Description Usage Arguments Details Value Note References See Also Examples

`WavTransf2D`

computes a forward intrinsic average-interpolation (AI) wavelet transform for a
rectangular surface in the manifold of HPD matrices equipped with a metric specified by the user, such as the
affine-invariant Riemannian metric, as described in Chapter 5 of \insertCiteC18pdSpecEst.

1 | ```
WavTransf2D(P, order = c(3, 3), jmax, metric = "Riemannian", ...)
``` |

`P` |
a ( |

`order` |
a 2-dimensional numeric vector |

`jmax` |
the maximum scale up to which the wavelet coefficients are computed. If |

`metric` |
the metric that the space of HPD matrices is equipped with. The default choice is |

`...` |
additional arguments for internal use. |

The 4-dimensional array `P`

corresponds to a discretized rectangular surface of *(d,d)*-dimensional
HPD matrices. The rectangular surface is of size *n_1* by *n_2*, where both *n_1* and
*n_2* are supposed to be dyadic numbers. `WavTransf2D`

then computes the intrinsic AI wavelet transform
of `P`

based on the given refinement orders and the chosen metric. The marginal refinement orders should be
smaller or equal to 9, and the function computes the wavelet transform using a fast wavelet refinement scheme based on weighted
intrinsic averages with pre-determined weights as explained in Chapter 5 of \insertCiteC18pdSpecEst. By default `WavTransf2D`

computes the intrinsic 2D AI wavelet transform equipping the space of HPD matrices with (i) the affine-invariant Riemannian metric as
detailed in e.g., \insertCiteB09pdSpecEst[Chapter 6] or \insertCitePFA05pdSpecEst. Instead, the space of HPD matrices
can also be equipped with one of the following metrics; (ii) the Log-Euclidean metric, the Euclidean inner product between matrix
logarithms; (iii) the Cholesky metric, the Euclidean inner product between Cholesky decompositions; (iv) the Euclidean metric and
(v) the root-Euclidean metric. The default choice of metric (affine-invariant Riemannian) satisfies several useful properties
not shared by the other metrics, see \insertCiteC18pdSpecEst for more details. Note that this comes at the cost of increased computation
time in comparison to one of the other metrics.

The function returns a list with three components:

`D ` |
the 2D pyramid of wavelet coefficients. This is a list of arrays, where each 4-dimensional array contains the
( |

`D.white ` |
the 2D pyramid of whitened wavelet coefficients. The structure of |

`M0 ` |
a numeric array containing the midpoint(s) at the coarsest scale |

The function does not check for positive definiteness of the input matrices, and (depending on the specified metric) may fail if matrices are close to being singular.

`InvWavTransf2D`

, `pdSpecEst2D`

, `pdNeville`

1 2 | ```
P <- rExamples2D(c(2^4, 2^4), 2, example = "tvar")
P.wt <- WavTransf2D(P$f)
``` |

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