Comparative Study of Semi-implicit schemes for Non-linear Diffusion in Hyperspectral imagery

Student: Julio Martin Duarte-Carvajalino e-mail: jmartin@ece.uprm.edu Advisor: Dr. Miguel Velez-Reyes e-mail: mvelez@ece.uprm.edu Assessor: Dr. Paul Castillo e-mail: castillo@math.uprm.edu

University of Puerto Rico at Mayaguez

ABSTRACT

STATE OF THE ART

Witkin (1983) introduces the Scale-Space concept: different objects appear at different image scales in the image. The scale-space of Witkin was based on the isotropic diffusion (Gaussian Blurring) of an image.

Perona and Malik (1990) present a nonlinear diffusion equation to generate a Scale-Space without blurring the edges. In fact, their intention was to enhance the image edges.

Alvarez, Lions, Morel and Guichard (1993) proved that the Perona Malik diffusion equation is ill-posed, due to unstable backward diffusion on the image edges. They also show how to obtain a continuous Scale-Space, by regularizing the Perona-Malik nonlinear equation, and formalize the concept of Scale-Space as a transformation with the following properties:

Architectural: recursivity, causality, regularity, locality and Consistency. Invariance: stability and shape-preserving.

Weickert (1996) establishes the requirements that a discretized diffusion equation must hold to constitute a scale-space with the same properties than the continuous transformation. He also introduces the concept of anisotropic diffusion using a tensor valued diffusion coefficient.

Jayant Shah (1996) introduces a common framework for curve evolution, segmentation and anisotropic diffusion.

Weickert (1998) introduces Additive Operator Splitting as a robust semi-implicit scheme to solve the regularized nonlinear diffusion equation.

Sapiro (1998) presents a relation between nonlinear diffusion and robust estimation statistics. Pollack and Willsky (2000) presents a PDE for image segmentation using stabilized backward

diffusion. Osher (2002) integrates level sets and nonlinear diffusion to smooth 3D surfaces. Gilboa (2004) introduces complex shock filters to obtain an adaptive, stabilized forward-backward

diffusion process. Lennon et al (2002) present a classification methodology using nonlinear diffusion and support vector

machines

REFERENCES

A. Witkin, “Scale-space filtering, ” in Int. Joint Conf. Artificial Intelligence, Karlsruhe, Germany, pp. 1019-1021, 1983.

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 7, pp. 629-639, July 1990.

L. Alvarez, F. Guichard, P. L. Lions, and J. M. Morel, “Axioms and fundamental equations of image processing,” Arch. Rational Mech. Anal., vol. 123, pp. 199-257, 1993.

J. Weickert, “Anisotropic diffusion in image processing,” PhD Thesis, Dept. of Mathematics, University of Kaiserslautern, Germany, January 1996.

J. Shah, “A common framework for curve evolution, image segmentation and anisotropic diffusion,” IEEE Conf. Computer Vision and Pattern Recognition, pp. 136 – 142, June 1996.

J. Weickert, B. M. ter Haar Romeny, and M. A. Viergever, “Efficient and reliable schemes for nonlinear diffusion filtering,” IEEE Tran. Image Processing, vol. 7, no. 3, pp. 398–410, March 1998.

M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, “Robust anisotropic diffusion,” IEEE Trans. Image processing, vol. 7, no. 3, pp. 421-431, 1998.

I. Pollack, A. S. Willsky, and H. Krim, “Image segmentation and edge enhancement with stabilized inverse diffusion equations,” IEEE Trans. Image Processing, vol. 9, no. 2, pp. 256-266, 2000.

T. Tasdizen, R. Whitaker, P. Burchard and S. Osher, “Geometric surface smoothing via anisotropic diffusion of normals,” Proc. IEEE conf. on Visualization, Boston-Massachusetts, pp. 125-132, 2002.

G. Gilboa, N. Sochen, and Y. Y. Zeevi, “Image enhancing and denoising by complex diffusion processes,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 8, pp. 1020-1036, 2004.

M. Lennon, G. Mercier, and L. Hubert-Moy, “Classification of Hyperspectral images with nonlinear filtering and support vector machines,” IEEE International Geoscience and Remote Sensing Symposium, 2002 vol. 3, pp. 1670 – 1672, June 2002.

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CHALLENGES AND SIGNIFICANCE

Extend nonlinear diffusion to smooth Hyperspectral imagery using explicit and semi-implicit schemes, and fast linear solvers (Preconditioned Conjugated Gradient, PCG).

Perform simulations on a synthetic Hyperspectral image to test the performance of the semi-implicit and PCG methods in terms of speedup and the square error, relative to the explicit methods.

Perform simulations on a real Hyperspectral image to test the performance of semi-implicit and PCG methods, relative to the explicit schemes in terms of classification accuracy and the reduction in the spectral and spatial variability.

TECHNICAL APPROACH

Classification of Hyperspectral imagery is still made on a pixel by pixel basis, with classification accuracies ranging between 80-85 % and they have not changed in the last two decades (Wilkinson, 2003).

