Signal- und Bildverarbeitung, 323.014 Image Analysis and Processing Arjan Kuijper 30.11.2006

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Signal- und Bildverarbeitung, 323.014 Image
Analysis and ProcessingArjan Kuijper30.11.2006

• Johann Radon Institute for Computational and
  Applied Mathematics (RICAM) Austrian Academy of
  Sciences Altenbergerstraße 56A-4040 Linz,
  Austria
• arjan.kuijper_at_oeaw.ac.at

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Last week

• Total variation minimizing models have become one
  of the most popular and successful methodology
  for image restoration.
• ROF (Rudin-Osher-Fatemi) is one of the earliest
  and best known examples of PDE based edge
  preserving denoising.
• It was designed with the explicit goal of
  preserving sharp discontinuities (edges) in
  images while removing noise and other unwanted
  fine scale detail.
• However, it has some drawbacks
• Loss of contrast
• Loss of geometry
• Stair casing
• Loss of texture

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Today

• Mean curvature motion
• Curve evolution
• Denoising
• Edge preserving
• Implementation
• Isophote vs. image implementation
• Taken from

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Overview of Evolution Equations

• We start with a generalized framework for a
  number of nonlinear evolution equations that have
  appeared in the literature.
• In Alvarez et al. (1993) the interested reader
  can find an extensive treatment on evolution
  equations of the general form
• Imposing various axioms on a multi-scale analysis
  the authors derive a number of evolution
  equations that are listed here.
• Here, we distinguish two approaches
• i) evolution of the luminance function and
• ii) evolution of the level sets of the image.
• These approaches are dual in the sense that one
  determines the other
• Alvarez, L., Guichard, F., Lions, P.L., and
  Morel, J.M. 1993. Axioms and fundamental
  equations of image processing. Arch. for Rational
  Mechanics, 123(3)199257.

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Choices for F

• F ? LIn this case we find the linear diffusion
  equation. The luminance L is conserved under the
  flow ? L.
• F c(? L) ? LThe heat conduction coefficient
  c is not a constant anymore but depends on local
  image properties (Perona and Malik, 1990)
  resulting in nonlinear or geometry-driven
  diffusion.
• F ? L/? LThis equation has been used in
  Rudin et al. (1992) to remove noise based on
  nonlinear total variation.

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Curve evolution

• Definition 8 (Curve Evolution). A general
  equation which evolves planar curves as a
  function of their geometry can be written as
• In other words, a curve evolves as a function of
  curvature (and derivatives of curvature with
  respect to arc-length) only. The definition above
  follows from the following considerations. A
  general evolution of a curve can be written
  aswhere T and N denote the tangential and
  normal unit vector to the curve respectively.
  However, the T component only affects the
  parameterization.

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• F. Cao, Geometric Curve Evolution and Image
  Processing, LNM 1805, Springer, 2002

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Choices for g

• g cThis choice results in normal
  motionIsophotes move in the normal direction
  with velocity c. This equation is equivalent to
  the morphological operation of erosion (or
  depending on the sign of c dilation) with a disc
  as structuring element.
• g k This equation evolves the curve as a
  function of curvature and is known as the
  Euclidean shortening flow. It implies the
  following evolution of the luminance function

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Choices for g

• This equation has been proposed since it is
  invariant under Euclidean transformations. An
  elegant generalization to find the flow which is
  invariant with respect to any Lie group action
  can be found is as follows The evolution given
  by
• where r denotes the arc-length which is
  invariant under the group, defines a flow which
  is invariant under the action of the group. These
  equations locally behave as the geometric heat
  equation
• where is the G-invariant metric. If r
  is Euclidean arc-length (re) we find the
  Euclidean shortening flow

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• This is not the second order derivative of the
  spatial coordinates (left), but of the
  parameterization (right)!

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Choices for g

• This evolution is known as the affine shortening
  flow. If we insert the affine arc-length (ra) we
  find
• g a b kA combination of normal motion and
  Euclidean shortening flow. Note that a and b have
  different dimensions (the value b/ a is not
  invariant under a spatial rescaling x -gt l x).
  We have to work in natural coordinates or
  multiply nth order derivatives with the nth power
  of scale. The luminance function evolves
  according to This is a Hamilton-Jacobi
  equation with parabolic right-hand side. Since
  there are two independent variables it generates
  a 2-dimensional Entropy Scale Space with a
  reaction axis (owing to the hyperbolic term) and
  a diffusion axis (owing to the parabolic
  right-handside).

