On Hamilton–Jacobi PDEs and Image Denoising Models with Certain Nonadditive Noise

Abstract

We consider image denoising problems formulated as variational problems. It is known that Hamilton–Jacobi PDEs govern the solution of such optimization problems when the noise model is additive. In this work, we address certain nonadditive noise models and show that they are also related to Hamilton–Jacobi PDEs. These findings allow us to establish new connections between additive and nonadditive noise imaging models. Specifically, we study how the solutions to these optimization problems depend on the parameters and the observed images. We show that the optimal values are ruled by some Hamilton–Jacobi PDEs, while the optimizers are characterized by the spatial gradient of the solution to the Hamilton–Jacobi PDEs. Moreover, we use these relations to investigate the asymptotic behavior of the variational model as the parameter goes to infinity, that is, when the influence of the noise vanishes. With these connections, some non-convex models for nonadditive noise can be solved by applying convex optimization algorithms to the equivalent convex models for additive noise. Several numerical results are provided for denoising problems with Poisson noise or multiplicative noise.

A Mathematical Background

In this section, several basic definitions and theorems in convex analysis are reviewed. All the results and notations can be found in [19, 23,24,25]. We use the angle bracket $\langle \cdot , \cdot \rangle$ to denote the inner product operator in any Euclidean space ${\mathbb {R}}^n$.

We denote $\Gamma _0({\mathbb {R}}^n)$ to be the set of proper, convex and lower semi-continuous functions from ${\mathbb {R}}^n$ to ${\mathbb {R}}\cup \{+\infty \}$. Recall that a function $f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\cup \{\pm \infty \}$ is called proper if there exists a point x such that $f(x)\in {\mathbb {R}}$. In this paper, we assume every function is proper if not mentioned specifically. The functions in $\Gamma _0({\mathbb {R}}^n)$ have good continuity properties, which are stated below.

Proposition 7

[23, Lem.IV.3.1.1 and Chap.I.3.1–3.2] Let $f\in \Gamma _0({\mathbb {R}}^n)$. If $x\in \mathrm {ri\ }\mathrm {dom}~f$, then f is continuous at x in $\mathrm {dom}~f$. If $x\in \mathrm {dom}~f {\setminus } \mathrm {ri\ }\mathrm {dom}~f$, then for all $y\in \mathrm {ri\ }\mathrm {dom}~f$,

$$\begin{aligned} f(x) = \lim _{t\rightarrow 0^+} f(x + t(y-x)). \end{aligned}$$

A vector p is called a subgradient of f at x if it satisfies

$$\begin{aligned} f(y)\ge f(x) + \langle p, y-x\rangle , \quad \forall y\in {\mathbb {R}}^n. \end{aligned}$$

The collection of all such subgradients is called the subdifferential of f at x, denoted as $\partial f(x)$. It is straightforward to check that $0\in \partial f(x)$ if and only if x is a minimizer of f. As a result, one can check whether x is a minimizer by computing the subdifferential.

Moreover, in most cases, the subdifferential operator commutes with summation. One set of assumption is given in the following proposition.

Proposition 8

[19, Proposition 5.6, Ch. I] Let $f,g\in \Gamma _0({\mathbb {R}}^n)$. Assume there exists a point $u\in \mathrm {dom}~f\cap \mathrm {dom}~g$ where f is continuous. Then, we have $\partial (f+g)(x) = \partial f(x) + \partial g(x)$ for all $x\in \mathrm {dom}~f \cap \mathrm {dom}~g$.

One important class of functions in $\Gamma _0({\mathbb {R}}^n)$ is called indicator functions. For any convex set C, the indicator function $I_C$ is defined by

$$\begin{aligned} I_C(x) := {\left\{ \begin{array}{ll} 0, &{} \text {if }x\in C,\\ +\infty , &{}\text {if }x\not \in C. \end{array}\right. } \end{aligned}$$

One can compute the subdifferential of the indicator function and obtain

$$\begin{aligned} \partial I_C(x) = N_C(x) , \end{aligned}$$

(98)

where $N_C(x)$ denotes the normal cone of C at x. Note that $N_C(x) = \{0\}$ when x is in the interior of the set C (see [23, p. 137] for details).

Next, we recall one important transform in convex analysis called Legendre–Fenchel transform. For all $f\in \Gamma _0({\mathbb {R}}^n)$, the Legendre–Fenchel transform of f, denoted as $f^*$, is defined by

$$\begin{aligned} f^*(p) := \sup _{x\in {\mathbb {R}}^n} \{\langle p, x\rangle - f(x)\}. \end{aligned}$$

(99)

The Legendre–Fenchel transform gives a duality relationship between f and $f^*$. In other words, if $f\in \Gamma _0({\mathbb {R}}^n)$ holds, then there hold $f^*\in \Gamma _0({\mathbb {R}}^n)$ and $f^{**} = f$. Similarly, along with this duality, some properties are dual to others, as stated in the following proposition. Recall that a function g is called 1-coercive if $\lim _{\Vert x\Vert \rightarrow +\infty } \frac{g(x)}{\Vert x\Vert } = +\infty$.

