Implementation of different “Mother Shearlets”¶

The classes, functions and objects in this module make it (comparatively) easy to define custom generating functions which act as “Mother shearlets”.

Warning

Implementing custom “Mother shearlets” is a complex task and thus should only be done if really needed. Proceed with caution.

To implement a custom set of generating functions, all one needs to do is to create a suitable object of the class MotherShearlet and pass it to the constructor of the AlphaTransform.AlphaShearletTransform, via the mother_shearlet parameter.

In a nutshell, in our implementation, a shearlet system is determined by

A low-pass filter which is given by $\phi \otimes \phi$ , where $\phi$ is given by the low_pass_function attribute of class MotherShearlet.
A mother shearlet $\psi$ , which in our case is (in Fourier domain) given by

$\widehat{\psi}(\xi) = \psi_1 (\xi_1) \cdot \psi_2 (\xi_2 / \xi_1).$

Here,
- we call $\psi_1$ the scale-sensitive generating function. It is given by the scale_function` attribute of class MotherShearlet.
- we call $\psi_2$ the direction sensitive generating function. It is given by the direction_function attribute of class MotherShearlet.

See the documentation of MotherShearlet below for more details.

The class `BumpFunction`¶

class MotherShearlets.BumpFunction¶

Each instance of this class encapsulates a specific (one-dimensional) bump function f.

Attributes:

support¶: A tuple (a,b) such that the bump function f modeled by this object satisfies $\mathrm{supp} f \subset (a,b)$ .

large_support¶: A tuple (a,b) such that the bump function f modeled by this object is “large” on $(a,b)$ , i.e., $|f| \geq c > 0$ on $(a,b)$ for a (reasonably large) constant $c>0$ , e.g. $c = \frac{1}{2}$ .

call¶: A callable object calculating f(x) given x. This should be vectorized, i.e., call() should accept numpy arrays (numpy.ndarray) and apply f componentwise.

theano_call¶: This should be a callable object doing the same as call, but working on theano.tensor.dmatrix objects.

Note

Currently, this attribute is not used anywhere and can thus be set arbitrarily (e.g. to None).

The class `MotherShearlet`¶

class MotherShearlets.MotherShearlet¶

Each instance of this class acts as a container for the three (main) components determining an alpha-shearlet system.

The actual mother shearlet $\psi$ is given by

$\psi(\xi) = \psi_1 (\xi_1) \cdot \psi_2 (\xi_2 / \xi_1),$

where:

$\psi_1$ is given by the scale_function attribute,
$\psi_2$ is given by the direction_function attribute.

Finally, the low frequency component of the shearlet system is given by $\phi \otimes \phi$ (tensor product), where $\phi$ is given by the low_pass_function attribute.

Note

All attributes are expected to be of type BumpFunction.

Finally,

if (a,b) = scale_function.large_support, then it is expected that:
1. a,b > 0
2. b/a >= 2.
if (a,b) = direction_function.large_support, it is expected that:
1. b >= 1/2,
2. a <= -1/2.

`BumpFunction`-related helper functions¶

MotherShearlets.translate(f, x_0)¶

This function translates a BumpFunction object f, i.e., it returns $L_{x_0} f$ , where $(L_y f)(x) = f(x - y)$ . The new support is thus $x_0 + \mathrm{supp} f$ .

Parameters:

Parameters:	f (BumpFunction) – The `BumpFunction` object which should be translated. x_0 (float) – The translation parameter.

Return value:

Returns:	A new `BumpFunction` object representing the translated function.

MotherShearlets.scale(f, a)¶

This function scales a BumpFunction object f, i.e., it returns $f \circ (x \mapsto x/a)$ . The new support is thus $a \cdot \mathrm{supp} f$ .

Parameters:

Parameters:	f (BumpFunction) – The `BumpFunction` object which should be translated. a (float) – The scale parameter (positive real number).

Return value:

Returns:	A new `BumpFunction` object representing the scaled function.

MotherShearlets.flip(f)¶

This function “flips” a BumpFunction object f, i.e., it returns $f \circ (x \mapsto -x)$ . The new support is thus $- \mathrm{supp} f$ .

Parameters:

Parameters:	f (BumpFunction) – The `BumpFunction` object which should be flipped.

Return value:

Returns:	A new ‘BumpFunction’ object representing the flipped function.

Meyer mother shearlet implementation¶

The object MotherShearlets.MeyerMotherShearlet is an instance of MotherShearlet.

It uses

meyer_low_pass() as the generating function for the low-pass part,
meyer_scale_function() as the scale-sensitive generating function.
meyer_direction_function() as the direction sensitive generating function.

MotherShearlets.meyer(A)¶

This function implements the function v from the paper Fast Finite Shearlet Transform.

It rises smoothly between 0 and 1. In particular, $v(x) = 0$ for $x \leq 0$ and $v(x) = 1$ for $x \geq 1$ .

Parameters:

Parameters:	A (numpy.ndarray) – The input (matrix with real entries).

Return value:

Returns:	$v(A)$ , i.e. the function `v` applied to each entry of `A`.

MotherShearlets.meyer_scale_function(R)¶

This is an implementation of the function $\tilde{W}$ from the paper Cartoon approximation with α-Curvelets.

The main properties of this function are the following:

$\mathrm{supp} \tilde{W} \subset [1/2, 2]$ .
$\tilde{W} \equiv 1$ on $[3/4, 3/2]$ .

Note

The quotient of the bounds of the interval where $\tilde{W}$ is 1 is precisely 3/2 * 4/3 = 2, so that the function is suitable as a scale_function for the MotherShearlet class.

