Comparative Study of Semi-implicit schemes for

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Comparative Study of Semi-implicit schemes for
Non-linear Diffusion in Hyperspectral imagery
Student Julio Martin Duarte-Carvajalino
e-mail jmartin_at_ece.uprm.edu
Advisor Dr. Miguel Velez-Reyes e-mail
mvelez_at_ece.uprm.edu Assessor Dr. Paul
Castillo e-mail castillo_at_math.uprm.edu Universit
y of Puerto Rico at Mayaguez
THE EXPLICIT AND SEMI-IMPLICIT SCHEMES
ABSTRACT
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.
Figure 1 represents the scheme used to discretize
the nonlinear diffusion equation, with column
major format and the following conventions
vk
The discretization of the nonlinear diffusion
equation, in matrix-vector notation, is given by
STATE OF THE ART
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• 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

SOLUTIONS TO THE SEMI-IMPLICIT SCHEME

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

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• Alternating Direction Implicit (ADI) here, we
  consider three ADI methods, based on G GxGy,

• Locally One-Dimensional (LOD)
• Peaceman-Rachford
• Douglas-Rachford

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• Preconditioned Conjugated Gradient (PCG)
  methods consists in solving AUn1 Un as
  M-1AUn1 M-1Un,
• where M ? A is the preconditioner matrix. We
  consider two methods here

• Successive Over-relaxation (SSOR) method
• Incomplete Cholesky Factorization

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EXPERIMENTS WITH HYPERSPECTRAL IMAGES (HIS)

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. 398410, 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.

Figure 2. Original Synthetic image
Figure 3. Smoothed Synthetic image
CHALLENGES AND SIGNIFICANCE

• 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.

Figure 5. Left ground truth, Right Training
Testing Samples
Figure 4. Left Original Indiana Pine image,
Right Smoothed Indiana Pine image
Reduction in the spectral/spatial variability,
Synthetic image
TECHNICAL APPROACH
Spectrum Mean variance Mean variance Variance reduction
Spectrum 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

• 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.

Figure 6. Classification accuracies, Indiana Pine
image
THREE-LEVEL DIAGRAM
FUTURE PLANS
THE REGULARIZED NONLINEAR DIFFUSION EQUATION

• 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

The regularized (scalar) nonlinear diffusion
equation is given by (Weickert, 1996)
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? Rectangular domain of the image, x (x,y)
coordinates of a pixel, t time, and T final
diffusion time.
"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.