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=12t=20t=16

t=24

t=40t=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

()u

()

∂2

div g

u

=

∇

u∇

t