Efficient Analysis and Detection of Edges Through Directional Multiscale Representations

Abstract

The analysis and detection of edges and interface boundaries is a fundamental problem in applied mathematics and image processing. In the study of the wave equation, for example, one is interested in the evolution of moving fronts; in image processing and computer vision, the detection and analysis of edges is an essential task for applications such as shape recognition, image enhancement, and classification. Multiscale methods and wavelets have been very successful in this area, due to a combination of useful micro-analytical properties and fast numerical implementations. The continuous wavelet transform in particular has the ability to signal the location of the singularities of functions and distributions through its asymptotic decay at fine scales. However, this approach is unable to provide additional information about the geometry of the singularity set, such as the edge orientation. This limitation can be overcome by using the continuous shearlet transform, an approach combining the analytical power of multiscale analysis and high directional sensitivity. This chapter gives an overview of the microlocal properties of the shearlet transform and illustrates its ability to provide a precise geometric characterization of edges and interface boundaries in images and other multidimensional data. These results provide the theoretical groundwork for innovative applications in problems of edge detection, feature extraction, and geometric separation.

Appendix

We recall some basic facts from Fourier analysis, including the Fourier transform of distributions. We refer the reader to [13, 14] for additional details.

4.1.1 The Fourier transform

$L^{1}(\mathbb{R}^{n})$ is the space of Lebesgue integrable function on $\mathbb{R}^{n}$ and $L^{2}(\mathbb{R}^{n})$ the Hilbert space of square Lebesgue integrable function on $\mathbb{R}^{n}$ endowed with the inner product $\langle f,g\rangle =\int _{\mathbb{R}^{n}}f\overline{g}$. The Schwartz space $\mathcal{S}(\mathbb{R}^{n})$ consists of those functions in $C^{\infty }(\mathbb{R}^{n})$ which, together with all their derivatives, vanish at infinity faster than any power of | x | . That is,

$$\displaystyle{\mathcal{S}(\mathbb{R}^{n}) =\{ f \in C^{\infty }(\mathbb{R}^{n}):\sup _{ x\in \mathbb{R}^{n}}(1 + \vert x\vert )^{N}\vert \partial ^{\alpha }f(x)\vert < \infty,\mbox{ for all }N,\alpha \},}$$

where N is any nonnegative integer and $\alpha = (\alpha _{1},\ldots,\alpha _{n}) \in \mathbb{N}^{n}$ is any multi-index.

Definition 4.11.

The Fourier transform is the operator $\mathcal{F}$ mapping a function $f \in L^{1}(\mathbb{R}^{n})$ into $\mathcal{F}f =\hat{ f}$ defined by

$$\displaystyle{\hat{f}(\xi ) =\int _{\mathbb{R}^{n}}f(x)e^{-2\pi ix\cdot \xi }dx.}$$

The inverse Fourier transform is the operator $\mathcal{F}^{-1}$ mapping a function $g \in L^{1}(\mathbb{R}^{n})$ into $\mathcal{F}^{-1}g =\check{ g}$, where

$$\displaystyle{\check{g}(x) =\hat{ g}(-x) =\int _{\mathbb{R}^{n}}g(\xi )e^{2\pi ix\cdot \xi }d\xi.}$$

It is a fact that $g = (\hat{g})^{\vee }$ for any function $g \in L^{1}(\mathbb{R}^{n})$ with $\hat{g} \in L^{1}(\mathbb{R}^{n})$.

The Fourier transform is a bijection of $\mathcal{S}$ onto itself and can be extended via an appropriate limit to a unitary map from L ² onto itself. Under this extension, the Fourier inversion theorem is valid and, for $f,g \in L^{2}(\mathbb{R}^{n})$, the Plancherel formula holds:

$$\displaystyle{\langle f,g\rangle =\langle \hat{ f},\hat{g}\rangle,}$$

and, in particular,

$$\displaystyle{\|f\|_{2} =\|\hat{ f}\|_{2}.}$$

Among the most important properties of Fourier analysis we recall the following list of results (cf. [13]).

Theorem 4.12.

