IEEE Geoscience and Remote Sensing Magazine - March 2021 - 60

[129]. Indeed, even if nonorthogonality could compromise the spectral quality of fused products [129], its beneficial characteristics, such as the shift-invariance property [130] and the capability of being easily matched to
the sensor MTF [47], produce accurate pansharpened images. The AWLP, originally proposed by [61], exploits the
" à trous " wavelet for details of extraction from the PAN
image using the fusion rule in (8) and injection coefficients estimated as
R
MS
G k = 1 N k . (9)
R
MS i
N i/
=1

	

To get high performance, a preliminary histogram-matching procedure is also required [31].
GENERALIZED LAPLACIAN PYRAMID
PL at the original MS resolution can be performed with
multiple fractional steps. This method is commonly referred to as pyramidal decomposition and dates back to the
seminal work of Burt and Adelson [36], which utilized
Gaussian low-pass filters to carry out the analysis steps.
The corresponding differential representation, achieved
by calculating the differences between the Gaussian pyramid levels, is named the Laplacian pyramid and later was
proved to be very valuable for pansharpening purposes
[43]. In [48], the authors showed that a single Gaussian

low-pass filter with a cut frequency equal to 1/R (where R is
the resolution ratio between PAN and MS) and decimating
by R is enough to get high performance. Gaussian filters
can be tuned to closely match the MS sensor's MTF. This
allows extracting from the PAN image those details that are
not seen by the MS sensor [47]. Since the Gaussian mask is
defined by a single parameter (i.e., its standard deviation),
its frequency response is fully specified by fixing that parameter. To this aim, the value of the amplitude response
at the Nyquist frequency is used, since it is commonly provided by the manufacturer.
In the literature, many instances of generalized Laplacian pyramid (GLP) approaches, relying upon filters that
exploit the MS sensor's MTF, have been proposed by changing the way to estimate the injection coefficients. High performance is usually expected by these methods, motivating
their inclusion into the proposed benchmark. Their brief
description is provided in the following:
◗◗ The classical MTF-GLP is based on MTF Gaussian filters
for detail extraction and an additive injection model, i.e.,
G k = 1 for each k = 1, f, N . To get high performance, a
histogram matching procedure between PL and each MS
band is required, as proposed in [31].
◗◗ MTF-GLP-HPM, where HPM stands for high-pass modulation, adopts the injection model exploited in the BT
(3) by substituting PL with I L , i.e.,
	

R
MS
G k = P k , (10)
L

	 where the division is intended as pixel-wise. Even in
MS

PAN

Interpolation
to PAN Scale

Equivalent
Filter
(Optional)
↑R

↑

MSk

this case, a preliminary histogram matching is required [31]. Improved versions are 1) MTF-GLP-HPMR [131], adopting a spectral matching procedure based
on the multivariate linear regression between each MS
band and a low-pass version of the PAN image; and 2)
MTF-GLP-HPM-H [59], a haze-corrected version like
the one in BT-H (see the " Brovey Transform With Haze
Correction " section), whose fusion rule is modified
as follows:

R
PL

Computation of
Injection Gains

Gk

−

*

+

	

+ P

◗◗ MTF-GLP-CBD (i.e., MTF-GLP with a context-based deci-

sion) adopts a projective, or regression-based, injection
model [47], [132] in conjunction with the MTF-GLP
framework. The injection coefficients are estimated
through multivariate linear regression on R
MS k and PL ,

+
MSk

MS

R
MS k - H
\
MS k = R
MS k + P - H k · ^P - PLh . (11)
L
p

	

Gk = gk 1 =

cov (R
MS k, PL)
var (PL) 1, (12)

	 in a way similar to that for the GS-based approaches
FIGURE 2. A flowchart of pansharpening methods belonging to the
MRA class. Some MRA approaches skip the dashed box.

60

(see the " Gram-Schmidt " section) but considering PL
instead of I L . A local version based on the use of a kmeans clustering to segment the MS image in spectrally
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IEEE Geoscience and Remote Sensing Magazine - March 2021

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