Automatic fidelity and regularization terms selection in variational image restoration

Abstract

This paper addresses the study of a class of variational models for the image restoration inverse problem. The main assumption is that the additive noise model and the image gradient magnitudes follow a generalized normal (GN) distribution, whose very flexible probability density function (pdf) is characterized by two parameters—typically unknown in real world applications—determining its shape and scale. The unknown image and parameters, which are both modeled as random variables in light of the hierarchical Bayesian perspective adopted here, are jointly automatically estimated within a Maximum A Posteriori (MAP) framework. The hypermodels resulting from the selected prior, likelihood and hyperprior pdfs are minimized by means of an alternating scheme which benefits from a robust initialization based on the noise whiteness property. For the minimization problem with respect to the image, the Alternating Direction Method of Multipliers (ADMM) algorithm, which takes advantage of efficient procedures for the solution of proximal maps, is employed. Computed examples show that the proposed approach holds the potential to automatically detect the noise distribution, and it is also well-suited to process a wide range of images.

A Proofs of the results

Proposition A.1

Let $s,\gamma \in {{\mathbb {R}}}_{++}$, $z \in {\mathbb {N}}$ be given constants, let $f_s: {{\mathbb {R}}}^z \rightarrow {{\mathbb {R}}}$ be the (parametric, not necessarily convex) function defined by $f_s(x) := \Vert x\Vert _2^s$ and let $\text {prox}_{f_s}^{\gamma }: {{\mathbb {R}}}^z \rightrightarrows {{\mathbb {R}}}^z$ be the proximal operator of function $f_s$ with proximity parameter $\gamma$, defined as the z-dimensional minimization problem

Then, for any $z \in {\mathbb {N}}, \, s,\gamma \in {{\mathbb {R}}}_{++}, \, y \in {{\mathbb {R}}}^z$, $\text {prox}_{f_s}^{\gamma }$ takes the form of a shrinkage operator:

$$\begin{aligned} \text {prox}_{f_s}^{\gamma }(y) \,\;{=}\;\,\, \xi ^* \big ( s,\rho \left( s,\gamma ,\Vert y\Vert _2\right) \big ) \, y \, , \;\;\,\text {with}\;\,\,\, \xi ^*\!: \, {{\mathbb {R}}}_{++} {\times } \,\, {{\mathbb {R}}}_+ \rightrightarrows \, [0,1) \, , \end{aligned}$$

and with function

$$\begin{aligned}&\rho : \, {{\mathbb {R}}}_{++}^2 {\times }\,\,\! {{\mathbb {R}}}_+ \rightarrow {{\mathbb {R}}}_+ \;\;\, defined \;\, by \;\;\;\, \rho (s,\gamma ,\Vert y\Vert _2) \\&\qquad {=}\;\, \left\{ \begin{array}{lll} \displaystyle { (\gamma /s) \Vert y\Vert _2^{2-s} \quad \text {if}\;\,\,\Vert y\Vert _2 \,{>}\; 0 \, , } \\ \displaystyle { 0 \qquad \qquad \qquad \!\! \text {if}\;\,\,\Vert y\Vert _2 \,{=}\; 0 \, , } \end{array} \right. \end{aligned}$$

In particular, for any $z \in {\mathbb {N}}$, the shrinkage coefficient function $\xi ^*(s,\rho )$ satisfies:

$$\begin{aligned} \begin{array}{llcl} s \;{>}\; 0, &{}\!\!\! \,\rho \;{=}\; 0 &{} \!\!\!{\Longrightarrow }\!\!\! &{} \xi ^*(s,\rho ) \,\;{=}\;\, 0 \, , \\ s \;{\le }\; 1, &{}\!\!\! \,\rho \;{<}\; {\bar{\rho }}(s) &{} \!\!\!{\Longrightarrow }\!\!\! &{} \xi ^*(s,\rho ) \,\;{=}\;\, 0 \, , \\ s \;{\le }\; 1, &{}\!\!\! \,\rho \;{=}\; {\bar{\rho }}(s) &{} \!\!\!{\Longrightarrow }\!\!\! &{} \xi ^*(s,\rho ) \,\;{\in }\; \{ 0, {\bar{\xi }}(s) \} \, , \\ s \;{\le }\; 1, &{}\!\!\! \,\rho \;{>}\; {\bar{\rho }}(s) &{} \!\!\!{\Longrightarrow }\!\!\! &{} \xi ^*(s,\rho ) \,\;{=}\;\, \text {unique sol. in}\,\;({\bar{\xi }}(s),1)\;\text {of}\;\, h(\xi ;s,\rho ) \;{=}\; 0 \, , \\ s \;{>}\; 1, &{}\!\!\! \,\rho \;{>}\; 0 &{} \!\!\!{\Longrightarrow }\!\!\! &{} \xi ^*(s,\rho ) \,\;{=}\;\, \text {unique sol. in}\,\quad \;\,(0,1)\;\text {of}\;\, h(\xi ;s,\rho ) \;{=}\; 0 \, , \end{array} \end{aligned}$$