Let $f,g \in L^{1}(\mathbb{R}^{n})$ . For $y \in \mathbb{R}^{n}$ , let T _y f(x) = f(x − y) and, for $M \in GL_{n}(\mathbb{C})$ , let $D_{M}f(x) = \vert \det M\vert ^{-1/2}f(M^{-1}x)$.

• (i) $(T_{y}f)^{\wedge }(\xi ) = e^{-2\pi i\xi \cdot y}\hat{f}(\xi )$ and $(D_{M}f)^{\wedge }(\xi ) = D_{N}\hat{f}(\xi ),$ where N = (M ^∗ ) ⁻¹ .

• (ii) $(f {\ast} g)^{\wedge } =\hat{ f}\,\hat{g}.$

• (iii) If x ^α f ∈ L ¹ for |α|≤ k, then $\partial ^{\alpha }\hat{f} = \left ((-2\pi ix)^{\alpha }f\right )^{\wedge }.$

• (iv) If f ∈ C ^k , $\partial ^{\alpha }\hat{f} \in L^{1}$ , for |α|≤ k, and $\partial ^{\alpha }\hat{f} \in C_{0}$ , for |α|≤ k − 1, then $(\partial ^{\alpha }\hat{f})^{\wedge }(\xi ) = (2\pi i\xi )^{\alpha }\hat{f}(\xi ).$

• (v) $\mathcal{F}\left (L^{1}(\mathbb{R}^{n})\right ) \subset C_{0}(\mathbb{R}^{n})$ .

The following proposition is a simple application of the Fourier transform showing that regularity on $\mathbb{R}^{n}$ implies decay in the Fourier domain.

Proposition 4.13.

Suppose that $\psi \in L^{2}(\mathbb{R}^{n})$ is such that $\hat{\psi }\in C_{c}^{\infty }(R)$ , where $R =\mathop{ \mathrm{supp}}\nolimits \hat{\psi } \subset \mathbb{R}^{n}$ . Then, for each $k \in \mathbb{N}$ , there is a constant C _k > 0 such that, for any $x \in \mathbb{R}^{n}$ , we have

$$\displaystyle{ \vert \psi (x)\vert \leq C_{k}\,(1 + \vert x\vert ^{2})^{-k}. }$$

In particular, $C_{k} = k\,m(R)\,{\bigl (\Vert \hat{\psi }\Vert _{\infty } +\Vert \bigtriangleup ^{k}\hat{\psi }\Vert _{\infty }\bigr )}$ , where $\bigtriangleup =\sum _{ i=1}^{n} \frac{\partial ^{2}} {\partial \xi _{i}^{2}}$ is the Fourier-domain Laplacian operator and m(R) is the Lebesgue measure of R.

Proof.

From the definition of the Fourier transform, it follows that, for every $x \in \mathbb{R}^{n}$,

$$\displaystyle{ \vert \psi (x)\vert \leq m(R)\,\Vert \hat{\psi }\Vert _{\infty }. }$$

(4.19)

An integration by parts shows that

$$\displaystyle{\int _{R}\bigtriangleup \hat{\psi }(\xi )\,e^{2\pi i\langle \xi,x\rangle }\,d\xi = -(2\pi )^{2}\,\vert x\vert ^{2}\,\psi (x).}$$

Thus, for every $x \in \mathbb{R}^{n}$,

$$\displaystyle{ (2\pi \,\vert x\vert )^{2k}\,\vert \psi (x)\vert \leq m(R)\,\Vert \bigtriangleup ^{k}\hat{\psi }\Vert _{ \infty }. }$$

(4.20)

Using (4.19) and (4.20), we have

$$\displaystyle{ {\bigl (1 + (2\pi \,\vert x\vert )^{2k}\bigr )}\,\vert \psi (x)\vert \leq m(R)\,{\bigl (\Vert \hat{\psi }\Vert _{ \infty } +\Vert \bigtriangleup ^{k}\hat{\psi }\Vert _{ \infty }\bigr )}. }$$

(4.21)

