000-01-14 2001-01-12 Utilisation de reseaux normaux dans l'imagerie 3d WO2001052194A1 (fr)

Priority Applications (1)

Application Number Priority Date Filing Date Title
AU2001229416A AU2001229416A1 (en) 2000-01-14 2001-01-12 Use of normal meshes in three-dimensional imaging

Applications Claiming Priority (4)

Application Number Priority Date Filing Date Title
US17636900P 2000-01-14 2000-01-14
US60/176,369 2000-01-14
US09/820,383 2000-07-26
US09/820,383 US7129947B1 (en) 2000-07-26 2000-07-26 Use of normal meshes in three-dimensional imaging

Publications (1)

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WO2001052194A1 true WO2001052194A1 (fr) 2001-07-19

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Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US5555356A (en) * 1992-10-29 1996-09-10 International Business Machines Corporation System and method for generating a trimmed parametric surface for display on a graphic display device
US6046744A (en) * 1996-01-11 2000-04-04 Microsoft Corporation Selective refinement of progressive meshes
US6108006A (en) * 1997-04-03 2000-08-22 Microsoft Corporation Method and system for view-dependent refinement of progressive meshes

Patent Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US5555356A (en) * 1992-10-29 1996-09-10 International Business Machines Corporation System and method for generating a trimmed parametric surface for display on a graphic display device
US6046744A (en) * 1996-01-11 2000-04-04 Microsoft Corporation Selective refinement of progressive meshes
US6108006A (en) * 1997-04-03 2000-08-22 Microsoft Corporation Method and system for view-dependent refinement of progressive meshes

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Publication number Publication date
AU2001229416A1 (en) 2001-07-24

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