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Overview
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Numerical stability

• When we approximate a partial differential
  equation with finite differences in the forward
  Euler scheme, we want to make large steps in
  evolution time (or scale) to reach the final
  evolution in the fastest way, with as little
  iterations as possible.
• How large steps are we allowed to make? In other
  words, can we find a criterion for which the
  equation remains stable?
• A famous answer to the question of stability was
  derived by Von Neumann, and is called the Von
  Neumann stability criterion.
• Alternative names
• Courant stability criterion
• CFL condition (Courant-Friedrichs-Lewy condition)
  

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• Consider the 1D linear diffusion equation
• This equation can be approximated with finite
  differences as
• We define , so rewrite to
• i.e. f(j,n)0

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• Let the solution Ljn of our PDE be a generalized
  exponential function, with k a general (spatial)
  wave number
• When we insert this solution in our discretized
  PDE, we get
• We want the increment function f(j,n) to be
  maximal on the domain j, so we get the condition

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• The amplitude xn of the solution x n e ? j k Dx
  should not explode for large n, so in order to
  get a stable solution we need the criterion
  x1. This means, because Cos(k Dx)-1 is always
  non-positive, that
• This is an essential result. When we take a too
  large step size for Dt in order to reach the
  final time faster, we may find that the result
  gets unstable. The Von Neumann criterion gives
  us the fastest way we can get to the iterative
  result. It is safe to stay well under the
  maximum value, to not compromise this stability
  close to the criterion.

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• The pixel step Dx is mostly unity, so the maximum
  evolution step size should be
• This is indeed a strong limitation, making many
  iteration steps necessary.
• Gaussian derivative kernels improve this
  situation considerably.
• We start again with a general possible solution
  for the luminance function L(x,j,n), where x is
  the spatial coordinate, j is the discrete spatial
  grid position, and n is the discrete moment in
  evolution time of the PDE.

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• The Laplacian in 1D is just the second order
  spatial derivative
• We recall that the convolution of a function f
  with a kernel g is defined as
• For discrete location j at time step n we get for
  the blurred intensity

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• If we divide this by the original intensity
  function, we find a multiplication factor
• We are looking for the largest absolute value of
  the factor, because then we take the largest
  evolution steps. Because the factor is negative
  everywhere we need to find the minimum of the
  factor with respect to j, i.e. -gt
• We then find for the maximum size of the time
  step factor

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• So we find for the Gaussian derivative
  implementation
• so x1 implies , thus Dt2ã s
• Introducing this in the time-space ratio R we get
  the limiting stepsize for a stable solution under
  Gaussian blurring
• Note that this enables substantially larger
  stepsizes then in the nearest neighbor case.

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• For the Gaussian blurring of an image with s0.8
  pixels for the Laplacian operator, we get Dsltã
  .821.74. We blur a test image to
  pixels (which is equal to s64, ) in
  two ways
• a) with normal Gaussian convolution and
• b) with the numerical implementation of the
  diffusion equation and Gaussian derivative
  calculation of the Laplacian.

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• Alvarez, Guichard, Lions and Morel realized that
  the PM variable conductance diffusion was
  complicated by the choice of the parameter k.
• They reasoned that the principal influence on the
  local conductivity should be to direct the flow
  in the direction of the gradient only
• They named the affine version (right-hand side to
  the power 1/3) the 'fundamental equation of
  image processingThis is the unique model of
  multi-scale analysis of an image, affine
  invariant and morphological invariant.
  L. Alvarez, F. Guichard, P-L. Lions, and J-M.
  Morel. Axioms and fundamental equations of image
  processing. Arch. Rational Mechanics and Anal.,
  16(9)200-257, 1993.

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Grayscale invariance

• The non-affine version can be written as
• There are a number of differences between this
  equation and the Perona Malik equation
• the flow (of flux) is independent of the
  magnitude of the gradient
• There is no extra free parameter, like the
  edge-strength turnover parameter k
• in the PM equation the diffusion decreases when
  the gradient is large, resulting in contrast
  dependent smoothing
• this equation is gray-scale invariant (the
  function does not change value when the grayscale
  function L is modified by a monotonically
  increasing or decreasing function f(L), f¹0.).
• This PDE is known as
• Euclidean shortening flow,
• curve shortening,
• Mean curvature Motion

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Numerical examples shortening flow

• Same test image. By blurring the image the noise
  is gone, but the edge is gone too. MCM deblurs
  and keeps the edge.

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• The noise gradually disappears in this nonlinear
  scale-space evolution, while the edge strength is
  well preserved.
• Because the flux term, expressed in Gaussian
  derivatives, is rotation invariant, the edges are
  well preserved irrespective of their direction
  this is edge-preserving smoothing.

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Why the name 'shortening flow'?

• For the metric of the curve, defined as where
  p is an arbitrary parametrization of the curve ,
  the evolution of the metric is equal to
• The total length of the curve evolves as so
  the length is always decreasing with time.

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Summary of Numerical Stability
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Summary

• There is a strong analogy between curve evolution
  and PDE based schemes. They can be related
  directly to one another.
• Euclidean shortening flow involves the diffusion
  to be limited to the direction perpendicular to
  the gradient only.
• The divergence of the flow in the equation is
  equal to the second order gauge derivative Lvv
  with respect to v, the direction tangential to
  the isophote.
• Implementation with Gaussian derivatives may
  allow larger time steps

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Next week

• Non-linear diffusionMumford Shah
• Diffusion - reaction equations
• Energy functional
• Edge set