(A.1)

with functions ${\bar{\rho }}: (0,1] \rightarrow [1,\infty )$, ${\bar{\xi }}: (0,1] \rightarrow [0,1)$, and $h: (0,1] \rightarrow {{\mathbb {R}}}$ defined by

$$\begin{aligned} {\bar{\rho }}(s) \,{=}\, \frac{(2-s)^{2-s}}{s \, (2-2s)^{1-s}} \, , \quad {\bar{\xi }}(s) \,{=}\, 2\,\frac{1-s}{2-s} \, , \quad h(\xi ;s,\rho ) \,{=}\, \xi ^{s-1} {+}\, \rho \left( \xi {-} 1 \right) \, . \end{aligned}$$

(A.2)

Finally, the function $\,\xi ^*$ exhibits the following regularity and monotonicity properties:

$$\begin{aligned} \begin{array}{c} \xi ^* \;{\in }\; C^{\infty }(S), \;\; S \;{=}\; S_0 \cup S_1, \;\;\, S_0 \;{=}\; (0,1]\,{\times }\,({\bar{\rho }}(s),\infty ), \; S_1 \;{=}\; (1,\infty )\,{\times }\,{{\mathbb {R}}}_{++} , \quad \\ \displaystyle { \frac{\partial \xi ^*}{\partial s}(s,\rho ) \;{>}\; 0 \, , \quad \frac{\partial \xi ^*}{\partial \rho }(s,\rho ) \;{>}\; 0 \, , \quad \forall \, (s,\rho ) \,\;{\in }\; S \, . } \end{array} \end{aligned}$$

(A.3)

Proof

The proof of (A.1)–(A.2) comes easily from Proposition 1 in [21], as (A.1)–(A.2) is there proved for $s<2$, but the proof of case c) in that proposition can be seamlessly extended to cover the case $s \ge 2$.

To prove (A.3), first we notice from (A.1)–(A.2) that $\xi ^*(s,\rho ) \,{\in }\, (0,1) \; \forall \, (s,\rho ) \,{\in }\, S$ and introduce the function $\,f\,{:}\;\, S_f \rightarrow {{\mathbb {R}}}$, $\,S_f = (0,1) \,{\times }\, S \;{\subset }\; {{\mathbb {R}}}_{++}^3$, defined by $\,f(\xi ,s,\rho ) \,{:}{=}\, h(\xi ;s,\rho )$ ${=}\; \xi ^{s-1} + \rho \, (\xi -1)$, $\forall \, (\xi ,s,\rho ) \,{\in }\, S_f$. The function f clearly satisfies

$$\begin{aligned} f \,{\in }\, C^{\infty }(S_f) \, , \quad \; \frac{\partial f}{\partial \xi } (\xi ,s,\rho )= (s-1) \, \xi ^{s-2} + \rho \;{=}\; 0 \;\; \text {for}\;\, s \,{<}\, 1 \;{\wedge }\; \xi ^{s-2} \,{=}\, \rho / (1-s) . \end{aligned}$$

We now demonstrate by contradiction that that if $s < 1$ then $\xi ^{s-2} \ne \rho /(1-s)$ for any $(\xi ,s,\rho ) \in S_f$. In fact, according to the definition of set $S_f$ given above, we have that $\xi > {\bar{\xi }}(s)$ and $\rho > {\bar{\rho }}(s)$ for $s<1$, with functions ${\bar{\xi }},{\bar{\rho }}$ defined in (A.2). But we have

$$\begin{aligned}&\xi \,\;{>}\;\, {\bar{\xi }}(s) \,\,\;{\Longleftrightarrow }\;\,\, \xi ^{s-2} \,{<}\, \left( {\bar{\xi }}(s)\right) ^{s-2}\\&\,{\Longleftrightarrow }\;\, \frac{\rho }{1-s} \,{<}\, \left( 2 \, \frac{1-s}{2-s} \right) ^{s-2} \,{\Longleftrightarrow }\;\, \rho \,\;{<}\;\, \frac{s}{2} \, {\bar{\rho }}(s) \, . \end{aligned}$$

