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VOXEL-BASED SURFACE FLATTENING

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VOXEL-BASED SURFACE FLATTENING Nahum Kiryati* Dept. of Electrical Engineering Systems Tel Aviv University * Joint work with Ruth Grossmann and Ron Kimmel. – PowerPoint PPT presentation

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Title: VOXEL-BASED SURFACE FLATTENING


1
VOXEL-BASED SURFACE FLATTENING
  • Nahum Kiryati
  • Dept. of Electrical Engineering Systems
  • Tel Aviv University

Joint work with Ruth Grossmann and Ron Kimmel.
2
SURFACE FLATTENING
Transform a surface from 3D space to 2D plane.
  • Some distortion is usually unavoidable
  • Need to define the most important property to
    preserve (area, angles,...)
  • Use cuts (interruptions) to selectively reduce
    distortion

Peters Projection area accuarte
Cuts reduce distortion
3
Applications Medical
  • Cortical surface flattening for visualization,
    registration, etc.
  • Virtual colonoscopy

Gray-white matter interface
Courtesy of Gabriele Lohmann, MPG Leipzig
4
Applications Computer Graphics
Texture Mapping Projecting a planar pattern onto
a 3D surface
  • Flatten the surface
  • Map the 2D pattern onto the flattened surface
  • Map the flattened surface back to 3D

Texture Mapping
From www.okino.com
5
Some previous works
  • E.L. Schwartz, A. Shaw and E. Wolfson, A
    Numerical Solution to the Generalized Mapmakers
    Problem Flattening Nonconvex Polyhedral
    Surfaces, IEEE Trans. Pattern Anal. Mach.
    Intell., Vol. 11, pp. 1005-1008, 1989.
  • C. Bennis, J.M. Vezien and G. Iglesias,
    Piecewise Surface Flattening for Non-Distorted
    Texture Mapping, Computer Graphics, Vol. 25, pp.
    237-247, 1991.
  • H.A. Drury, D.C. Van Essen, C.A. Anderson, C.W.
    Lee, T.A. Coogan and J.W. Lewis, Computerized
    Mappings of the Cerebral Cortex A
    Multiresolution Flattening Method and
    Surface-Based Coordinate System, J. of Cognitive
    Neuroscience, Vol. 8, pp. 1-28, 1996.
  • S. Angenent, S. Haker, A. Tanenbaum and R.
    Kikinis, Conformal Geometry and Brain
    Flattening, Proc. 2nd Intl. Conf. on Medical
    Image Computing and Computer Assisted
    Intervention (MICCAI), Cambridge, England, pp.
    269-278, 1999.
  • A.M. Dale, B. Fischl and M.I. Sereno, Cortical
    Surface-Based Analysis. II Inflation, Flattening
    and a Surface-Based Coordinate System,
    NeuroImage, Vol. 9, pp. 195-207, 1999.
  • S. Haker, S. Angenent, A. Tanenbaum, R. Kikinis,
    G. Sapiro and M. Halle, Conformal Surface
    Parameterization for Texture Mapping, IEEE
    Trans. on Visualization and Computer Graphics,
    Vol. 6, No. 2, 2000.

6
New Method Unique Features
  • Operates on voxel data Triangulation not
    necessary
  • Optimal (global preservation of geodesic
    distances)
  • Solid mathematical basis. Analytic solution (in
    principle)
  • Computationally efficient

Voxel data
From www.chez.com
7
Our Approach to Surface Flattening
  • Mark N (very many) points on the 3D surface.
  • Estimate the geodesic distances between each
    point and all others (how?)
  • Optimally map the N points onto the 2D plane,
    such that the 2D Euclidean interpoint distances
    will be as close as possible to the corresponding
    3D geodesic distances (how?)
  • Regard the N points as control points, and map
    the rest by interpolation.

8
Crucial problem 1
Estimate the minimal (geodesic) distance between
points on a 3D surface.
Kiryati Szekely, 1993
Accurate and efficient solution for surface
represented by voxels.
  • Represent the surface as a weighted graph
  • The weight of an edge depends on the link type
  • Minimal geodesic distance shortest path in
    graph
  • Graph is sparse very fast algorithm

direct, minor diagonal, major diagonal
Question what weights should we use?
9
Crucial problem 1
Estimate the minimal (geodesic) distance between
points on a 3D surface.
Kiryati Székely, 1993
Accurate and efficient solution for surface
represented by voxels.
  • Represent the surface as a weighted graph
  • The weight of an edge depends on the link type
  • Minimal geodesic distance shortest path in
    graph
  • Graph is sparse very fast algorithm

direct, minor diagonal, major diagonal
Question what weights should we use?
Wrong answer
10
Bias in length estimation (2D example)
  • Continuous curve (length L)
    Digital curve
  • Digital curve Chain code
    (8-connected)
  • Estimate length


Unbiased, minimum MSE length estimation
Unbiased and minimum MSE for straight lines.
Unbiased for lines ? wonderful for curves
(errors cancel out).
11
Kiryati Kübler, 1992
Unbiased, minimum MSE length estimation for 3D
curves
  • RMS error for straight lines 2.88
  • Straight lines are the worst case
  • (much better with curves)

