Bayesian Image and Video Enhancement Using a Non-Gaussian Prior - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 309

shape to the neighbor interaction function, such as the edge penalty function
proposed by Geman and Reynolds, 22

1

ρ s (ξ ) =− , (9.34)

s

1 +

in which s serves as a scaling parameter for the data.
Convex edge penalty functions have also been investigated, primarily be-
cause of the difficulty surrounding nonconvex optimization. The generalized
Gauss-Markov random field (GGMRF) model was introduced by Bouman
and Sauer 23 for edge-preserving image restoration and tomographic image
reconstruction. This model is characterized by the edge penalty function,

p ,

ρ (ξ ) = | ξ | ≤ ≤

for 1 p 2 , (9.35)

p

so that the GGMRF model is a Laplacian density for p = 1 and a Gaussian

density for p 2. The parameter, p , controls the overall shape of this edge

penalty function, as shown in Figure 9.6. Schultz and Stevenson 1 proposed
the Huber-Markov random field (HMRF) model, which utilizes the piecewise
convex function first introduced by Huber to preserve discontinuities,

=

ξ

2 ,

| ξ | ≤

T H ,

ρ T H (ξ ) = (9.36)

| ξ | −

T H ,

| ξ | > .

2 T H T H

The Huber threshold parameter, T H , separates the quadratic and linear regions
of this edge penalty function, serving to control the size of discontinuities
within the data. The Huber edge penalty function is displayed in Figure 9.7,
superimposed on the quadratic to show the reduced penalty assigned to large
discontinuities.

100

90

80

70

60

50

40

30

20

10

0

− 10 − 8 − 6 − 4 − 2 0 2 4 6 8 10

FIGURE 9.6
Family of generalized Gaussian edge penalty functions, with p = 1 (absolute value), p = 1 . 2,

and p = 2 (quadratic).