Hence, $\,\partial f / \partial \xi \ne 0 \;\, \forall \, (\xi ,s,\rho ) \in S_f$ and it follows form the implicit function theorem that, for any $(s,\rho ) \in S$, the shrinkage coefficient $\xi ^* \in (0,1)$ is given by the infinitely many times differentiable function of $(s,\rho )$, denoted $\xi ^*(s,\rho )$, solution of equation

$$\begin{aligned} f\left( \xi ^*(s,\rho ),s,\rho \right) = 0 \,\;\;{\Longleftrightarrow }\;\; \left( \xi ^*(s,\rho )\right) ^{s-1} + \rho \, \left( \xi ^*(s,\rho ) - 1\right) = 0 \, . \end{aligned}$$

(A.4)

Taking the partial derivatives of both sides of (A.4) with respect to s and $\rho$, and recalling that for any function $\,c(x) \;{=}\; w(x)^{a(x)}$ with $\,w(x) \;{>}\; 0 \;\, \forall \, x \in {{\mathbb {R}}}$, it holds that

$$\begin{aligned} c'(x) \,{=}\, c(x) \left[ \, a'(x) \ln w(x) + a(x) w'(x) / w(x) \, \right] \,, \end{aligned}$$

after simple manipulations we have

$$\begin{aligned}&\frac{\partial }{\partial s} f\left( \xi ^*(s,\rho ),s,\rho \right) \;{=}\; 0 \,{\Longleftrightarrow }\, \frac{\partial \xi ^*}{\partial s}(s,\rho ) \,{=}\, \frac{ - \left( \xi ^*(s,\rho )\right) ^{s-1} \, \ln \, \xi ^*(s,\rho ) }{ (s-1) \left( \xi ^*(s,\rho )\right) ^{s-2} + \rho } \, , \qquad \quad \end{aligned}$$

(A.5)

$$\begin{aligned}&\frac{\partial }{\partial \rho } f\left( \xi ^*(s,\rho ),s,\rho \right) \;{=}\; 0 \,{\Longleftrightarrow }\, \frac{\partial \xi ^*}{\partial \rho }(s,\rho ) \,\;{=}\;\, \frac{ 1 - \xi ^*(s,\rho ) }{ \;\, (s-1) \left( \xi ^*(s,\rho )\right) ^{s-2} + \rho \;} \, . \qquad \quad \end{aligned}$$

(A.6)

Since $\,(s,\rho ) \in S \,\;{\Longrightarrow }\;\, \rho > 0, \, \xi ^*(s,\rho ) \in (0,1)$, then both the numerators in the definitions of $\partial \xi ^* / \partial s$ and $\partial \xi ^* / \partial \rho$ in (A.5)–(A.6) are positive quantities. The denominator in (A.5)–(A.6) is also clearly positive for $s \ge 1$; for $s<1$ it is positive for

$$\begin{aligned} \left( \xi ^*(s,\rho )\right) ^{2-s} \,{>}\;\, (1-s)/\rho \,\;{\Longleftarrow }\; \left( \xi ^*(s,\rho )\right) ^{2-s} \,{>}\;\, (1-s)/{\bar{\rho }}(s) \,\;{=}\;\, (s/2) \, \left( {\bar{\xi }}(s)\right) ^{2-s}\,, \end{aligned}$$

which is always verified since $\,\xi ^*(s,\rho ) \;{>}\; {\bar{\xi }}(s)$ for $s<1$. This proves (A.3). $\square$

Corollary A.1

Under the setting of Proposition A.1, let $(s,\rho ) \in S$. Then, the Newton–Raphson iterative scheme applied to the solution of $h(\xi ;s,\rho ) = 0$, namely

$$\begin{aligned} \xi _{k+1} \,\;{=}\;\,\, \xi _k - \frac{h\left( \xi _k;s,\rho \right) }{h'\left( \xi _k;s,\rho \right) } \,\;{=}\;\,\, \frac{ \rho \;{+}\; (s-2) \, \xi _k^{s-1} }{ \rho \;{+}\; (s-1) \, \xi _k^{s-2} } \, , \;\;\;\; k \;{=}\; 1,2,\ldots , \end{aligned}$$