Use the weights (0.9016, 1.289, 1.615) in the
algorithm for finding shortest paths (geodesics)
and minimal distances on digitized surfaces.
Verwer, Beckers Smeulders
12
Crucial problem 1
Estimate the minimal distance between points on a
digitized 3D surface.
  • Solved (Kiryati Székely, 1993)


13
Reminder
Our Approach to Surface Flattening
Mark N (very many) points on the 3D
surface. Estimate the geodesic distances
between each point and all others
(how?) Optimally map the N points onto the 2D
plane, such that the 2D Euclidean interpoint
distances will be as close as possible to the
corresponding 3D geodesic distances
(how?) Regard the N points as control points,
and map the rest by interpolation.
14
Crucial problem 2
Given N points on a surface in 3D, map them onto
the 2D plane, such that the 2D Euclidean
interpoint distances will be as close as possible
to the corresponding geodesic distances on the 3D
surface.

15
Crucial problem 2
Given N points on a surface in 3D, map them onto
the 2D plane, such that the 2D Euclidean
interpoint distances will be as close as possible
to the corresponding geodesic distances on the 3D
surface.

Special case of Multi-Dimensional Scaling (MDS)
16
Goal
Place N points in the 2D plane, such that the
sets and are as similar as
possible.

Objective functions
Name Definition Algorithm
Stress iterative
SStress iterative
Strain something else analytic solution!
The analytic solution for minimal strain is
called classical scaling (Borg Groenen, 1997).
17
Classical Scaling
X Coordinate Matrix (N x 2) matrix with the
(x,y) coordinates of N points in the plane1
(unknown) B Scalar Product Matrix
(N x N) matrix
(unknown) d Squared Distances Matrix
(N x N) matrix containing the squared
Euclidean distances between points in the
plane (unknown)
Suppose that d is known. It can be shown that
where and 1 is a vector of 1s.
Having B, X can be found via eigendecomposition

holds exactly two (positive) non-zero
eigenvalues. So
where and are (2 x 2) and (N x 2)
submatrices.
1Column centered coordinate system origin at the
point-set centroid.
18
Classical Scaling (contd)
But we dont have the Squared Distances Matrix
d, since interpoint distances in the plane are
unknown. We do have the interpoint
geodesic distances on the 3D surface.
Algorithm
  • Create D (Squared Geodesic Distance Matrix)
  • Compute the estimated Scalar Product Matrix
  • Compute the eigendecomposition of up to rank
    2
  • The flattened Coordinate Matrix is

  • Classical scaling minimizes the error metric
    (strain).
  • Zero for a true Scalar Product Matrix
    obtained from a true Squared Distance Matrix d
    containing
  • squared planar Euclidean distances.
  • Partial eigendecomposition (up to rank 2) is
    easy and cheap (power method).

19
Reminder
Our Approach to Surface Flattening
Mark N (very many) points on the 3D
surface. Estimate the geodesic distances
between each point and all others. Optimally
map the N points onto the 2D plane, such that the
2D Euclidean interpoint distances will be as
close as possible to the corresponding 3D
geodesic distances. Regard the N points as
control points, and map the rest by interpolation.
We used N 1000 control points, and mapped the
rest using radial basis function
interpolation. Any reasonable interpolation
method can be used.
20
Unfolded
Curled planar surface (synthetic)
Euclidean distance (2D) vs. geodesic distance (3D)
21
Unfolded
Curled planar surface (synthetic)
Can you figure out why errors over small
distances are larger than errors over large
distances in this case?
Euclidean distance (2D) vs. geodesic distance (3D)
22
Unfolded
Curled planar surface (synthetic)
Can you figure out why errors over small
distances are larger than errors over large
distances in this case?
On this surface, many short geodesic paths are
straight, or nearly straight. Straight lines are
the worst case for the geodesic distance
estimator. Over long distances, paths are not
straight, and errors cancel out.
Euclidean distance (2D) vs. geodesic distance (3D)
23
A depth image of a human face.
Euclidean distance (2D) vs. geodesic distance (3D)
24
A depth image of a human face.
Euclidean distance (2D) vs. geodesic distance (3D)
The flattened surface is not shown, because the
depth image has no texture.
25
A depth image of a human face.
Euclidean distance (2D) vs. geodesic distance (3D)
Errors are larger than in the previous example,
because this surface is not developable it
cannot be flattened without some distortion.
26
Textures
27
Texture Mapping via Surface Flattening
28
Texture Mapping via Surface Flattening
29
Highlights
  • New surface flattening algorithm
  • Operates on voxel data. Triangulation is not
    necessary.
  • Distance measurements on 3D surface classical
    scaling.
  • Global distance preservation
  • Essentially analytic solution Global optimum
    guaranteed
  • Surface flattening, texture mapping

30
Appeared
31
Companion paper
  • Flattening with triangulated data
  • Geodesic distance measurements via Fast
    Marching on Triangulated Domains classical
    scaling
  • State of the art texture mapping quality
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