各项异性扩散由来及原理

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

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

???=D j

Physical background of diffusion (2)

Diffusion is mass transport without destroying mass or

creating new mass

Continuity equation

t

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

??+???=?=?y x u t j j j div

Physical background of diffusion (3)

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

()

u u t ??=?D div

Physical background of diffusion(4)

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

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 y x I y x u u u t ==?

Linear isotropic diffusion

(2)

t=4t=8t=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

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

()u

()

?2

div g

u

=

?

u?

t

u for

1u for 0→?→∞→?→g g Nonlinear diffusion (2)

Conditions for g

Perona & Malik:

()2

2

2

11λu

u

g ?+

=

?

Nonlinear diffusion (3)

t=4t=8t=12

t=20t=16

t=24

t=40

t=0

Nonlinear diffusion (4)t=4

t=20

t=40

t=40

t=20

t=4

Linear diffusion

Nonlinear diffusion

Nonlinear diffusion(5)

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

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

Nonlinear anisotropic diffusion(3)

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 v v u u v 1121Nonlinear anisotropic diffusion (4)

u

v u v ?⊥?21||

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

1u

g ?=λNonlinear anisotropic diffusion (5)

1

2=λ

Now, we obtain D

as

Result is an edge-enhancing anisotropic diffusion

T

? ? =||||00||||212121

v v v v D λλNonlinear anisotropic diffusion (6)