Parameters:

Parameters:	R (numpy.ndarray) – The input (matrix with real entries).

Return value:

Returns:	$\tilde{W}(R)$ , i.e., $\tilde{W}$ applied componentwise.

MotherShearlets.meyer_low_pass(A)¶

This function uses the meyer() function to construct a bump function $\phi$ with the following properties:

$\mathrm{supp} \phi \subset [-2,2]$ ,

$\phi(x) = 1$ for $x \in (-3/2, 3/2)$ .

Parameters:

Parameters:	A (numpy.ndarray) – The input (matrix with real entries).

Return value:

Returns:	$\phi(A)$ , i.e. the function $\phi$ applied to each entry of `A`. Precisely, $\phi(x) = 4 - 2 * v(\|x\|)$ , where `v` is the function `meyer()`.

MotherShearlets.meyer_direction_function(A)¶

This is an implementation of a direction function f, i.e., it can be used for the attribute direction_function of the class MotherShearlet.

The main properties of this function are the following:

$\mathrm{supp} f \subset (-1,1)$ ,
$f \equiv 1$ on $(-1/2, 1/2)$ ,
f is symmetric.

The implementation uses the meyer() function.

Parameters:

Parameters:	A (numpy.ndarray) – The input (matrix with real entries).

Return value:

Returns:	`f(A)`, i.e. `f` is applied componentwise.

Haeuser mother shearlet implementation¶

The Häuser mother shearlet is the one which is used in the paper Fast Finite Shearlet Transform by S. Häuser and G. Steidl. In our implementation, it is represented by the object MotherShearlets.HaeuserMotherShearlet, which is an instance of MotherShearlet.

It uses

meyer_low_pass() as the generating function for the low-pass part,
haeuser_scale_function() as the scale-sensitive generating function,
haeuser_direction_function() as the direction sensitive generating function.

MotherShearlets.haeuser_scale_function(A)¶

This is an implementation of the function b from the paper Fast Finite Shearlet Transform, but with support restricted to positive numbers, whereas in the paper, the function is symmetric.

This function is supported in [1,4], is “large” on [2,3] and satisfies $\sum_{j=-\infty}^\infty b(2^j x) = 1$ for all $x > 0$ .

Furthermore, $b(x) \geq \frac{1}{\sqrt{2}} \geq \frac{1}{2}$ for $x \in [1.5, 3]$ . Thus, b is a good candidate for a scale_function, cf. MotherShearlet.

Parameters:

Parameters:	A (numpy.ndarray) – The input (a matrix with real entries).

Return value:

Returns:	`b(A)`, i.e., `b` is applied componentwise.

MotherShearlets.haeuser_direction_function(A)¶

This is an implementation of the function $\widehat{\psi_2}$ from the paper Fast Finite Shearlet Transform (cf. eq. (6) in that paper).

This function has support in [-1,1], is “large” on [-1/2,1/2] and satisfies $\sum_{j=-\infty}^\infty |\widehat{\psi_2}(x + j)|^2 = 1$ for all x.

Thus, $\widehat{\psi_2}$ is a good candidate for a direction_function, cf. MotherShearlet.

Parameters:

Parameters:	A (numpy.ndarray) – The input (a matrix with real entries).

Return value

Returns:	$\widehat{\psi_2}(A)$ , i.e., $\widehat{\psi_2}$ is applied componentwise.

Indicator mother shearlet implementation¶

The “indicator” mother shearlet is a mother shearlet which is used purely for testing purposes. The low-pass part and the mother shearlet determined by its generating functions are indicator functions in the Fourier domain. Hence, a shearlet system using this mother shearlet will have very bad time localization.

The “indicator” mother shearlet is represented by the object MotherShearlets.IndicatorMotherShearlet, which is an instance of MotherShearlet. It uses

$\chi_{[-1,1]}$ as the low-pass part of the set of generating functions,
$\chi_{[1,2]}$ as the scale-sensitive generating function, and
$\chi_{[-1/2, 1/2]}$ as the direction sensitive generating function.

MotherShearlets.indicator_scale_function(A)¶

Vectorized implementation of the indicator function $\chi_{[1,2]}$ , which can be used as a scale_function, mainly for testing purposes (since the space localization is horrible).

Parameters:

Parameters:	A (numpy.ndarray) – The input (matrix with real entries).

Return value:

Returns:	$\chi_{[1,2]}(A)$ , applied componentwise.

MotherShearlets.indicator_low_pass_function(A)¶

Vectorized implementation of the indicator function $\chi_{[-1,1]}$ , which can be used as a low_pass_function, mainly for testing purposes (since the space localization is horrible).

Parameters:

Parameters:	A (numpy.ndarray) – The input (matrix with real entries).

Return value:

Returns:	$\chi_{[-1,1]}(A)$ , applied componentwise.

Table Of Contents

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Implementation of different “Mother Shearlets”¶

The class `BumpFunction`¶

The class `MotherShearlet`¶

Meyer mother shearlet implementation¶

Haeuser mother shearlet implementation¶

Indicator mother shearlet implementation¶

Implementation of different “Mother Shearlets”¶

The class BumpFunction¶

The class MotherShearlet¶

BumpFunction-related helper functions¶

Meyer mother shearlet implementation¶

Haeuser mother shearlet implementation¶

Indicator mother shearlet implementation¶

The class `BumpFunction`¶

The class `MotherShearlet`¶

`BumpFunction`-related helper functions¶