Description Usage Arguments Details Value References See Also Examples

`InvWavTransf2D`

computes the inverse intrinsic average-interpolation (AI) wavelet
transform mapping an array of coarsest-scale HPD midpoints combined with a 2D pyramid of Hermitian
wavelet coefficients to a surface in the manifold of HPD matrices equipped with a metric specified by the
user, as described in Chapter 5 of \insertCiteC18pdSpecEst. This is the inverse operation of the
function `WavTransf2D`

.

1 2 | ```
InvWavTransf2D(D, M0, order = c(3, 3), jmax, metric = "Riemannian",
...)
``` |

`D` |
a list of arrays containing the 2D pyramid of wavelet coefficients, where each array contains the
( |

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

`order` |
a 2-dimensional numeric vector |

`jmax` |
the maximum scale (resolution) up to which the 2D surface of HPD midpoints (i.e. scaling coefficients) are
reconstructed. If |

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

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

The input list of arrays `D`

and array `M0`

correspond to a 2D pyramid of wavelet coefficients and
the coarsest-scale HPD midpoints respectively, both are structured in the same way as in the output of
`WavTransf2D`

. As in the forward AI wavelet transform, 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 inverse 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.

Returns a (*d, d, n_1, n_2*)-dimensional array corresponding to a rectangular surface of size *n_1* by
*n_2* of (*d,d*)-dimensional HPD matrices.

`WavTransf2D`

, `pdSpecEst2D`

, `pdNeville`

1 2 3 4 | ```
P <- rExamples2D(c(2^4, 2^4), 2, example = "tvar")
P.wt <- WavTransf2D(P$f) ## forward transform
P.f <- InvWavTransf2D(P.wt$D, P.wt$M0) ## backward transform
all.equal(P.f, P$f)
``` |

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