Projects:SphericalWaveletsInITK

Description

For a general description of how we are using spherical wavelets for shape analysis, see 3D Shape Analysis Using Spherical Wavelets

This page outlines the steps we will take to code the Spherical Wavelet transform in ITK.

• The best place to learn about the spherical wavelet transform that we will implement is in this paper, that also has pseudo-code.

Spherical Wavelets: Texture Processing (1995) Peter Schröder, Wim Sweldens

http://citeseer.ist.psu.edu/oder95spherical.html

In this paper, it is shown how to do decompose a scalar signal defined on a spherical mesh into spherical wavelet coefficients (analysis step, also called forward transform), and vice-versa (synthesis step, also called inverse transform).

• With the goal of analysizing the scalar function defined on a spherical topological surface, the spherical parametrization is needed as the first step. Then, we can deem the function to be defined on a sphere. (If the function is a vector one, we can apply the analysis component by component. Thus the scalar function is used for simplicity in the future.)

• For our case of SHAPE representation, the function we are going to analize are the x, y and z coordinates of the original triangulated surface.

• The numerical computation of spherical wavelet analysis requires the scalar function to be defined on a special triangulated sphere, the one generated by a recursive subdivision process. Also for a better performance, the base shape of the triangulation is selected as an icosahedron. The scalar function defined on the original spherical mesh is then interpolated onto the mesh generated by the recursive subdivision using bi-linear interpolation.

• The forward wavelet transformation is carried out on the configuration of the previous step and a set of wavelet coefficients are obtained, called Gamma coefficients. The backward transformation can recover the Gamma coefficients into an approximation of the original scalar function.

the In our implementation, we will generalize this code in 2 ways:

• 1. The signal can be decomposed on any surface mesh (not necessarily a sphere) that has a spherical parametrization. This input surface mesh will be called the ParametrizedMesh Internally in our code, this surface mesh will be retriangulated to have a particular triangulation structure needed for the spherical wavelet analysis (built from recursively subdividing an icosahedron). This retriangulated mesh will be called the TemplateMesh (that will be available to the user by a get method).
  2. the signal can be N-dimensional (so for example, a 3D signal is a vector field defined on the spherical mesh).

• We will have 2 types of itk classes, inspired by the ITK Spherical Harmonics classes:
  1. an itk object that is a container for the template mesh, the signal defined on the mesh and the spherical wavelet coefficients. This object will be called itkSphericalWaveletObject and its subclass will be the itkSurfaceWaveletObject:
     • If the base mesh is a sphere, our object will be called a SphericalWaveletObject. There will be no need to input a ParametrizedMesh since it is built internally by the object by recursively subdividing an icosahedron. The subdivided icosahedron will be the TemplateMesh.
     • If the base mesh is an arbitrary surface, our object will be called a SurfaceWaveletObject and there will be a need to input a ParametrizedMesh, as in the ITK Spherical Harmonics case, that represents the surface. The object will internally derive a TemplateMesh from the ParametrizedMesh. (Note: we have not determined yet if this input will be two separate inputs, the surface mesh and its spherical parametrization, or a single input that consists of the surface mesh with the spherical parametrization somehow saved inside the mesh object. This will be determined and made consistent for both the ITK Spherical Harmonics and Spherical Wavelets).
  2. 2 filters that take the itkSphericalWaveletObject as input and process it to calculate either the coefficients from the signal (itkSphericalWaveletsSignalToCoefficients) or the signal from the coefficients (itkSphericalWaveletsCoefficientsToSignal). The output of both filters is an itkSphericalWaveletObject.

For a preliminary API, see ITK Spherical Wavelets API

Implementation

• Before the spherical wavelet transformation, the scalar function is interpolated onto a sphere which is generated from recursively subdividing an icosahedron and which has the least number of vertexes more than that of the original mesh.

• The whole process is done in one ITK filter, itkSWaveletFilter inherited from itkMeshSource class.

• The processes going within the class are:
  1. The hierarchy of the recursive subdivision is built from the icosahedron. The parameters needed for wavelet decomposition are calculated and stored in the internally. The level of the subdivision is chosen so that the finest sphere in the subdivision coincides with the one in the previous interpolation step.

• 1. The scalar function is set using the API SetScalarFunction().

• 1. The wavelet coefficients are calculated from the finest level and downward. It's and implementation of the lifting scheme of the second generation of wavelet transformatiom.

• 1. The coefficients of all levels are stored in one structure of vector and can be exported to a file if needed.

• 1. If needed, the reconstruction of function can be carried out from the coefficients. The difference between the reconstructed function and the original function is around [math]10^{-16}[/math] .

Current Status

• We have created the API with Martin Styner (UNC), using as a motivation the ITK Spherical Harmonics ITK classes
• We have started to code the template mesh creation using an icosahedron subdivision with a specific ordering of the vertices of the mesh and a record of the hierarchical structure between vertices needed for the transform. We are using as a starting point the code in Code/Algorithms/RegularSphereMeshSource

Next Steps

• We will code the Base class itkSphericalWaveletsObject (that will include the template mesh creation code we have been working on)
• We will code the filters
• Finally we will code the Base class itkSurfaceWaveletsObject that has more complex template creation

Members

• Xavier LeFaucheur (Gatech)
• Yi Gao (Gatech)
• Delphine Nain (Gatech)
• John Melonakos (Gatech)
• Jim Miller (GE)
• Luis Ibanez (Kitware)
• Martin Styner (UNC)

Links

• 3D Shape Analysis Using Spherical Wavelets
• ITK Spherical Wavelets API