Patch-Based Filtering

Finally, we investigate the more complicated functionals (8.14) introduced for the variational description of the patch-based filtering method. Here, let X be a

rectangular domain inR^m, p∈[2,∞), andJ : L^p(X)→R∪ {∞}be a functional of the form

J(u) =α

X

X

G(Hu(x,y))k(|x−y|)dx dy+

X

|u(x)−u₀(x)|^pdx, where G :[0,∞)→[0,∞)is a Borel measurable function, k :[0,∞)→[0,∞)is a measurable function, u0∈L^p(X),α∈(0,∞), and

Hu(x,y) =

X

h(t)|u(x−t)−u(y−t)|²dt

with some non-negative function h∈L^∞(X). Moreover, we consider u∈L^p(X)to be periodically continued outside the domain X .

This type of functionals does not fit into our Definition8.1of a non-local func-tional since the integrand does not only depend on two values of the function u, but rather on all values of u in some neighbourhoods of two points. Nevertheless, we may try to find a sufficient condition for the functions G, k, and h such thatJ has a minimising point by requiring as before thatJ shall be coercive and sequentially lower semi-continuous with respect to the weak topology on L^p(X).

The coercivity ofJ follows directly from the fact thatJ(u)≥ u−u0^pp. And for the sequential lower semi-continuity ofJ we get the following result.

Proposition 8.11. Let p∈[2,∞), h∈L^∞(X)and k∈L^∞(X×X)be non-negative functions, and G :[0,∞)→[0,∞)be a monotonically increasing, convex function.

Then the functional

R: L^p(X)→R∪ {∞}, R(u) =

X

X

G(Hu(x,y))k(x,y)dx dy

is sequentially lower semi-continuous with respect to the weak topology on L^p(X).

Proof. First, we remark thatRis a convex functional. Indeed, the map L^p(X)→R, u→ Hu(x,y) is for all x,y∈X convex and so is the map L^p(X)→R, u→ G(Hu(x,y)), since G is monotonically increasing and convex. Thus, by the mono-tonicity and linearity of the integral,Ris convex.

Moreover, the functionalRis sequentially lower semi-continuous with respect to the strong topology. To see this, let(u)∈N⊂L^p(X)be a sequence converging strongly to u∈L^p(X). Then for every point(x,y)∈X×X , the functions X→R, t →u(x−t)−u(y−t)converge with →∞strongly to the function X →R, t→u(x−t)−u(y−t). Since p≥2, we therefore have that

→∞limHu(x,y) =Hu(x,y)

for all x,y∈X . So, we get with Fatou’s Lemma lim inf

→∞ R(u)≥

X

X

lim inf

→∞ G(Hu(x,y))k(x,y)dx dy=R(u).

Since a convex functional on L^p(X)which is lower semi-continuous with respect to the strong topology on L^p(X)is also sequentially lower semi-continuous with respect to the weak topology, we can conclude thatRis sequentially lower semi-continuous with respect to the weak topology on L^p(X).

Fig. 8.3 Results obtained with the fixed point iterations of the non-local functional and the patch-based functional. The kernel used in the functionals are defined by G(ξ) =1−e^−ξ/λ, k=χ[−σk,σk]

and h=χ[−σh,σh]. The balanceα between the regularisation term and the data term has been selected a posteriori. The initial image is a 243×270 fraction of the famous “boat” image whose intensities range from 0 to 255. The noisy version has been obtained by adding a Gaussian noise of standard deviationu₀−f2=10. Finally, the computation times tcobtained on a 8×3.40 GHz computer are indicated in seconds

8.4.3.1 Numerical Implementation

We consider the numerical minimisation of the non-local functionalJ: L²(X)→R defined in (8.14) with G(ξ) =1−e^−ξ/λ, k(x,y) =χ_[−σk,σk](|x−y|), and h= χ_[−σh,σh]. This functional has thus four parametersλ,α,σhandσk.

Since the function G is not convex, we cannot apply Proposition8.11to guarantee the existence of a minimising point of the functionalJ in this case. Nevertheless, we are trying to minimise the functional numerically.

Similarly to the derivation of (8.13), we find the fixed point iteration u(x) =u0(x) +α_X^g˜(x,y,u₋₁)k(x,y)u₋₁(y)dy

1+α_Xg(x,˜ y,u₋₁)k(x,y)dy , ∈N, for the minimisation ofJ where ˜g is given by (8.10).

We remark that this expression corresponds to the block implementation of the non-local means filter described in [6] if we consider a single iteration and let α→∞. As in the previous section, we initialise with u0and iterate the fixed point equation until u−u₋₁2<ε where ε∈R is a sufficiently small parameter.

Figure8.3e, f illustrates the result of this procedure for two different values of the parameterσkwhich defines the local character of the functional.

Acknowledgements The work of CP and OS has been supported by the Austrian Science Fund (FWF) within the research networks NFNs Industrial Geometry, Project S09203, and Photo-acoustic Imaging in Biology and Medicine, Project S10505-N20.

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Opial-Type Theorems and the Common Fixed