geometric (bio-) modeling and visualization
TRANSCRIPT
Geometric (Bio-) Modeling and Visualizationhttp://www.cs.utexas.edu/~bajaj/c3s84R10/
Lecture 19b
Maps IIb: Image/Surface - Filtering
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Center for Computational Visualization http://www.ices.utexas.edu/CCVInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin May 2008
Bilateral & Anisotropic FilteringBilateral filtering
where and are parameters
and f(.) is the image intensity value.
Anisotropic diffusion filtering
where a stands for the diffusion tensor
determined by local curvature estimation.W. Jiang, M. Baker, Q. Wu, C. Bajaj, W. Chiu, Journal of Structural
Biology, 144, 5,(2003), Pages 114-122
C. Bajaj, G. Xu, ACM Transactions on Graphics, (2003),22(1), pp. 4- 32.
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2007
Image Denoising (Filtering): A Variational Approach
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2007
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2007
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2007
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2007
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2007
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2007
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2007
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2007
Derivative Relationships
H(x) is the Mean Surface Curvature
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2007
Anisotropic Diffusion Filtering
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2007
be principal curvature directions of point
Shape Modulation via Diffusion Tensor
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2007
Functions on Surface: Texture
Initial data After 1 iteration After 4 iterations
Center for Computational Visualization http://www.ices.utexas.edu/CCVInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin May 2008
where is a decreasing function is the angle between the central pixel and its surrounding pixels.
• Gradient vector diffusion:- smoothing the vector fields
- diffusion to flat regions
• For smooth data:- zeroes of the gradient vector field
- simple, easy to implement
• For noisy data:- Gradient vector diffusion
- higher time complexity but robust to noise
Filtering Gradient Map Critical Points
minimum maximum saddle (0) (3) (1, 2)
Y.Zu, C. Bajaj IEEE Transactions on Image Processin, 2005, 14, 9, 1324-1337
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2007
• Surface: S: z = f(x,y) – explicit form• F(x,y,z) = 0 – implicit form
Principal Curvatures - I
•Tangent Space Basis:
•Normal:
•1st Fundamental Form: Measures the squared length of a tangent vector u by
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2007
Principal Curvatures - II•2nd Fundamental Form: Measures the change of unit normal along a unit tangent vector u via
•Normal change along a direction u: Curvature of the curve on the surface passing through the point p and having tangent u
•Principal Curvatures: Maximum and Minimum Normal Curvatures with the corresponding directions as principal curvature directions.
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2007
•Principal Curvatures are the directions in the tangent space that optimize the quantity
Principal Curvatures - III
•They are the Eigenvalues/Eigenvectors of the matrix i.e. The Shape Operator S =
•S is also known as Weingarten Matrix W corresponding to the Weingarten Map
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2007
Level Set of Implicit Fn. (3D)
• Gradient:
• Normal:
• Level Set:
• Hessian:
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2007
Curvature
• Jacobian of Normal:
• Nonzero Eigenvalue/vectors of C gives principal curvatures.
• Surface M in 3D is the level set F(x,y,z) = 0 of the 3D Map F. Shape Operator S is given by
• Eigenvalues of S are c1, c2, 0 where c1, c2 are principal curvatures.
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2007
Summary of Curvature Computations for • Surface M in 3D is the level set F(x,y,z) = 0 of the 3D Map F. The Shape Operator is extended by adding the extra constraint S(n) = 0 and thus S: R3 -> R3 is given by
• Surface M in 3D is explicitly given by z = f(x,y), equivalently by the level set F(x,y,z) = f(x,y) – z = 0.• The Shape Operator S: R2 -> R2 is given by
• Basis of Tangent Space
Additional Reading • The references given below include the ones cited in
the lecture slides. Please check for pdf’s of these additional references on university computers from http://cvcweb.ices.utexas.edu/cvc/papers/papers.php
• C.Bajaj Tutorial Notes on “Multiscale, Bio-Modeling and Visualization”, Chap 2, 2010.