Proposition 9

[25, Propositions E.1.3.8 and E.1.3.9] Let $f\in \Gamma _0({\mathbb {R}}^n)$. Then, f is finite-valued if and only if $f^*$ is 1-coercive.

In particular, the subdifferential set of $f^*$ is characterized by the maximizers in (99), as stated in the following proposition.

Proposition 10

[25, Corollary E.1.4.4] Let $f\in \Gamma _0({\mathbb {R}}^n)$ and $p,x \in {\mathbb {R}}^n$. Then, $p\in \partial f(x)$ holds if and only if $x\in \partial f^*(p)$ holds, which is equivalent to $f(x) + f^*(p) = \langle p, x\rangle$.

Except from the Legendre–Fenchel transform, there is another operator defined on convex functions, called inf-convolution. Given two functions $f,g\in \Gamma _0({\mathbb {R}}^n)$, assume there exists an affine function l such that $f(x)\ge l(x)$ and $g(x)\ge l(x)$ for all $x\in {\mathbb {R}}^n$. Then, the inf-convolution between f and g, denoted by $f\square g$, is a convex function taking values in ${\mathbb {R}}\cup \{+\infty \}$ defined by

$$\begin{aligned} (f\square g)(x) := \inf _{u\in {\mathbb {R}}^n} \{f(u) + g(x-u)\}. \end{aligned}$$

(100)

In the following proposition, the relation between Legendre–Fenchel transform and inf-convolution is stated.

Proposition 11

[25, Theorem E.2.3.2] Let $f,g \in \Gamma _0({\mathbb {R}}^n)$. Assume the intersection of $\mathrm {ri\ }\mathrm {dom}~f$ and $\mathrm {ri\ }\mathrm {dom}~g$ is non-empty. Then, there hold $f^*\square g^*\in \Gamma _0({\mathbb {R}}^n)$ and $f^*\square g^* = (f+g)^*$. Moreover, the minimization problem

$$\begin{aligned} \min _{u\in {\mathbb {R}}^n} \{f^*(u) + g^*(x-u)\} \end{aligned}$$

(101)

has at least one minimizer whenever x is in $\mathrm {dom}~f^*\square g^*=\mathrm {dom}~f^* + \mathrm {dom}~g^*$.

Let f be a function in $\Gamma _0({\mathbb {R}}^n)$ and $x_0$ be an arbitrary point in $\mathrm {dom}~f$. The asymptotic function of f, denoted by $f'_\infty$, is defined by

$$\begin{aligned}&f'_\infty (d) = \sup _{s>0} \frac{f(x_0 + sd) - f(x_0)}{s} \nonumber \\&\quad = \lim _{s\rightarrow +\infty } \frac{f(x_0 + sd) - f(x_0)}{s}, \end{aligned}$$

(102)

for all $d\in {\mathbb {R}}^n$. In fact, this definition does not depend on the point $x_0$.

Proposition 12

[25, Example B.3.2.3] Let $f\in \Gamma _0({\mathbb {R}}^n)$. Take $x_0\in \mathrm {dom}~f$. Then, the function defined by

$$\begin{aligned} {\mathbb {R}}^n\times {\mathbb {R}}\ni (d,t)\mapsto {\left\{ \begin{array}{ll} t\left( f\left( x_0+\frac{d}{t}\right) - f(x_0)\right) , &{} \text {if }t>0,\\ f'_\infty (d), &{}\text {if } t=0,\\ +\infty , &{}\text {if }t<0, \end{array}\right. } \nonumber \\ \end{aligned}$$

(103)

is in $\Gamma _0({\mathbb {R}}^n\times {\mathbb {R}})$. As a result, we have

$$\begin{aligned} \liminf _{t\rightarrow 0^+, d\rightarrow d_0} t\left( f\left( x_0+\frac{d}{t}\right) - f(x_0)\right) \ge f'_\infty (d_0). \end{aligned}$$

(104)

Proposition 13

[25, Proposition E.1.2.2] Let $f\in \Gamma _0({\mathbb {R}}^n)$. Then, the support function of $\mathrm {dom}~f$ is given by the asymptotic function of $f^*$ as follows:

$$\begin{aligned} I_{\mathrm {dom}~f}^* = (f^*)'_\infty . \end{aligned}$$

(105)

There is another important class of functions called Legendre functions, whose definition is given as follows. Note that here we use an equivalent definition to simplify our proofs.

Definition 1

[37, Section 26] Suppose f is a function in $\Gamma _0({\mathbb {R}}^n)$. Then, f is Legendre (or a Legendre function or a convex function of Legendre type), if f satisfies

• 1. $\mathrm {int\ }\mathrm {dom}~f$ is non-empty.

• 2. f is differentiable on $\mathrm {int\ }\mathrm {dom}~f$.

