LINEBURG

| v|

i.e., we still have motion by mean curvature followed by the slightly

modi¬ed projection on the constraint set. 0

The maximum principle is a mixed blessing in this case. If J — u = u,

i.e., we are doing the pure denoising case, then the fact that extrema

are not ampli¬ed is a good thing, and we may take as the initial data ’50

0 50 100 150 200 250 300

u(x, y, 0) = u0 (x, y); i.e., the noisy image is a good initial guess. This is not

a good choice for the deblurring case. There we merely use the linear decon- Figure 11.1. Original versus noisy signal in one spatial dimension.

volved approximation in equation (11.20) where µ is chosen to match the

constraint equation (11.21). Although this introduces spurious oscillations

Figures 11.4 and 11.5 show an original image and a noisy (signal-to-noise

in the initial guess, they seem to disappear rapidly when equation (11.41)

ratio approximately 3) image, respectively. Figure 11.6 shows the usual TV

is used.

recovered image, while Figure 11.7 uses the method in [111] and seems to

An interesting feature of this new approach is that we can view the right-

do better in recovering smooth regions.

hand side of equation (11.41) as consisting of an elliptic term added to a

Figure 11.8 represents an image blurred by a discrete Gaussian blur

Hamilton-Jacobi term. The elliptic term uses standard central di¬erences,

obtained by solving the heat equation with Figure 11.4 as initial data and

while the Hamilton-Jacobi term is upwinded according to the direction

computing the solution at t = 10 on a 128 — 128 grid. Figure 11.9 shows

of characteristics. What is a bit unusual here is that there should be no

the result of an approximate linear deconvolution. Note that in the absence

entropy ¬x in the approximate Hamiltonian. The viscosity criterion does

of noise the result is oscillatory but greatly improved. Figure 11.10 shows

not apply, and Roe™s entropy-violating scheme is used. For details see [111].

the result of using Figure 11.9 as an initial guess for our improved TV

To repeat, one consequence of this approach is that although the steady

restoration. Notice the good resolution without spurious oscillations.

states of equations (11.24) and (11.41) are the same, the numerical solutions

Our most demanding experiment was performed on the blurry and noisy

di¬er, and staircasing, as described in [42], seems to be minimized using

image obtained from the original image represented in Figure 11.11. The

equation (11.41).

experimental point-spread function j(x, y) is shown in Figure 11.12, and we

We demonstrate the results of our improved algorithms with the following

add Gaussian noise, signal-to-noise ratio approximately 5. The blurry noisy

experiments. Figure 11.1 shows a noisy piecewise linear one-dimensional

image is shown in Figure 11.13. The linear recovery is shown in Figure 11.14.

signal with a signal-to-noise ratio approximately equal to 3. Figure 11.2

Finally, the improved TV restoration using Figure 11.14 as initial guess is

shows the recovered signal wtih denoised edges, but also with staircasing

shown in Figure 11.15.

e¬ects based on the original TV method developed in [142]. Figure 11.3

shows the improved result without staircase e¬ects in the linear region; see

[111].

11.3. Numerical Implementation of TV Restoration 107 108 11. Image Restoration

200 200

180

160 150

140

120 100

100

80 50

60

40 0

20

0 ’50

0 50 100 150 200 250 300 0 50 100 150 200 250 300

Figure 11.2. TV recovery using the original method of Rudin et al. [142]. Figure 11.3. Improved TV recovery.

11.3. Numerical Implementation of TV Restoration 109 110 11. Image Restoration

50

50

100

100

150

150

200

200

250

250

50 100 150 200 250

50 100 150 200 250

Figure 11.4. Original image.

Figure 11.5. Noisy image, signal-to-noise ratio approximately equal to 3.

11.3. Numerical Implementation of TV Restoration 111 112 11. Image Restoration

50

50

100

100

150

150

200 200

250 250

50 100 150 200 250 50 100 150 200 250

Figure 11.6. Usual TV recovered image.

Figure 11.7. Improved TV recovered image.

11.3. Numerical Implementation of TV Restoration 113 114 11. Image Restoration

50 50

100 100

150 150

200 200

250 250

50 100 150 200 250 50 100 150 200 250

Figure 11.8. Image blurred by convolution with a Gaussian kernel. Figure 11.9. Linear deconvolution applied to Figure 11.8.

11.3. Numerical Implementation of TV Restoration 115 116 11. Image Restoration

50

50

100

100

150

150

200

200

250

250

50 100 150 200 250

50 100 150 200 250

Figure 11.10. Improved TV restoration of Figure 11.8 using Figure 11.9 as an

Figure 11.11. Original image.

initial guess.

11.3. Numerical Implementation of TV Restoration 117 118 11. Image Restoration

20

50

40

100

60

150

80

200

100

120

250

50 100 150 200 250

20 40 60 80 100 120

Figure 11.14. Linear restoration of Figure 11.13.

Figure 11.12. Experimental point-spread function.

50

50

100

100

150

150

200

200

250

250 50 100 150 200 250

50 100 150 200 250

Figure 11.15. Improved TV restoration of Figure 11.13 using Figure 11.14 as an

Figure 11.13. Blurry noisy version of Figure 11.11. initial guess.

120 12. Snakes, Active Contours, and Segmentation

A typical example is

1

g( u0 (x)) =

12 u 0 |p

1 + |J —

for p ≥ 1, where J is a Gaussian of variance σ.

