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 C18pdSpecEst. This is the inverse operation of the
function WavTransf2D.
InvWavTransf2D(D, M0, order = c(3, 3), jmax, metric = "Riemannian",
...)a list of arrays containing the 2D pyramid of wavelet coefficients, where each array contains the
(\(d,d\))-dimensional wavelet coefficients from the coarsest wavelet scale j = 0 up to the finest
wavelet scale j = jmax. This is the same format as the $D component given as output by
WavTransf2D.
a numeric array containing the midpoint(s) at the coarsest scale j = 0 in the 2D midpoint pyramid.
This is the same format as the $M0 component given as output by WavTransf2D.
a 2-dimensional numeric vector \((1,1) \le\) order \(\le (9,9)\) corresponding to the marginal
orders of the intrinsic 2D AI refinement scheme, defaults to order = c(3, 3).
the maximum scale (resolution) up to which the 2D surface of HPD midpoints (i.e. scaling coefficients) are
reconstructed. If jmax is not specified it is set equal to the resolution in the finest wavelet scale
jmax = length(D).
the metric that the space of HPD matrices is equipped with. The default choice is "Riemannian",
but this can also be one of: "logEuclidean", "Cholesky", "rootEuclidean" or
"Euclidean". See also the Details section below.
additional arguments for internal use.
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.
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 C18pdSpecEst. 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., B09pdSpecEst[Chapter 6] or PFA05pdSpecEst.
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 C18pdSpecEst for more details. Note that this
comes at the cost of increased computation time in comparison to one of the other metrics.
# NOT RUN {
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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