Software Project: 3D Shape Compression

1
Software Project 3D Shape Compression

• Spring 2006

This presentation is based on the pdf document
2
Introduction

• 3D Shape compression

3
Introduction

• 3D Shapes in Computer Science
• Graphic Library
• Shape Compression
• Mathematical (Algebraic) Tools

4
Overview

• 3D Shapes contain millions of primitives

424,376 triangles
6.8 million triangles
8.2 million triangles
5
Overview

• 3D Shape representation
• vertices (x,y,z) - floating precision
• faces (a,b,c) integer precision

6
Overview

• Compress reduce amount of data in a lossy way
• In our case reduce vertices data similar to jpeg
  technique
• Compute a Laplacian matrix
• Decompose matrix into a subset of significant
  eigenvectors
• Represent compressed vertices using eigenvectors
  and a set of coefficients

7
Overview

• Project suggested stepping stones
• Parse 3d shape file into 3D shape data
• Interface with the supplied graphic library
• Implement the compression algorithm
• Link whole project and test

8
3D Shape Representation

• 3D shapes are represented by vertices and faces
• A vertex holds the 3D coordinates of a point
  x,y,z.
• A face is a triangle and holds 3 indexes to 3
  different vertices a, b, c (starting at zero)

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Object File Format (off)

• OFF n m 0
• x1 y1 z1
• x2 y2 z2
• ...
• xn yn zn
• 3 a1 b1 c1
• 3 a2 b2 c2
• ...
• 3 am bm cm
• write a simple parser that reads off files

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Off example

• OFF 4 4 0
• -0.500000 -0.500000 -0.500000
• 0.500000 -0.500000 -0.500000
• -0.500000 -0.500000 0.500000
• -0.500000 0.500000 0.500000
• 3 0 1 3
• 3 0 2 3
• 3 2 1 3
• 3 0 1 2

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Graphic Library Interface

• Visualization and control of the 3D shapes.
  Functionality
• left button rotation
• right button translation
• middle button zooms
• 'q' quits program running
• p' reads a number (number of coeff to use)
  from the user(input) and calls compress
• 'r' reads a number (compression number k) from
  the user(input) and calls compute
• 's' saves the screen into an image

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Graphic Lib. Interface

• void startLib() - starts the graphic library.
• Once this function is called, your system steps
  into interactive mode.
• You should call function at the end of your
  main() function.

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Graphic Lib. Interface

• void charge_arrays(float vertexarr, int
  nvertex, int facearr, int nface)
• This function charges your model into the graphic
  library for display.
• The only functionality that this function should
  perform, is to fill the parameters sent to this
  function
• (vertexarr)3i xi, (vertexarr)3i1 yi,
  (vertexarr)3i2 zi
• (facearr)3i ai, (facearr)3i1 bi,
  (facearr)3i2 ci,

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Graphic Lib. Interface

• void compute(int nbasis) computes the
  compression coefficients.
• When the 'r' key is pressed, the user is prompted
  to enter the number of bases to use in this
  compression (compression number k).
• After the user enters a number, the system calls
  compute(int nbasis)
• Note you should call your algorithm
  implementation from this function call.
• At the end, you should print to the screen the
  algorithm duration in milliseconds

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Graphic Lib. Interface

• void compress(int ncoeff) computes the
  compressed vertices.
• When the p' key is pressed, the user is prompted
  to enter the number of coeff. to use in this
  compression
• After the user enters a number, the system calls
  compress(int ncoeff)
• If the user enters a non-valid number of
  coefficients (ncoeff lt 0 or ncoeff gt nbasis), no
  computation should take place and charge_arrays()
  should fill vertexarr with the original vertices
  (default).
• Otherwise, the new vertices should be charged in
  subsequent calls to charge_arrays() after
  compress() finished.

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Graphic Lib. Interface

• void myexit() - performs a clean exit.
• It is called upon pressing the 'q' key.
• You should implement this function to make a
  clean and valid exit of your program (most
  important releasing all allocated memory).

