Generalized 2-Microlocal Frontier Prescription

Abstract

The characterization of local regularity is a fundamental issue in signal and image processing, since it contains relevant information about the underlying systems. The 2-microlocal frontier, a monotone concave downward curve in $\mathbb {R}^2$, provides a useful way to classify pointwise singularity. In this paper we characterize all functions whose 2-microlocal frontier at a given point $x_0$ is a given line. Further, for a general concave downward curve, we obtain a large family of functions (or distributions) for which the 2-microlocal frontier is the given curve. This family contains—as special cases—the constructions given in Meyer (CRM monograph series, 1998), Guiheneuf et al. (ACHA 5(4):487–492, 1998) and Lévy Véhel et al. (Proc Symp Pure Math 72:319–334, 2004). Moreover, following Lévy Véhel et al. (2004), we extend our results to the prescription on a countable dense set.

Appendix

In this Appendix we complete the technical details required to show that the functions obtained in [22] and [21] can be obtained by appropriately choosing the coefficients as indicated in Sect. 2.3.

1.1 Computation for the Choice of Coefficients of Meyer

Y. Meyer chooses a countable dense set $\{s'_m\}$ in $\mathbb {R}$ and defines

$$\begin{aligned} p_m= - \frac{dA}{dt}(s'_m)\quad \quad and \quad \quad \tau _m=A(s'_m)-\frac{dA}{dt}(s'_m)s'_m. \end{aligned}$$

(48)

He proves that the function f whose wavelet coefficients are given by

$$\begin{aligned} c_{j,k}= 2^{-j\tau _m} \quad \text {if} \quad j\in {\Lambda }_m ~\text {and} ~k_j=\left[ 2^{jp_m}\right] , \end{aligned}$$

(49)

with ${\Lambda }_m$ as in (9), and 0 otherwise, has $s= A(s')$ as its 2-microlocal frontier at $x_0=0$.

Using Theorem 2.1, we obtain the function f proposed by Y. Meyer, by selecting on one hand $\mathscr {C}_{j,k}=1$ and on the other, $\lambda _{j,k}$ such that

$$\begin{aligned} \lambda _{j,k}= \left\{ \begin{array}{lll} \frac{1+\left| k\right| }{2^{jp_m}}&{}\quad \text { if }&{}\quad j\in {\Lambda }_m \text { and } k=\left[ 2^{jp_m}\right] \\ \\ 1&{}\quad \text { otherwise. }&{} \end{array} \right. \end{aligned}$$

To see that for $j\in {\Lambda }_m$ and $k=\left[ 2^{jp_m}\right]$,

$$\begin{aligned} \mathscr {C}_{j,k} \inf _{\sigma \in \mathbb {R}} \left\{ ~2^{-jS(\sigma )}\left( \frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{S (\sigma )-\sigma }\right\} \end{aligned}$$

are the coefficients given in (49), we replace $\lambda _{j,k}= \frac{1+\left| k\right| }{2^{jp_m}}$ and $\mathscr {C}_{j,k}=1$. We have

$$\begin{aligned} \mathscr {C}_{j,k} \inf _{\sigma \in \mathbb {R}}\left\{ ~2^{-jS(\sigma )}\left( \frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{S (\sigma )-\sigma }\right\} = \inf _{\sigma \in \mathbb {R}} \left\{ ~2^{-jS(\sigma )-p_m(S (\sigma )-\sigma )}\right\} . \end{aligned}$$

To compute the infimum, recall that $A(s')$ is downward concave and differentiable. Hence the tangent line at $s'_m$ is on top of the graph of $A(s')$, e.g., for all $s'$,

$$\begin{aligned} s=A(s')\le A(s'_m)+\frac{dA}{dt}(s'_m)(s'-s'_m). \end{aligned}$$

Hence, using (48) we have that for all $s'$ $s+p_ms'\le \tau _m .$ Consequently, since $S(\sigma )=s$ if and only if $s=A(\sigma -s)$ we have

$$\begin{aligned} 2^{-j\tau _m}\le 2^{-j(s+p_m(\sigma -s))}= 2^{-j(S(\sigma )-p_m(S(\sigma )-\sigma ))}\quad \forall \sigma \in \mathbb {R}. \end{aligned}$$

(50)

Further, the equality $2^{-j\tau _m}= 2^{-j(S(\sigma )-p_m(S(\sigma )-\sigma ))}$ is true if $\sigma =\sigma _m=s_m+s'_m$ with $s_m=A(s'_m)$.

Therefore,

$$\begin{aligned} \inf _{\sigma \in \mathbb {R}} \left\{ ~2^{-jS(\sigma )-p_m(S (\sigma )-\sigma )}\right\} =2^{-j\tau _m}=c_{j,k}.\end{aligned}$$

For all indexes j, k not yet considered, we define $c_{j,k}=0$, and so they clearly satisfy (12) for $\mathscr {C}_{j,k}=1$ and $\lambda _{j,k}=1$.

With these choices, the sequences $\mathscr {C}_{j,k}$ and $\lambda _{j,k}$ satisfy (i) y (ii) of Theorem  2.1 since,

$$\begin{aligned} \mathscr {C}_{j,k}=1\; \quad \text {and} \quad 1\le \lambda _{j,k}\le 2 \end{aligned}$$

for j, k such that $j\ge 0$ and $| k|<2^j$.