Observe that, for each $k \in \mathbb{N}$,

$$\displaystyle{(1 + \vert x\vert ^{2})^{k} \leq {\bigl ( 1 + (2\pi )^{2}\,\vert x\vert ^{2}\bigr )}^{k} \leq k\,{\bigl (1 + (2\pi \,\vert x\vert )^{2k}\bigr )}.}$$

Using this last inequality and (4.21), we have that for each $x \in \mathbb{R}^{n}$

$$\displaystyle{ \vert \psi (x)\vert \leq k\,m(R)\,(1 + \vert x\vert ^{2})^{-k}\,{\bigl (\Vert \hat{\psi }\Vert _{ \infty } +\Vert \bigtriangleup ^{k}\hat{\psi }\Vert _{ \infty }\bigr )}. }$$

 □ 

Under the same assumptions of Proposition 4.13, we can derive a similar estimate valid for ψ _M, t  = T _t D _M ψ. Using a change of variables in the last step of the proof above, we have that for all k > 0 there is a C _k  > 0 such that

$$\displaystyle{\vert \psi _{M,t}(x)\vert \leq C_{k}\,\vert \det M\vert ^{-\frac{1} {2} }\,(1 + \vert M^{-1}(x - t)\vert ^{2})^{-k}.}$$

4.1.2 Distributions and the Fourier transform of distributions

The space $\mathcal{D}(\mathbb{R}^{n})$ of test functions is the space of all $C^{\infty }$ functions whose support is compact. A sequence {ϕ _j } in $\mathcal{D}(\mathbb{R}^{n})$ converges in $\mathcal{D}$ to ϕ if the supports of all ϕ _j are contained in a fixed compact subset of $\mathbb{R}^{n}$ and if $\partial ^{\alpha }\phi _{j} \rightarrow \partial ^{\alpha }\phi$ uniformly for all multi-indices α.

Definition 4.14.

A distribution is a continuous linear functional on $\mathcal{D}$ and the space of distributions is denoted by $\mathcal{D}'$. We impose the weak^∗ topology on $\mathcal{D}'$, that is, the topology of pointwise convergence on $\mathcal{D}$.

If $F \in \mathcal{D}'(\mathbb{R}^{n})$ and $\phi \in \mathcal{D}(\mathbb{R}^{n})$, we denote the value of F at ϕ by F(ϕ) or $\langle F,\phi \rangle$. The latter notation conflicts with the notation of inner product but its meaning will be clear by the context.

Given two distribution F and G, we say that F = G if $\langle F,\phi \rangle =\langle G,\phi \rangle$, for all $\phi \in \mathcal{D}(\mathbb{R}^{n})$.

Example 4.15.

Every $f \in L_{loc}^{1}(\mathbb{R}^{n})$ defines a distribution by $\phi \in \mathcal{S}(\mathbb{R}^{n}) \rightarrow \int _{\mathbb{R}^{n}}f\phi$.

Example 4.16.

The Dirac’s impulse δ is defined by $\delta (\phi ) =\langle \delta,\phi \rangle =\phi (0)$, $\phi \in \mathcal{S}(\mathbb{R}^{n})$. This is an example of a distribution which is not a function.

Example 4.17.

The distribution $\mathop{\mathrm{pv}}\nolimits (\frac{1} {x})$ is defined by

$$\displaystyle{\langle \mathop{\mathrm{pv}}\nolimits (\tfrac{1} {x}),\phi \rangle = P.V.\int _{\mathbb{R}} \frac{\phi (x)} {x} \,dx =\lim _{\varepsilon \rightarrow 0}\int _{-\varepsilon }^{\varepsilon }\frac{\phi (x)} {x} \,dx,}$$

for $\phi \in \mathcal{S}(\mathbb{R})$, where P.V. is the principal value of the integral.

There is a general procedure for extending many linear operations from functions to distributions.

