Aniso Diff Filtering

1Segmentierung medizinischer BilddatenStefan Burkhardt

8 Anisotropic diffusion filtering


� Images contain of the image itself and noise

�Noise: random, little disturbances of the image

�To improve the segmentation: the noise should be reduced

�Condition:

•Remove noise

•But: keep the image information unchanged



Diffusion

�Physical process for balancing concentration changes

�Now:

• the image intensity can be seen as a “concentration”

•The noise can be modelled as little concentration inhomogeneities

�These inhomogeneities could be smoothed by diffusion

�Diffusion should only be perpendicular e.g. to edges


8.1 Physical background

Physical background of diffusion

�Given: a concentration distribution u

�Fick’s law:

�Concentration gradient causes a flux j

� j aims to compensate the gradient

�D: diffusion tensor, in general a positive definite, symmetric matrix

u∇⋅−= Dj



Physical background of diffusion (2)

�Diffusion is mass transport without destroying mass or creating new mass

�Continuity equation

� t denotes the time, �tu the deviation of u with respect to t

��

��

�

∂∂+

∂∂−=−=∂

yxut

jjjdiv




�diffusion equation

�Application in many physical transport process, e.g. for heat transfer it is called heat-transfer-equation

� Image processing:

� Identify the concentration with the grey value at a certain location

( )uut ∇⋅=∂ Ddiv




�Diffusion tensor:

•Constant over the whole image: homogeneous (or linear) diffusion

•Often it is a function of the structure of the image itself: nonlinear diffusion

•Isotropic: j and the concentration gradient are parallel

•Anisotropic: otherwise


8.2 Linear diffusion

Linear isotropic diffusion

�Mostly used for smoothing images

�The image I itself is the initial starting for the diffusion process

�We use D = 1 since D only influences the speed of the diffusion

),()0,,(

div

yxIyxu

uut==∂



Linear isotropic diffusion (2)

t=4 t=8 t=12

t=20t=16 t=24 t=40

t=0



Linear isotropic diffusion (3)

�Advantages:

•Continuously simplifying of the image

•Reducing the noise in the image

�Disadvantages:

•Linear isotropic diffusion does not only reduce noise

•It also blues important features like edges

•No a-priori knowledge is taken into account

•Result: it makes edges harder to identify


8.3 Nonlinear diffusion

Nonlinear diffusion

� Improvement:

•Preservation of the edges, only smooting between edges

•Need to assign a position specific diffusivity

�Adapting the diffusivity g to the gradient in the actual image u(x,y,t)

�We obtain the equation

( )( )uuu ∇∇=∂ 2div gt



0ufor1

ufor0

→∇→∞→∇→

g

g

Nonlinear diffusion (2)

�Conditions for g

�Perona& Malik:

( )2

2

2

1

1

λu

ug∇

+=∇




t=4 t=8 t=12

t=20t=16 t=24 t=40

t=0




t=4 t=20 t=40

t=40t=20t=4

Linear diffusion

Nonlineardiffusion




�Advantages

•Steering of the diffusion at each point of the image is possible

•So diffusion can be reduced on edges

•The result: edge-preserving image smoothing

•But: the smoothing of the edges cannot be completely circumvented

�Next: anisotropic approaches


8.4 Nonlinear anisotropic diffusion

Nonlinear anisotropic diffusion

�Until now: determination of the amount of diffusion depending on local features (here: the gradient).

�Other features could be possible

� Idea: we could define the diffusion tensor so that the diffusion goes around some structures

�Here: diffusion should preserve edges

•On edges: no diffusion over edges, but diffusion parallel to edges should be enabled



Nonlinear anisotropic diffusion (2)

�Combination of two features

•Non-linearity

-The diffusion at border is much less than the diffusion elsewhere

•Anisotropy

-Diffusion should be perpendicular to edge

-No diffusion over edges




�How to define the diffusion tensor D?

�Remember: D is a positive definite symmetric matrix

•It has two different eigenvalues and two eigenvectors

•Therewith the eigenvectors are perpendicular

•The eigenvectors gives the main diffusion direction

•The corresponding eigenvalues the strength of diffusion in the direction of the eigenvectors.


�Definition of the eigenvectors

�We want to stop the diffusion over the edge. Therewith we need as one eigenvector the direction of the gradient.