(A.7)

converges to $\xi ^*(s,\rho )$ if the initial iterate $\xi _0$ is chosen, dependently on $s,\rho$, as follows:

$$\begin{aligned} \begin{array}{llcl} s \;{\le }\; 1, &{} \rho \;{>}\; {\bar{\rho }}(s) &{} {\Longrightarrow } &{} \xi _0 \;\;{\in }\;\, \big [ {\bar{\xi }}(s),1 \big ] \, , \\ s \;{\in }\; (1,2), &{} \rho \;{>}\; 2-s &{} {\Longrightarrow } &{} \xi _0 \;\;{\in }\;\, (0,1] \, , \\ s \;{\in }\; (1,2), &{} \rho \;{\le }\; 2-s &{} {\Longrightarrow } &{} \xi _0 \;\;{\in }\;\, \big ( 0,\xi ^*(s,\rho ) \big ] \, , \\ s \;{\ge }\; 2 &{} \rho \;{>}\; 0 &{} {\Longrightarrow } &{} \xi _0 \;\;{\in }\;\, [0,1] \, . \end{array} \end{aligned}$$

(A.8)

Hence, based on (A.3), $\xi _0$ for computing $\xi ^*(s,\rho )$ by (A.7) can be chosen among the solutions $\xi ^*({\tilde{s}},\rho )$ for different s values according to the following strategy:

$$\begin{aligned} \begin{array}{llcl} s \;{\le }\; 1, &{} \rho \;{>}\; {\bar{\rho }}(s) &{} {\Longrightarrow } &{} \xi _0 \;\;{=}\;\, \xi ^*({\tilde{s}},\rho ), \;\; {\tilde{s}} \;{\ge }\; s \, , \\ s \;{\in }\; (1,2), &{} \rho \;{>}\; 2-s &{} {\Longrightarrow } &{} \xi _0 \;\;{=}\;\, \xi ^*({\tilde{s}},\rho ), \;\; {\tilde{s}} \;{>}\; 1 \, , \\ s \;{\in }\; (1,2), &{} \rho \;{\le }\; 2-s &{} {\Longrightarrow } &{} \xi _0 \;\;{=}\;\, \xi ^*({\tilde{s}},\rho ), \;\; {\tilde{s}} \;{\in }\; (1,s] \, , \\ s \;{\ge }\; 2 &{} \rho \;{>}\; 0 &{} {\Longrightarrow } &{} \xi _0 \;\;{=}\;\, \xi ^*({\tilde{s}},\rho ), \;\; {\tilde{s}} \;{>}\; 0 \, . \end{array} \end{aligned}$$

(A.9)

Proof

According to Proposition A.1, for any given pair $(s,\rho ) \in S$, the shrinkage coefficient $\xi ^*(s,\rho )$ is given by the unique root of nonlinear equation $h(\xi ;s,\rho ) = 0$ in the open interval $({\bar{\xi }}(s),1)$ for $s\le 1$, (0, 1) for $s>1$. Convergence of the Newton–Raphson method applied to finding such roots depends on the initial guess as well as on the first- and second-order derivatives of the function h, which read

$$\begin{aligned} h'(\xi ;s,\rho ) \,\;{=}\;\, (s-1) \, \xi ^{s-2} + \rho \, , \quad \; h''(\xi ;s,\rho ) \,\;{=}\;\, (s-1)(s-2) \, \xi ^{s-3} \, . \end{aligned}$$

In particular, it is immediate to verify that $h \;{\in }\, C^{\infty }\left( (0,1]\right)$ for any $(s,\rho ) \in S$ and that, depending on s, the function h and its derivatives $h',h''$ satisfy