• Differentiation. For any $F \in \mathcal{D}'(\mathbb{R}^{n})$, the derivatives $\partial ^{\alpha }F \in \mathcal{D}'(\mathbb{R}^{n})$ are given by

  $$\displaystyle{\langle \partial ^{\alpha }F,\phi \rangle = (-1)^{\vert \alpha \vert }\langle F,\partial ^{\alpha }\phi \rangle.}$$

• Multiplication by a smooth function. Given $g \in C^{\infty }(\mathbb{R}^{n})$, for any $F \in \mathcal{D}'(\mathbb{R}^{n})$, the product $gF \in \mathcal{D}'(\mathbb{R}^{n})$ is given by

  $$\displaystyle{\langle gF,\phi \rangle =\langle F,g\phi \rangle.}$$

• Convolution. Let $g \in C^{\infty }(\mathbb{R}^{n})$. For any $F \in \mathcal{D}'(\mathbb{R}^{n})$, the convolution $F {\ast} g \in \mathcal{D}'(\mathbb{R}^{n})$ is given by

  $$\displaystyle{\langle F {\ast} g,\phi \rangle =\langle F,\phi {\ast}\tilde{g}\rangle,}$$

  where $\tilde{g}(x) = g(-x)$.

For example, let H be the one-dimensional Heaviside function, defined by H(x) = 0 if x < 0, H(x) = 1 if x ≥ 0. A direct computation shows that, for any $\phi \in \mathcal{S}(\mathbb{R})$,

$$\displaystyle{\langle H',\phi \rangle = -\langle H,\phi '\rangle = -\int _{0}^{\infty }\phi '(x)\,dx =\phi (0) =\langle \delta,\phi \rangle.}$$

Hence H′ = δ.

The following class of distributions are useful to extend the Fourier transform beyond the realm of classical functions.

Definition 4.18.

A tempered distribution is a continuous linear functional on $\mathcal{S}$ and the space of tempered distributions is denoted by $\mathcal{S}'$. We impose the weak^∗ topology on $\mathcal{S}'$, that is, the topology of pointwise convergence on $\mathcal{S}$.

Example 4.19.

Every $f \in L_{loc}^{1}(\mathbb{R}^{n})$ such that $\int _{\mathbb{R}^{n}}(1 + \vert x\vert )^{N}\,\vert f(x)\vert \,dx < \infty$ defines a tempered distribution by $\phi \rightarrow \int f\phi$.

Example 4.20.

Any distribution with compact support is tempered.

The Fourier transform extends to a continuous linear map from $\mathcal{S}'$ to itself by defining

$$\displaystyle{\langle \hat{F},\phi \rangle =\langle F,\hat{\phi }\rangle,\quad F \in \mathcal{S}',\phi \in \mathcal{S}.}$$

This definition agrees with the classical definition when $F \in L^{1} \cap L^{2}$. Furthermore, it is easy to verify that the basic properties of the Fourier transform continue to hold. In particular, for $F \in \mathcal{S}'(\mathbb{R}^{n})$, we have the following formulas.

• (i) $(T_{y}F)^{\wedge } = e^{-2\pi i\xi \cdot y}\hat{F}$ and $(D_{M}F)^{\wedge } = D_{N}\hat{F},$ where N = (M ^∗)⁻¹.

• (ii) $(F {\ast} g)^{\wedge } =\hat{ F}\,\hat{g},$ for $g \in \mathcal{S}(\mathbb{R}^{n})$.

• (iii) $\partial ^{\alpha }\hat{F} = \left ((-2\pi ix)^{\alpha }F\right )^{\wedge }.$

• (iv) $(\partial ^{\alpha }\hat{F})^{\wedge } = (2\pi i\xi )^{\alpha }\hat{F}.$

Similarly, the inverse Fourier transform is defined on $\mathcal{S}'$ by

$$\displaystyle{\langle \check{F},\phi \rangle =\langle F,\check{\phi }\rangle,\quad F \in \mathcal{S}',\phi \in \mathcal{S},}$$

and, for all $F \in \mathcal{S}'(\mathbb{R}^{n})$, $F = (\check{F})^{\wedge } = (\hat{F})^{\vee }.$

Example 4.21.