1.2 Computation for the Choice of Coefficients of Lévy Véhel and Seuret

To proof that the function given in [21] is in fact a special case of our formula is rather delicate.

We will exhibit here only the computation for the case that g is a line and strictly increasing. The non-linear case can be found in section 4.3.3 of [23].

In [21], the function whose 2-microlocal frontier at $x_0=0$ is $g(s')=Ms' +d$, with $0<M \le 1$ and $d>0$, is given by its wavelet coefficients $c_{j,k}$. These are defined as 0 if $k<0$ or $j=0$ and for $j>0$ and $k \ge 0$ as

$$\begin{aligned} c_{j,k} = \left\{ \begin{array}{lll} 2^{-j^2} &{}\quad \text { if }&{}\quad k=[2^{j(1-M)}] \,\text { and } j\le d\\ 2^{-jd} &{}\quad \text { if }&{}\quad k=[2^{j(1-M)}]\,\text { and } j> d \\ 2^{-j^2} &{}\quad \text { if }&{}\quad k\ne [2^{j(1-M)}]. \end{array} \right. \end{aligned}$$

(51)

The Legendre transform of g is

$$\begin{aligned}g^*(\rho )=\inf _{s'}\{\rho s'-Ms'-d\}=\left\{ \begin{array}{ccc} -d &{}\quad \text { if }&{}\quad \rho =M\\ -\infty &{}\quad \text { if }&{}\quad \rho \ne M. \end{array} \right. \end{aligned}$$

In the $(\sigma ,s)$-plane, the curve $s=g(s')$ is

$$\begin{aligned} S(\sigma )=d+ \frac{1-M}{M}(d-\sigma ). \end{aligned}$$

We choose the sequences $\mathscr {C}_{j,k}$ and $\lambda _{j,k}$ in (31) and (32) as:

$$\begin{aligned} \lambda _{j,k}=\frac{1+|k|}{2^{j(1-M)}} \;\text { and }\; \mathscr {C}_{j,k}=\left\{ \begin{array}{lll} 1 &{}\quad \text { if }&{}\quad k=[2^{j(1-M)}]\,\text { and } j> d \\ 2^{-j^2+jd} &{}\quad \text { if }&{}\quad k=[2^{j(1-M)}] \,\text { and } j\le d \\ 2^{-j^2+jd} &{}\quad \text { if }&{}\quad k\ne [2^{j(1-M)}]. \end{array} \right. \end{aligned}$$

(52)

Replacing $\lambda _{j,k}$ and $S(\sigma )$ in the equation

$$\begin{aligned} |c_{j,k}|=\mathscr {C}_{j,k}\;. \displaystyle {\inf _{\sigma \in \mathbb {R}} \left\{ ~2^{-jS(\sigma )}\left( \frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{S (\sigma )-\sigma }\right\} }\end{aligned}$$

we obtain $|c_{j,k}|=\mathscr {C}_{j,k}\;2^{-jd}.$ Therefore, if in (12) we consider equality and insert $\mathscr {C}_{j,k}$ given above by (52), we obtain exactly the wavelet coefficients (12) for $j>0$ and $k\ge 0$. For $k<0$ or $j=0$ we define $|c_{j,k}|=0$, which also satisfies (12).

It only remains to check that the sequences $\mathscr {C}_{j,k}$ and $\lambda _{j,k}$ satisfy (i) and (ii) of Theorem  2.1. In this case, the index set I is

$$\begin{aligned} I = \{(j,k): j\in \Lambda _m \text { and } |k_j|=[2^{jr_m}]\}, \end{aligned}$$

for $r_m= 1-M$ for all m, where $1-M$ is the only element in the range of $\frac{S'(\sigma )}{S'(\sigma )-1}$.

If $(j,k_j)\in I$ we have that $k_j =[2^{j(1-M)}]$ and hence $\mathscr {C}_{j,k_j}=1$ or $\mathscr {C}_{j,k_j}= 2^{-j^2+jd}$ and $1\le \lambda _{j,k_j}\le 2$, and hence the conditions are satisfied.

If $k\ne [2^{j(1-M)}]$ we have $\mathscr {C}_{j,k}=2^{-j^2+jd}$. Further $\lambda _{j,k}$ is bounded by

$$\begin{aligned} \frac{1}{2^{j(1-M)}}\le \frac{1+| k|}{2^{j(1-M)}}\le \frac{2^j}{2^{j(1-M)}}=2^{jM},\end{aligned}$$

and so $\frac{\log _2(\lambda _{j,k})}{j}$ is bounded.

This proves that for $(j,k_j)\in I$ or $(j,k_j)\notin I$, (i) is satisfied, i.e. for any sequence $(k_j)_j$ such that $|k_j|<2^j$,

$$\begin{aligned} \displaystyle {\lim _{\begin{array}{c} j\rightarrow +\infty \end{array}}{\left( \frac{\log _2\left( \mathscr {C}_{j,k_j}\right) }{j} +C \; \frac{\log _2\left( \lambda _{j,k_j}\right) }{j}\right) }\;\le 0}.\end{aligned}$$