IEEE Geoscience and Remote Sensing Magazine - September 2014 - 17

compression algorithm that extends CCSDS-IDC to
multi-component images, supporting several transforms in the spectral dimension. A first attempt to assess
what the performance could be for such an extension
was provided in [31]. However, the JPEG2000 standard
does provide a multi-component extension in its Part
2 [32], which has been exploited by several authors to
design compression algorithms for multispectral and
hyperspectral images. Such algorithms are not amiable to on-board compression due to the complexity
and memory requirements of a JPEG2000 encoder, but
they can be used for on-the-ground image compression
and delivery. In the following we will describe a typical
setting for multi-component image compression based
on JPEG2000; it is expected that many of the principles
on which this algorithm is based will carry over to the
multi-component extension of CCSDS-IDC, when this
will become available.
The key ingredient of lossy multi-component image
compression is the choice of transform to be applied along
the spectral dimension, as it is known that this dimension
exhibits a very high degree of correlation. There is not a
single best choice, but several options that one can choose
from depending on specific complexity and performance
requirements, e.g., a spectral discrete cosine transform,
wavelet transform, Karhunen-Loève transform (KLT), or
some approximation of the KLT. Common to all these
choices is the notion that the spectral transform should
be applied separately from the spatial transform; since the
nature of the correlation in the spectral and spatial dimensions is different, there is no benefit in employing a transform that is isotropic in all dimensions. E.g., if one chooses
a spectral wavelet transform, the best results are obtained
applying first all levels of the spectral transform, and only
later applying the spatial transform to the spectral wavelet
coefficients. Having said that, the transform that typically
provides the best results is the spectral KLT. This transform
has been employed as a spectral decorrelator by many
authors [33]-[42], and is known to be optimal for decorrelation of Gaussian processes [43]. KLT has a non-negligible
computational cost and several approaches have been proposed to alleviate this complexity [44], [45].
Defining the covariance matrix C X of a random column
vector X with mean value n X as C X = E [(X - n X)(X - n X) T ],
the KLT transform matrix V is obtained by aligning
columnwise the eigenvectors of C X . It can be shown that
the transformed random vector Y = V T (X - n X) has uncorrelated components, i.e., C Y = V T C X V is a diagonal matrix.
It should be noted that the KLT transform coincides with
principal component analysis, which is a well known tool
for dimensionality reduction. In many papers employing
the KLT for compression, they use it to reduce the number
of components by zeroing out a given number of least significant transformed bands. However, in general it is more
appropriate to keep all components and represent them
with different accuracy via rate-distortion optimization
SEPTEMBER 2014

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techniques. As far as multi-component image compression
is concerned, for calculating the KLT one considers each
spectral vector (i.e., each one-dimensional vector obtained
fixing the spatial coordinates i, j and letting the spectral
coordinate k vary) as a sample realization of a random process. From the set of sample realizations the mean vector n X
and covariance matrix C X can be calculated; diagonalizing

(a)

(b)

(a) Original, (b) 0.1 bpppb, and (c) 1.0 bpppb.
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Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - September 2014