$$\begin{aligned}&\left\{ \begin{array}{l} s \;{<}\; 1 \\ \rho \;{>}\; {\bar{\rho }}(s) \end{array} \right. \nonumber \\&\quad {\Rightarrow }\left\{ \begin{array}{l} h \,{\in }\, C^{\infty } \left( \left[ {\bar{\xi }}(s),1\right] \right) , \;\, h\!\left( {\bar{\xi }}(s)\right) \,{=}\, \frac{s}{2-s}\left( {\bar{\rho }}(s)-\rho \right) \,{<}\, 0, \;\, h(1) \,{=}\, 1, \quad \\ h'\!\left( {\bar{\xi }}(s)\right) {=}\, \rho \,{+}\, \frac{s}{2}{\bar{\rho }}(s), \;\,h'(1) \,{=}\, \rho \,{+}\, s \,{-}\,\! 1, \;\,h'(\xi ),h''(\xi ) \,{>}\, 0 \; \forall \xi \,{\in }\, [{\bar{\xi }}(s),1], \end{array} \right. \quad \nonumber \\&\left\{ \begin{array}{l} s \;{\in }\, (1,\!2) \\ \rho \;{>}\; 0 \end{array} \right. \nonumber \\&\quad {\Rightarrow }\left\{ \begin{array}{l} h \,{\in }\, C^0\big ([0,1]\big ) \,{\cap }\, C^{\infty }\big ((0,1]\big ), \;\, h(0) \,{=}\, {-}\rho \,{<}\, 0, \;\, h(1) \,{=}\, 1, \;\, h'(0^+) \,{=}\,\! {+}\infty , \quad \\ h'(1) \,{=}\, \rho \,{+}\, s \,{-}\,\! 1, \;\, h'(\xi ) \,{>}\, 0 \;\, \forall \xi \,{\in }\, (0,1], \;\, h''(\xi ) \,{<}\, 0 \;\, \forall \xi \,{\in }\, (0,1], \end{array} \right. \quad \nonumber \\&\left\{ \begin{array}{l} s \;{>}\, 2 \\ \rho \;\!{>}\; 0 \end{array} \right. \nonumber \\&\quad {\Rightarrow }\left\{ \begin{array}{l} h \,{\in }\, C^1\big ([0,1]\big ) \,{\cap }\, C^{\infty }\big ((0,1]\big ), \;\, h(0) \,{=}\, {-}\rho \,{<}\, 0, \;\, h(1) \,{=}\, 1, \;\, h'(0) \,{=}\, \rho , \quad \\ h'(1) \,{=}\, \rho \,{+}\, s \,{-}\,\! 1, \;\, h'(\xi ) \,{>}\, 0 \;\, \forall \xi \,{\in }\, [0,1], \;\, h''(\xi ) \,{>}\, 0 \;\, \forall \xi \,{\in }\, (0,1], \end{array} \right. \end{aligned}$$

Properties of $\,h,h',h''$ for $s\,{<}\,1 \,{\wedge }\, \rho \,{>}\,{\bar{\rho }}(s)$ indicate that in this case the iterative scheme (A.7) is guaranteed to converge to the unique root $\xi ^*(s,\rho )$ of $h(\xi ;s,\rho ) = 0$ in the open interval $({\bar{\xi }}(s),1)$ if $\xi _0 \in \big [\xi ^*(s,\rho ),1]$. But, since it can be proved (we omit the proof for shortness) that setting $\xi _0 = {\bar{\xi }}(s)$ the first iteration of (A.7) yields $\xi _1 \in \big [\xi ^*(s,\rho ),1\big )$, then (A.7) converges under the milder condition $\xi _0 \in [{\bar{\xi }}(s),1]$.

Properties of $\,h,h',h''$ for $s\,{\in }\,(1,2) \,{\wedge }\, \rho \,{>}\,0$ lead to the convergence condition $\xi _0 \,{\in }\, \big (0,\xi ^*(s,\rho )\big ]$ for (A.7), with $\xi _0 \,{=}\, 0$ excluded since $h'(0^+) \,{=}\; {+}\infty \;{\Longrightarrow }\; \xi _k \,{=}\, 0 \;\, \forall k$. It can be proved that setting $\xi _0 = 1$, then (A.7) yields $\xi _1 \in \big (0,\xi ^*(s,\rho )\big ]$ if and only if $\rho \,{>}\, 2-s$. Hence, for $\rho \,{>}\, 2-s$ we have the milder convergence condition $\xi _0 \,{\in }\, (0,1]$.

Properties of $\,h,h',h''$ for $s\,{>}\,2 \;{\wedge }\; \rho \,{>}\,0$ indicate that (A.7) is guaranteed to converge to the desired $\,\xi ^*(s,\rho )\,$ if $\,\xi _0 \,{\in }\, \big [\xi ^*(s,\rho ),1]$. However, as it is easy to prove that $\xi _0\,{=}\,0$ in (A.7) yields $\xi _1\,{=}\,1$, then (A.7) converges for any $\xi _0\,{\in }\,[0,1]$ in this case.

