IEEE Geoscience and Remote Sensing Magazine - September 2014 - 13

In this vector, the first three components stand for
spatial directional local differences, where the superscript identifies the cardinal orientation (North, West,
or Northwest). The other components stand for spectral
local differences.
If reduced prediction mode is employed, the spatial
directional local difference components, i.e., the first three
components of U z, y, x, are not used.
The prediction itself is a little bit more convoluted.
First, the local difference vector U i, j, k is further scaled
through an inner product by a weight vector W i, j, k, yielding dt i, j, k = W Ti, j, k $ U i, j, k, an estimation of the local differences
d i, j, k . The weight vector is adaptive and is employed to
account for the usefulness of each component in the local
difference vector when predicting the pixel to-be-coded.
These weight vectors can be initialized by default or by a
custom user-defined method.
The estimation of the local differences dt i, j, k and the local
sum v i, j, k are then finally employed to produce a scaled predicted sample value, Kx i, j, k through a rounding operation that
takes into account the magnitude of the interval where
these coefficients lie and the precision of the registers used
for storing them.
Next, as explained previously, a scaled prediction error is
defined as e i, j, k = 2 $ x i, j, k - Kx i, j, k . However, contrary to what
happens in most prediction-based lossless compression
methods, these scaled prediction errors are not being sent
to the entropy coder, but only used to adapt the weight vector considering their sign.
In the end, the scaled predicted sample value Kx i, j, k are
employed to compute a predicted sample value, x{ i, j, k = 7Kx i, j, k /2A,
which, in their turn, are used to compute the prediction residual K i, j, k = x i, j, k - x{ i, j, k .
The prediction residuals are signed integer values and
need to be translated to non-negative integer values, producing the mapped prediction residual m i, j, k . In practice,
the mapped non-negative integers are computed from the
signed prediction residuals K i, j, k, the predicted pixel x{ i, j, k,
and the least significant bit of the scaled predicted pixel
x{ i, j, k . The non-negative mapped residuals m i, j, k are the final
output of the predictor, sent to the next stage, the entropy
encoder. At the decoder side, the original pixels can be
recovered without loss from m i, j, k .
2) EncodEr
The second functional part, the encoder, encodes the
mapped prediction residual m i, j, k without loss. Recall that
the entropy encoding can be applied on a sample basis or
on a block basis, with commonly a higher performance
for sample-adaptive encoders. In addition, the user can
select the order in which the prediction residuals are
encoded, either in Band Interleaved by Line (BIL), in Band
Interleaved by Pixel (BIP), or in Band Sequential (BSQ)
order. This encoding order might be independent of the
order with which the sensed pixels are captured, and
also independent of the order with which the predicted
SEPTEMBER 2014

ieee Geoscience and remote sensing magazine

residuals are produced. The encoding order will not affect
the coding performance if sample-adaptive encoding is
used, though it usually affects the coding performance of
block-adaptive encoding. Notice that the encoding order
can impact the memory requirements, as extra buffering
might be needed.
As mentioned, block-adaptive encoding is based on
the previous CCSDS standard [13] for mono band lossless
compression, and is not further discussed here. For sample-adaptive encoding, variable-length binary codewords
are used to encode each mapped prediction residual, similar to the process in JPEG-LS standard [14], [15], which the
reader may be more familiar with.
In a nutshell, two internal variables are used in the
sample-adaptive encoder, an
The key ingredienT of
accumulator R i, j, k and a counter
lossy mulTi-componenT
C i, j . After encoding a given
image compression is The
mapped predicted residual
choice of Transform To
m i, j, k, this m i, j, k is added to the
be applied along The
accumulator and the counter
specTral dimension.
is incremented by 1. The quotient R i, j, k /C i, j estimates the
average value of the mapped
prediction residual, and is used to select the parameter of a
Golomb-power-of-two (GPO2) code [16]. Each component
or band in the multi-component image can have its own
initial accumulator and counter values to determine a different GOP2 code for each band, with the goal to improve
the overall coding performance.
As a final remark regarding practical issues, we note
that whenever the counter reaches a given limit (the socalled rescaling counter size), both the counter and the accumulator are halved, and that the length of each encoded
sample is also tested against another given limit (the socalled unary length limit), and in case of a codeword exceeding this limit length, a unary sequence is signalled and the
mapped prediction residual is simply written in unsigned
binary form. This control may prove useful when the first
functional part, the predictor, is not able to provide a good
enough estimate.
B. Performance assessment
We now here report the lossless coding performance
of CCSDS-123 for the images in the considered data
set. Table 2 provides a comparison among three coding
standards: classical JPEG2000 [17], RKLT+JPEG2000-
described in Section III-, and CCSDS-123, along with
comparison against M-CALIC [18], which is a multi-component-only extension of CALIC [19]. CALIC was devised
as a proposal for the ISO standard for lossless and nearlossless compression, JPEG-LS [14], and although it provides a higher coding performance, the less computationally complex LOCO-I [15] was finally selected.
For JPEG2000, no multi-component transform is
applied along the spectral dimension and 5 levels of the
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Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - September 2014