Snakes, Active Contours, and Rather than using the energy de¬ned in equation (12.1), we can de¬ne a

compact version as in Caselles et al. [28] or Kichenassamy et al. [95] via

Segmentation 1

F2 (C) = g( (u0 (s))) ds. (12.2)

0

Using the variational level set formulation of Zhao et al. [175], we arrive

at

φ

φt = | φ| · g( u0 ) (12.3)

| φ|

φ

= | φ| g( u0 )κ + g( u0 ) ·

| φ|

= | φ|g( u0 )κ + g( u0 ) · φ.

This is motion of the curve with normal velocity equal to its curvature

12.1 Introduction and Classical Active Contours times the edge detector plus convection in the direction that is the gradient

of the edge detector. Thus, the image gradient determines the location of

The basic idea in active contour models (or snakes) is to evolve a curve, the snakes.

subject to constraints from a given image u0 , in order to detect objects in The level set formulation of this came after the original snake model was

that image. Ideally, we begin with a curve around the object to be detected, invented in [94]. This was ¬rst done in Caselles et al. [27] (without the

and the curve then moves normal to itself and stops at the boundary of the convection term), next by Malladi et al. [109], and the variational formula-

object. Since its invention by Kass et al. [94] this technique has been used tion used above came in [28] and [95]. Of course, this level set formulation

both often and successfully. The classical snakes model in [94] involves an allows topological changes and geometrical ¬‚exibility, and has been quite

edge detector, which depends on the gradient of the image u0 , to stop the successful in two and three spatial dimensions. Most models have the same

evolving curve at the boundary of the object. general form as equation (12.3), involving an edge detector times curvature

Let u0 (x, y) map the square 0 ¤ x, y ¤ 1 into R, where u0 is the image plus linear advection. The higher-order terms coming from the term multi-

and C(I) : [0, 1] ’ R2 is the parametrized curve. The snake model is to plying β in equations (12.1) and (12.2) are usually omitted. We note that

minimize this model depends on the image gradient to stop the curve (or surface)

1 1

evolution.

2

| u0 (C(s))|2 ds, (12.1)

|C (s)| ds + β |C (s)| ds ’ »

F1 (C) = ±

In a sequence of papers beginning with Chan and Vese [35] (see also [34]

0 0 0

and [32]) the authors propose a di¬erent active contour model without a

where ±, β, and » are positive parameters. The ¬rst two terms control the

stopping i.e. edge function, i.e., a model that does not use the gradient

smoothness of the contour, while the third attracts the contour toward the

of the image u0 for the stopping process. The stopping term is based on

object in the image (the external energy). Observe that by minimizing the

the Mumford-Shah segmentation technique, which we describe below. The

energy, we are trying to locate the curve at the points of maximum | u0 |,

model these authors develop can detect contours both with and without

which act as an edge detector, while keeping the curve smooth.

gradients, for instance objects that are very smooth, or even have discon-

An edge detector can be de¬ned by a positive decreasing function g(z),

tinous boundaries. In addition, the model and its level set formulation are

depending on the gradient of the image u0 , such that

such that interior contours are automatically detected, and the initial curve

lim g(z) = 0. can be anywhere in the image.

|z|’∞

12.2. Active Contours Without Edges 121 122 12. Snakes, Active Contours, and Segmentation

the level set function associated with „¦, as

E(C1 , C2 , φ) = µ δ(φ)| φ| dx (12.5)

+ν H(φ) dx

|u0 (x) ’ C1 |2 H(φ) dx

+ »1

|u0 (x) ’ C2 |2 (1 ’ H(φ)) dx.

+ »2

This involves the four nonnegative parameters µ, ν, »1 , and »2 .

The classical Mumford-Shah functional is a more general segmentation

de¬ned by

EM S (“, u) = µ length(“) (12.6)

|u ’ u0 |2 dx

+»

Figure 12.1. A simple case, showing that the ¬tting term E1 (“) is minimized

when the curve is on the boundary of the object.

| u|2 dx.

+ν

“c

Here u is the cartoon image approximating u0 , u is smooth except for jumps

12.2 Active Contours Without Edges on the set “ of boundary curves, and “ segments the image into piecewise

smooth regions. The method de¬ned in equation (12.5) di¬ers from that

De¬ne the evolving curve “ as the boundary of a region „¦. We call „¦ the in equation (12.6) in that only two subregions are allowed in which u is

inside of “ and the complement of „¦ = „¦c the outside of “. The method piecewise constant, so we may write

is the minimization of an energy-based segmentation. Assume that u0 is

formed by two regions of approximately piecewise constant intensities of

u = C1 H(φ) + C2 (1 ’ H(φ)). (12.7)

distinct values ui and uo . Assume further that the object to be detected

0 0

is represented by the region with value ui . Denote its boundary by “0 .

0 This was generalized considerably by Chan and Vese [36, 37]. We also

Then we have u0 ≈ ui inside “0 and u0 ≈ uo outside “0 . Now consider the

0 0 mention the approach of Koep¬‚er et al. [99], which approximates equa-

“¬tting” term

tion (12.6) by letting ν = 0 and » = 1, where µ is the scale parameter.

Again, u is piecewise constant, although many constants are allowed; i.e.,

the averages of u will generally be di¬erent in di¬erent segments of the

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