For any $\phi \in \mathcal{S}(\mathbb{R})$,

$$\displaystyle{\langle \hat{\delta },\phi \rangle =\langle \delta,\hat{\phi }\rangle =\hat{\phi } (0) =\int _{\mathbb{R}}\phi (x)\,dx =\langle 1,\phi \rangle.}$$

Hence $\hat{\delta }= 1$. It follows that, for any $y \in \mathbb{R}$, the Fourier transform of δ _y  = T _y δ is $\hat{\delta }_{y}(\xi ) = e^{-2\pi iy\xi }$ and, for any $k \in \mathbb{N}$, $\widehat{\delta ^{(k)}}(\xi ) = (2\pi i\xi )^{k}.$

Example 4.22.

We will show that $\widehat{\mathop{\mathrm{sgn}}\nolimits }(\xi ) = \frac{1} {i\pi } \mathop{ \mathrm{pv}}\nolimits (\frac{1} {\xi } )$, where $\mathop{\mathrm{sgn}}\nolimits$ is the signum function, that is defined as $\mathop{\mathrm{sgn}}\nolimits (x) = -1$ of x < 0, $\mathop{\mathrm{sgn}}\nolimits (x) = 0$ of x ≥ 0. In order to derive this result, we consider first the functions f _n defined by $f_{n}(x) = \left \{\begin{array}{@{}l@{\quad }l@{}} -e^{x/n}\quad &\mbox{ if }x < 0 \\ e^{-x/n}\quad &\mbox{ if }x > 0 \end{array} \right.$, where $n \in \mathbb{N}$. An application of Lebesgue Dominated Convergence theorem shows that f _n converges to $f =\mathop{ \mathrm{sgn}}\nolimits$ as $n \rightarrow \infty$ in the sense of tempered distributions, that is, $\langle f_{n},\phi \rangle \rightarrow \langle \mathop{\mathrm{sgn}}\nolimits,\phi \rangle$ as $n \rightarrow \infty$, for all $\phi \in \mathcal{S}(\mathbb{R}).$ A direct computation (note that $f_{n} \in L^{1}(\mathbb{R})$) shows that

$$\displaystyle{\hat{f}_{n}(\xi ) =\int _{ 0}^{\infty }e^{-x/n}e^{-2\pi ix\xi }\,dx -\int _{ -\infty }^{0}e^{x/n}e^{-2\pi ix\xi }\,dx = \frac{1} { \frac{1} {n} + 2\pi i\xi } - \frac{1} { \frac{1} {n} - 2\pi i\xi }.}$$

Finally, we use that fact that if $(F_{n}),F \in \mathcal{S}'$ and $\langle F_{n},\phi \rangle \rightarrow \langle F,\phi \rangle$ for all $\phi \in \mathcal{S},$ then $\langle \hat{F}_{n},\phi \rangle \rightarrow \langle \hat{ F},\phi \rangle$ for all $\phi \in \mathcal{S}.$ Since

$$\displaystyle{\lim _{n\rightarrow \infty }\langle \hat{f}_{n},\phi \rangle = \frac{1} {i\pi } P.V.\int \frac{\phi (\xi )} {\xi } d\xi,}$$

we conclude that $\widehat{\mathop{\mathrm{sgn}}\nolimits }(\xi ) = \frac{1} {i\pi } \mathop{ \mathrm{pv}}\nolimits (\frac{1} {\xi } )$.

Example 4.23.

The one-dimensional Heaviside function H(x) can be written as $H(x) = \frac{1} {2} + \frac{1} {2}\,\mathop{ \mathrm{sgn}}\nolimits (x)$. It follows that $\hat{H}(\xi ) = \frac{1} {2}\delta (\xi ) + \frac{1} {2\pi i}\,\mathop{ \mathrm{pv}}\nolimits (\frac{1} {\xi } ).$

Example 4.24.

