geometric (bio-) modeling and visualization

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.