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Shape Compression

• Compression algorithm
• Build the Laplacian matrix based on shape
  connectivity (neighborhood between vertices).
• Decompose Laplacian matrix into first k
  significant eigenvectors.
• Project vertices on eigenvectors and obtain a set
  of compressing coefficients.

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Laplacian Matrix

• Assume a shape of n vertices where edges connect
  neighboring vertices
• Adjacency matrix A is defined
• Diagonal matrix D is defined s.t. (di is the
  degree of vi )

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Laplacian Matrix

• Laplacian matrix is L I DA

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Matrix Decomposition

• Goal given a symmetric matrix Anxn , find its k
  smallest eigenpairs (eigenvectors and
  eigenvalues).
• Power Method finds the dominant eigenvector
• Graham Schmidt Orthogonalization finds the first
  k eigenvectors

21
Power Method

• A method that finds the dominant eigenpair of a
  non-singular matrix A.
• Basic idea is that multiplication of any random
  vector with a high power of the matrix A, results
  in an approximation of the dominant eigenvector.

22
Power Method

• Given a symmetric matrix Anxn with ordered
  eigenvalues
• Any vector Xn can be written as a linear
  combination of n linearly independent eigen
  vectors V1,, Vn

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Power Method

• Thus
• Applying matrix powers
• Dividing by ?1 we get

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Power Method

• To get ?1
• And the corresponding eigenvector V1
• Therefore

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ConvergenceExample

• Convergence condition
• Example

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Matrix Decomposition

• Overview
• start with k random vectors and iteratively
  multiply them by the A (power method).
• at each iteration orthogonalize the k vectors
  using the Graham Schmidt

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Matrix Shifting

• Goal obtain the k vectors of the k smallest
  eigenvalues
• Preconditioning for symmetric matrices
• invert eigenvalue order (x-1)
• translate eigenvalues to be positive (A1I)

-1
A1I
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Graham Schmidt orthog.

• Start with k random vectors (X1,X2,Xk)
• At each iteration (AmX) orthogonalize each vector
  with its previous vectors

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Vertex Compression

• Goal given a set of eigenvectors of the k
  smallest eigenvalues of the Laplacian matrix L
  compress vertices
• Project vertices on eigenvectors and obtain a set
  of 3k coefficients.
• Coefficients together with adjacency matrix
  encode 3D shape

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Coefficients Computation

• Denote V1, V2,...,Vk the set of eigenvectors
• Denote X,Y and Z the vectors of the x,y and z
  coordinates

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Coefficients Computation

• Projection of vertices onto the set of
  eigenvectors bases

• Compressed vertices are obtained by
  multiplication of coefficients x vectors

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Implementation Requirements

• Power Method Implementation and Testing
• Simple Power Method
• Matrix Decomposition
• Compress Input/Output
• Error Handling and Robustness
• Coding and Compilation

33
Power Method Implementation and Testing

• Simple Power Method
• void dominant_eig(float matrix, int size, float
  precision, floateig_vec, float eig_val)
• dominant_eig_test.exe dim epsilon
• Matrix decomposition
• void matrix_decomp(float matrix, int size,
  float precision, float eig_vec, float
  eig_val, int k)
• matrix_decomp_test.exe dim k epsilon

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Compression Project

• compress.exe filename.off
• You can not assume that the file exists/opens
• You can assume that the syntax is valid
• handle all possible errors (memory allocation,
  bad calculation results etc.)

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Compression Project

• Your code should be compilable and executable on
  the UNIX machines in CS
• Your code must comply the ANSI C specification
  and should be compiled using the standard gcc
• Your code should compile with no errors together
  with the graphics library
• Code should be gracefully parted into files and
  functions.
• Code line length should not extend 80 characters.
  
• Code should be commented at critical points in
  the code and at function declarations.
• Programs should reside in different
  subdirectories /dominant_eig/ /matrix_decomp/
  /compress/