Let us consider the two-dimensional Heaviside function $H_{1}(x_{1},x_{2}) =\chi _{x_{1}>0}(x_{1},x_{2})$. Since it can be written as the tensor product $H_{1}(x_{1},x_{2}) = H(x_{1})1(x_{2})$, it follows that $\hat{H}_{1}(\xi _{1},\xi _{2}) = \frac{1} {2}\delta (\xi _{1})\delta (\xi _{2}) + \frac{\delta (\xi _{1})} {2\pi i} \,\mathop{ \mathrm{pv}}\nolimits (\frac{1} {\xi } _{1}).$

4.1.3 Singular support and wavefront set

The notion of singular support is introduced to describe the location where a distribution fails to be smooth. Since a distribution is not defined at a single point, this definition requires to deal with open sets containing the point of interest.

For a distribution F, we say that $x_{0} \in \mathbb{R}^{n}$ is a regular point of F if there exists a function $g \in C^{\infty }(U)$, where $U \subset \mathbb{R}^{n}$ is an open neighborhood of x ₀ and g(x ₀) = 1, such that $gF \in C^{\infty }(U)$. The complement of the set of the regular points of F is called the singular support of F and is denoted by $\mathop{\mathrm{sing}}\nolimits \mathop{\mathrm{supp}}\nolimits (F)$. It is easy to see that the singular support of F is a closed set.

For example, $\mathop{\mathrm{sing}}\nolimits \mathop{\mathrm{supp}}\nolimits (\delta ) =\{ 0\}$. Also $\mathop{\mathrm{sing}}\nolimits \mathop{\mathrm{supp}}\nolimits (\mathop{\mathrm{pv}}\nolimits (\frac{1} {x})) =\{ 0\}$.

Note that the condition $gF \in C^{\infty }(U)$ is equivalent to $(gF)^{\wedge }$ being rapidly decreasing, i.e., for all N > 0 there exists a C _N  > 0 such that

$$\displaystyle{(gF)^{\wedge }(\xi )\vert \leq C_{ N}\,(1 + \vert \xi \vert )^{-N}.}$$

If a function or distribution fails to be smooth, we can look not only for the location of the singularity in space, but also for the orientation of the singularity.

We shall say that a set $\varGamma \in \mathbb{R}^{n}\setminus \{0\}$ is conic if $\xi \in \varGamma$ implies that $\lambda \xi \in \varGamma$ for all $\lambda > 0$. A conic neighborhood of a point is an open conic set containing it. For a distribution F, the point $(x,\xi ) \in \mathbb{R}^{n} \times \mathbb{R}^{n}\setminus \{0\}$ is a regular directed point of F if there exists a function $g \in C^{\infty }(U)$, where $U \subset \mathbb{R}^{n}$ is an open neighborhood of x and g(x) = 1, such that $gF \in C^{\infty }(U)$ and, for al N > 0, there exists a C _N  > 0 such that

$$\displaystyle{\vert (gF)^{\wedge }(\xi )\vert \leq C_{ N}\,(1 + \vert \xi \vert )^{-N},}$$

for all $\xi$ is a conic neighborhood containing the direction $\xi _{0}$. The complement in $\mathbb{R}^{n} \times \mathbb{R}^{n}\setminus \{0\}$ of the set of regular directed points of F is called the wavefront set of F and is denoted by WF(F).

Example 4.25.

Let x = (x′, x″) be a splitting of the coordinates and define the distribution F by

$$\displaystyle{\langle F,\phi \rangle =\int \phi (0,x'')\,dx''.}$$

It is easy to see that $\mathop{\mathrm{sing}}\nolimits \mathop{\mathrm{supp}}\nolimits (F) =\{ (x',x''): x' = 0\}$. To compute the wavefront set, observe that, for $\gamma \in C^{\infty }(U)$, where U is a neighborhood of a point x ₀ = (x ₀′, x ₀″), we have:

$$\displaystyle{\langle \gamma F,\phi \rangle =\int \gamma (0,x'')\,\phi (0,x'')\,dx''.}$$

Thus, $(\gamma F)^{\wedge }(\xi ) =\hat{\gamma } _{0}(\xi '')$, where γ ₀(x″) = γ(0, x″). Since γ ₀ is $C^{\infty }$ and compactly supported, its Fourier transform has rapid decay as a function of $\xi ''$ but is constant as a function of $\xi '$. Hence we conclude that $WF(F) =\{ (0,x'',\xi ',0)\}$.