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Cone-adapted bandlimited shearlet system

The `shearletSystem`

object represents a cone-adapted
bandlimited shearlet system. After you create the shearlet system, you can use `sheart2`

to
obtain the shearlet transform of a real-valued 2-D image. You can also use `isheart2`

to
obtain the inverse transform. Additional Object Functions are
provided.

creates a cone-adapted
real-valued bandlimited shearlet system for a real-valued image of size 128-by-128 with
the number of scales equal to 4. The system `sls`

= shearletSystem`sls`

is a nondecimated
shearlet system. Shearlets extending beyond the 2-D frequency bounds are periodically
extended. Using real-valued shearlets with periodic boundary conditions results in
real-valued shearlet coefficients.

The implementation of `shearletSystem`

follows the approach described in Häuser and Steidl [6]

creates a cone-adapted bandlimited shearlet system with Properties specified by one or
more `sls`

= shearletSystem(`Name,Value`

)`Name,Value`

pairs. For example,
`shearletSystem('ImageSize',[100 100])`

creates a shearlet system for
images of size 100-by-100. Properties can be specified in any order as
`Name1,Value1,...,NameN,ValueN`

. Enclose each property name in single
quotes (`' '`

) or double quotes (`" "`

).

**Note**

Property values of a shearlet system are fixed. For example, if the shearlet
system `SLS`

is created with an `ImageSize`

of [128
128], you cannot change that `ImageSize`

to [200 200].

`sheart2` | Shearlet transform |

`isheart2` | Inverse shearlet transform |

`framebounds` | Shearlet system frame bounds |

`filterbank` | Shearlet system filters |

`numshears` | Number of shearlets |

Boundary effects of a real-valued shearlet transform of a non-square image can result in complex-valued coefficients. As implemented,

`shearletSystem`

constructs shearlets in the 2-D Fourier domain. For a real-valued shearlet transform, the shearlets in the 2-D Fourier domain should be symmetric in the positive and negative 2-D frequency plane. Shearlets constructed for square images are symmetric. However, as the image aspect ratio increases, the shearlets constructed become less symmetric. If the support of the lowpass filter in the 2-D frequency plane is too large, boundary effects can increase. Whenever possible, use square images. See Boundary Effects in Real-Valued Bandlimited Shearlet Systems for additional information and strategies to mitigate boundary effects.

[1] Guo, K., G. Kutyniok, and D.
Labate. "Sparse multidimensional representations using anisotropic dilation and shear
operators." In *Wavelets and Splines: Athens 2005* (G. Chen, and M.-J.
Chen, eds.), 189–201. Brentwood, TN: Nashboro Press, 2006.

[2] Guo, K., and D. Labate. "Optimally
Sparse Multidimensional Representation Using Shearlets." *SIAM Journal on
Mathematical Analysis*. Vol. 39, Number 1, 2007, pp. 298–318.

[3] Kutyniok, G., and W.-Q Lim.
"Compactly supported shearlets are optimally sparse." *Journal of Approximation
Theory*. Vol. 163, Number 11, 2011, pp. 1564–1589.

[4] *Shearlets: Multiscale
Analysis for Multivariate Data* (G. Kutyniok, and D. Labate, eds.). New York:
Springer, 2012.

[5] *ShearLab*.
`https://www3.math.tu-berlin.de/numerik/www.shearlab.org/`

.

[6] Häuser, S., and G. Steidl. "Fast Finite Shearlet Transform: a tutorial." arXiv preprint arXiv:1202.1773 (2014).