�The second is perpendicular and can be seen as the tangential vector to the edge


[ ][ ] ��

��

�

−=

∇∇=

x

y

v

vv

u

uv

1

12

1


uv

uv

∇⊥∇

2

1 ||


�Definition of the eigenvalues

�First: We want to stop the diffusion over the edge (direction v1). We use the non-linearity and define

�Perpendicular to the edge the diffusion should not be stopped


( )2

1 ug ∇=λ


12 =λ


�Now, we obtain D as

�Result is an edge-enhancing anisotropic diffusion


T

��

�

�

��

�

�

⋅��

��

�⋅��

�

�

��

�

�

=||

||

0

0

||

||

2

12121 vvvvD

λλ





t=4 t=8 t=12

t=20t=16 t=24 t=40

t=0




t=4 t=20 t=40

t=40t=20t=4

Nonlineardiffusion

Nonlinearanisotropicdiffusion


8.5 Numerical implementations

Numerical implementation

�We have

�Looking for a solution u(x,y,t)

�Either for a specific t = t0

�Or for t��

�Problem: the equation cannot be analytically solved

�Numerical solution is necessary

),()0,,( yxIyxu =


�1. step: discretizing the time axis

•Determination of a time step ��

-�� dependson thenumerical stability

•only the times

are taken into account

�We obtain u(x,y,ti)


Numerical implementation (2)

τ∆⋅== itt i


�2. step: finding an approximation for the derivative with respect to t

�The Taylor serie for u(x,y,t) leads to

�Neglecting the remain and transform the equation leads to


( )ζττ Rtyxut

tyxutyxu +∂∂⋅∆+=∆+ ),,(),,(),,(


),,(),,(),,( 1 iii tyxut

tyxutyxu∂∂⋅∆+=+ τ


�Conditions to the step size:

�Dimensions: m, grid width: h

�Costs per iteration: very low

�Efficiency: low (many iterations needed to get a result)


m

h

2

2

<∆τ



�Until now we used the so-called explicit scheme

�Other notation

with

•I: identity matrix

•ui values of u(x,y,ti) stacked in vector form

•Al(uk) matrix version of the diffusion tensors


Notation

( ) im

l

il

i uuAIu ⋅��

��

� ⋅∆+= �=

+

1

1 τ


�Semi-implicit scheme

�Additive operator splitting


( ) im

l

il

i uuAIu ⋅��

��

� ⋅∆−=−

=

+ �1

1

1 τ

Improvements

( )( ) im

l

il

i mm

uuAIu ⋅⋅∆⋅−=−

=

+ �1

1

1 1 τ


�Semi-implicit scheme: We can write the derivative of uwith respect to t as

� Implicit: because ui+1 is used

�Semi-implicit: operator Al(ui) is combined with ui+1

�Transformation of the equation leads to


( ) 1

1

1+

=

+

⋅=∆

−� i

m

l

il

ii

A uuuu

τ

Semi-implicit scheme

( ) im

l

il

i uuAIu ⋅�

��

⋅∆−=−

=

+ �1

1

1 τ


�Features:

•Stable for ��< �, but: convergenceusually becomesslower for too large ��

•Cost per iteratorion: high, because thematrix must bebuilt and inverted in each iteration, O(n2)

•Efficiency: medium (better than theexplicit schemebecauseof thepossible larger �, but further improvementis possible)

�Question: How could the invertion of thematrix bemoreeffective

•Hint: each Al is a simple tri-diagonal matrix


Semi-implicit scheme (2)


�Objective: split the derivation operator for each spatial direction

�Then we get for each direction a simple tri-diagonal matrix

�Such matrices can be very efficiently inverted

�Advantage:

•Keep the unrestrictedness of ��

•Make thecalculation moreeffective


Additive operator splitting (AOS)


�One can write

as


( )�

��

⋅∆− �=

m

l

il

1

uAI τ

Additive operator splitting (2)

( )[ ]�=

⋅⋅∆−m

l

imm 1

1

1uAI τ


�For thecomputation of the inversematrix weuse theapproximation

�Thenow occuring error is

•only of second and higher order

•This means: the first derivatives haveno errors


( ) 111 −−− +≈+ BABA



�And weapproximate the inversematrix by

�Therewith weobtain theequation


( ) ( )[ ]1

11

1

1

1−

=

−

=�� ⋅⋅∆−≈�

��

⋅∆−m

l

im

l

il m

muAIuAI ττ


( )[ ] im

l

ii mm

uuAIu1

11

1 1−

=

+ � ⋅⋅∆−= τ


�Differencesand improvements

•A split in theoperator for each spatial direction has beenperformed

•Thepixels can bearranged for each operator separately

•As result each operator is tri-diagonal matrix and isinverted separately.

•Such 3-diagonal matrices can be inverted in linear time O(n), not in O(n2)




�Features

•Stable for ��< �

•Cost per iteratorion: low, but a littlehigher than for theexplicit scheme

�Efficiency: high





highlow��<�AOS

mediumhigh��<�Semi-implicit

lowvery lowexplicit

efficiencyCost per iteration

StabilityScheme

Conclusion

m

h

2

2

<∆τ

Aniso Diff Filtering

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