Image Processing
Diffusion Filters
Diffusion – Background
Motivation: a physical process – concentration balancing
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Diffusion – Background
Concentration is a real-valued function in space, i.e.
For instance in physic it is often
Spatial concentration gradient
Flux
(vector field)
leads to the
Fick’s law:
is a positive definite symmetric matrix – Diffusion Tensor
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Diffusion – Background
It follows from the mass conservation (second Fick’s law)
with divergence
(a real-valued function)
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Diffusion for images – Idea
The image is interpreted as the initial concentration distribution
The “image” is changed (in time) according to
The diffusion tensor
controls the process
Cases with respect do :
is a scalar
→ isotropic
is a general tensor
→ anisotropic
independent on
dependent on
→ linear
→ non-linear
(all four combinations are possible)
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Homogenous diffusion
Special case of the linear isotropic diffusion.
Diffusion tensor is a “constant”, i.e.
( is a unit matrix)
with the Laplace Operator
There exists the analytical solution (for
i.e. the convolution of the image
with the Gaussian of variance
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→ basically smoothing
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Numerical Schemes
For homogenous diffusion as the example:
Approximate derivatives (continuous) by finite differences (discrete)
is the step in time, − spatial resolution.
are left out → approximation.
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Numerical Schemes
All together:
This was an explicit schema:
the new values are computed from the old ones directly
It is stable (converges) if all “weights” are non-negative, i.e.
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Numerical Schemes
Implicit Schema: Divergences in the next time step are used:
becomes
New values can not be computed from the old ones directly,
because they depend on each other. However, they depend linearly
→ system of linear equations.
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Numerical Schemes
System of linear equation:
Huge, but sparse:
Special iterative methods
(Jakobi …)
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Numerical Schemes – explicit vs. implicit
Stability (an oversimplified example):
Explicit:
Implicit:
less stable, fast
more stable, slow (solve a system at each time)
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Linear isotropic diffusion
The Idea – smooth dependent on edge information
With a pre-computed
Very often
is a positive decreasing
function (Diffusivity) of the squared length of image gradients
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Non-linear isotropic diffusion
The idea – edges are more distinctive in the “de-noised” image
becomes
(the diffusion tensor depends on )
A special case – TV-flow:
• No further contrast-dependent parameters
• Piecewise constant grey-value profiles (similar to segmentation)
Problem:
at
→ regularization
Implicit numerical schema leads to a system of non-linear equation.
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Non-linear isotropic diffusion
Some examples:
Original
Image Processing: Diffusion Filters
Gaussian smoothing
Non-linear diffusion
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Shock-Filter
The Idea – dilation close to the local maximums and erosion close to
the local minimums:
Joachim Weickert, Coherence-Enhancing Shock Filters, DAGM2003
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Shock-Filter
Usual Shock-Filter
Original
Coherence-Enhancing
← Different filter parameters →
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Shock-Filter
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Literature
Joachim Weickert: Anisotropic Diffusion in Image Processing
http://www.mia.uni-saarland.de/weickert/book.html
Further names:
Tomas Brox, Daniel Cremers, Andrés Bruhn …
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