• 3. The subdifferential of f satisfies

  $$\begin{aligned} \partial f(x) = {\left\{ \begin{array}{ll} \emptyset , &{} \text {if }x\in (\mathrm {dom}~f){\setminus } (\mathrm {int\ }\mathrm {dom}~f),\\ \{\nabla f(x)\}, &{} \text {if }x\in \mathrm {int\ }\mathrm {dom}~f. \end{array}\right. }\nonumber \\ \end{aligned}$$

  (106)

• 4. f is strictly convex on $\mathrm {int\ }\mathrm {dom}~f$.

For a Legendre function f, we have the following property which is used in proofs of this paper. For more properties of Legendre functions, we refer readers to [37, Section 26] and [5].

Proposition 14

[37, Theorem 26.5] Let f be a function in $\Gamma _0({\mathbb {R}}^n)$. Then, f is a Legendre function if and only if $f^*$ is a Legendre function.

Now, we are going to define the primal–dual Bregman distance denoted by d and the (primal–primal) Bregman distance denoted by D. For properties of Bregman distances, we refer readers to [5] and [9], for instance. For $f\in \Gamma _0({\mathbb {R}}^n)$, the primal–dual Bregman distance with respect to $f$ between a vector x in the primal space ${\mathbb {R}}^n$ and a vector p in the dual space ${\mathbb {R}}^n$, denoted by $d_f(x,p)$, is defined as follows:

$$\begin{aligned} d_f(x,p) := f(x) + f^*(p) - \langle p,x\rangle . \end{aligned}$$

(107)

When we have $f\in \Gamma _0({\mathbb {R}}^n)$, since the equality $(f^*)^* = f$ holds, we have

$$\begin{aligned} d_{f^*}(p,x) = f^*(p) + (f^*)^*(x) - \langle x,p\rangle = d_{f}(x,p).\nonumber \\ \end{aligned}$$

(108)

Additionally, for two points $x,u\in \mathrm {dom}~f$, let $f$ be differentiable at u. The Bregman distance $D_f$ between x and u with respect to the function $f$ is defined as

$$\begin{aligned} D_f(x,u)= f(x)- f(u) - \langle \nabla f(u),x-u\rangle . \end{aligned}$$

(109)

These two definitions are related to each other by the following result.

Proposition 15

Let $f\in \Gamma _0({\mathbb {R}}^n)$. Then, for all $x,u\in \mathrm {dom}~f$ such that $f$ is differentiable at u, we have

$$\begin{aligned} d_f(x,\nabla f(u)) = D_f(x,u). \end{aligned}$$

(110)

Proof

For all $x,u\in \mathrm {dom}~f$, we have

$$\begin{aligned} \begin{aligned}&d_f(x,\nabla f(u)) = f(x) + f^*(\nabla f(u)) - \langle \nabla f(u),x\rangle \\&\quad = f(x) + \langle \nabla f(u), u\rangle - f(u) - \langle \nabla f(u), x\rangle \\&\quad = D_f(x,u), \end{aligned} \end{aligned}$$

(111)

where the second equality holds by Proposition 10. $\square$

We also have the following equality for $D_f$.

Proposition 16

Let $f\in \Gamma _0({\mathbb {R}}^n)$. Let $t>0$ and $\frac{x}{t}, \frac{u}{t}\in \mathrm {dom}~f^*$. Assume $f^*$ is differentiable at $\frac{u}{t}$. Then, we have

$$\begin{aligned} tD_{f^*}\left( \frac{x}{t}, \frac{u}{t}\right) = D_{(tf)^*}(x,u). \end{aligned}$$

(112)

Proof

By definition of $D_{f^*}$ in (109), it follows that

$$\begin{aligned}&tD_{f^*}\left( \frac{x}{t}, \frac{u}{t}\right) = t\left( f^*\left( \frac{x}{t}\right) -f^*\left( \frac{u}{t}\right) - \left\langle \nabla f^*\left( \frac{u}{t}\right) , \frac{x-u}{t}\right\rangle \right) \nonumber \\&\quad = tf^*\left( \frac{x}{t}\right) -tf^*\left( \frac{u}{t}\right) - \left\langle \nabla f^*\left( \frac{u}{t}\right) , x-u\right\rangle . \end{aligned}$$

(113)

Define $g:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\cup \{+\infty \}$ by $g(w):= tf^*(\frac{w}{t})$ for all $w\in {\mathbb {R}}^n$, then there hold $\nabla g(u) = \nabla f^*(\frac{u}{t})$ and $g= (tf)^*$. And hence by (113) we obtain

$$\begin{aligned}&tD_{f^*}\left( \frac{x}{t}, \frac{u}{t}\right) = g(x) - g(u) - \langle \nabla g(u), x-u\rangle \nonumber \\&\quad = D_{g}(x,u) = D_{(tf)^*}(x,u), \end{aligned}$$

(114)

which proves the statement. $\square$