This completes the proof of (A.8), whereas (A.9) comes easily from (A.8) and from the monotonicity properties of function h given in (A.3).$\square$

Proof of Proposition 4.2

Based on Proposition 2.1, the function T is infinitely differentiable. Hence, we impose a first order optimality condition on T with respect to $\sigma$:

$$\begin{aligned} \frac{\partial T}{\partial \sigma } = (n-d{+}1)/\sigma -s(\phi (s))^{-s}\Vert x\Vert _s^s(1/\sigma )^{s+1}+(d{-}1)m^{-s}\sigma ^{s-1}=0\,.\qquad \end{aligned}$$

(A.10)

Equation (A.10) admits the following closed-form solution:

$$\begin{aligned} \sigma ^{\text {MAP}}(s) = m\,\left[ \left( -\alpha +\sqrt{\alpha ^2+4\beta m^{-s} (\sigma ^{\text {ML}})^{s}}\right) /(2\beta )\right] ^{1/s}\,, \end{aligned}$$

(A.11)

where the expression of $\sigma ^{\text {ML}}(s)$ is given in (2.7). Note that (A.11) can be further manipulated so as to give

$$\begin{aligned}&\sigma ^{\text {MAP}}(s) = m\alpha ^{1/s}[(-1+\sqrt{1+c(s)})/(2\beta )]^{1/s}\,,\nonumber \\&\quad \text {with}\quad c(s):=4(\beta /\alpha ^2) (m\phi (s))^{-s}(s/n)\Vert x\Vert _s^s\,. \end{aligned}$$

(A.12)

It is easy to verify that the second derivative of T with respect to $\sigma$ computed at $\sigma ^{\text {MAP}}(s)$ is strictly positive, hence the stationary point in (A.10) is a minimum. Finally, plugging (A.12) into the expression of function T in (4.13), after a few simple manipulations, the s estimation problem takes the form:

Based on Proposition 2.1, it is easy to verify that $f^{\text {MAP}}\in C^{\infty }(B)$ and that the following limits hold:

$$\begin{aligned} \lim _{s\rightarrow +\infty } f_1(s)= n\ln \sqrt{3} - \ln d \in {{\mathbb {R}}}\,,\quad \lim _{s\rightarrow +\infty } \Vert x\Vert _s/(m\phi (s)) = \Vert x\Vert _{\infty } / (m\sqrt{3}) =: \nu \,. \end{aligned}$$

Depending on $\nu$, the three following scenarios arise:

• (a) $\nu =1$. Based on the properties of function $\phi$ and $\Vert \cdot \Vert _{\infty }$, $\nu$ tends to 1 from the right. If $\nu \rightarrow 1^+$ slower than $s\rightarrow +\infty$, case (a) leads to case (c), otherwise, $c(s)\sim O(s)$ and $f_2(s)$ vanish so that $f^{\text {MAP}}(s)$ tends to a finite value when s goes to $+\infty$.

• (b) $\nu < 1$.

  For large s, the first term in $f_2(s)$ vanishes, while for the logarithmic term we consider the Maclaurin series expansion of $\sqrt{1+c(s)}$, so that:

  $$\begin{aligned}&\lim _{s\rightarrow +\infty } f_2(s) = \lim _{s\rightarrow +\infty } (n\alpha /s)[\ln (\alpha /(2\beta ))+\ln (-1+1+(1/2)c(s)+o(c(s)))]\\&\quad =\lim _{s\rightarrow +\infty } (n\alpha /s) \ln ((1/2)c(s)) = \lim _{s\rightarrow +\infty } (n\alpha /s)\ln (2(\beta /\alpha ^2)(s/n)) \nonumber \\&\qquad + (n\alpha )\ln \nu = n\alpha \ln \nu \in {{\mathbb {R}}}\,. \end{aligned}$$

  Therefore, function $f^{\text {MAP}}(s)$ tends to a finite value when $s\rightarrow +\infty$.

• (c) $\nu > 1$. For large s, we have $(n\alpha /s)\ln (-1+\sqrt{1+c(s)})\sim (n\alpha /s)\ln \sqrt{c(s)}\rightarrow k\in {{\mathbb {R}}}_{+}$ as $s\rightarrow + \infty$. Hence:

  $$\begin{aligned} \lim _{s\rightarrow +\infty } f_2(s) = \lim _{s\rightarrow +\infty } (n\alpha /s)\sqrt{1+c(s)} + k = + \infty \longrightarrow \lim _{s\rightarrow +\infty }f^{\text {MAP}}(s)=+\infty . \end{aligned}$$

We conclude that $f^{\text {MAP}}$ admits either a finite minimizer, that can be plugged into (A.11) thus returning $\sigma ^{\text {MAP}}$, or $s^{\text {MAP}}=+\infty$, i.e. the samples in x are drawn from a uniform distribution with $\sigma ^{\text {MAP}}=m\nu$. $\square$