Object-based segmentation of Hyperspectral imagery can enhance image classification, by classifying objects rather than image pixels, with improved statistical information.

Nonlinear image diffusion can be used to enhance and even segment Hyperspectral imagery, but it requires: robust noise estimation, local adaptivity, incorporation of spectral similarity measures, edge enhancing and proper segmentation of the image, and the use of state of the art numerical methods and parallel processing.

EXPERIMENTS WITH HYPERSPECTRAL IMAGES (HIS)

FUTURE PLANS

THREE-LEVEL DIAGRAM

"This work was supported in part by CenSSIS, the Center for Subsurface Sensing and Imaging Systems, under the Engineering Research Centers Program of the National Science Foundation (Award Number EEC-9986821)." Julio M. Duarte was supported by a fellowship from the PR NSF-EPSCOR program.

Automatic selection of the threshold parameters in the nonlinear diffusion equation.

Analyze and adapt state of the art, modified nonlinear diffusion equations to obtain both image smoothing and segmentation in Hyperspectral imagery.

Optimization of the code using High Performance architectures and state of the art numerical methods

Nonlinear diffusion has been successfully employed since the past two decades to enhance images by reducing undesirable intensity variability (noise) within the semantically meaningful objects in the image, while enhancing the contrast of the boundaries (edges) in scalar, and more recently in vector valued images such as color, Multi and Hyperspectral imagery. In this presentation, we show that nonlinear diffusion can improve the classification accuracy of Hyperspectral imagery by reducing the spatial and spectral variability of the image, while preserving the semantically meaningful boundaries of the objects. We also show that semi-implicit schemes and fast linear solvers as Preconditioned Conjugated Gradient (PCG) methods can significantly speedup the evolution of the nonlinear diffusion equation with respect to traditional explicit schemes.

THE REGULARIZED NONLINEAR DIFFUSION EQUATION

83.31480

,, 0,

, 0 ;

, 0 0, ,

u K

n

u tg u u t on T

tu t u on g u e

u t on T

xx

x

x

The regularized (scalar) nonlinear diffusion equation is given by (Weickert, 1996)

THE EXPLICIT AND SEMI-IMPLICIT SCHEMES

The discretization of the nonlinear diffusion equation, in matrix-vector notation, is given by:

[1]

Figure 1 represents the scheme used to discretize the nonlinear diffusion equation, with column major format and the following conventions:

1 12

n n nk k kg g bands

v v

[2] 1 1 1 2Explicit: , Semi-Implicit: , wherex y

T

n n n n n n m m U I G U I G U U U v v v

vk

vk+1

vk-myvk+my

vk-1

1nkg

1nkg

y

nk mg

y

nk mg

Figure 1 Discretization Scheme

1, , where :pixels along the x-axis, :pixels along the y-axisx y x yk m m m m

th:Spectral vector of pixel k at time step nnkv

The Additive Operator Splitting (AOS): using the decomposition, G = Gx+Gy, where Gx,Gy are both tri- diagonal matrices, AOS approximates the solution to the semi-implicit scheme as,

Figure 4. Left: Original Indiana Pine image, Right: Smoothed Indiana Pine image

Figure 5. Left: ground truth, Right: Training & Testing Samples

Figure 2. Original Synthetic image Figure 3. Smoothed Synthetic image

SpectrumMean variance Variance reduction

Original image Smoothed image (%)

Hay-windrowed 2.42E-04 7.81E-07 99.68

Soybeans-min 8.92E-05 5.70E-07 99.36

Corn-min 8.92E-05 4.45E-07 99.50

Soybeans-notill 3.23E-04 3.85E-06 98.81

Reduction in the spectral/spatial variability, Synthetic image

Figure 6. Classification accuracies, Indiana Pine image

1 1 1 1 1

1; where 2 , 2

2x y x x y y

n n n n n n n n n U U U I G U U I G U U

Alternating Direction Implicit (ADI): here, we consider three ADI methods, based on G = Gx+Gy,

Locally One-Dimensional (LOD):

Peaceman-Rachford:

Douglas-Rachford:

1x yn n n n I G I G U U

1 2 ,n n n n n nx y x y I G I G U I G G U

[3]

[4]

11 1 1 1.

2 2 2 2n n n n n nx y x y

I G I G U I G I G U

[5]

Preconditioned Conjugated Gradient (PCG) methods: consists in solving AUn+1= Un as M-1AUn+1= M-1Un,

where M A is the preconditioner matrix. We consider two methods here:

SOLUTIONS TO THE SEMI-IMPLICIT SCHEME

Successive Over-relaxation (SSOR) method:

Incomplete Cholesky Factorization: 1

, , 0 22

T T

M I L I L A I L L

[6]

1,T T M L L A LL

[7]

: Rectangular domain of the image, x =(x,y) coordinates of a pixel, t: time, and T: final diffusion time.

[8]