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i Compression
of Color-mapped A.C.
University
Images*
Hadenfeldt
of Nebraska
Omaha,
20k73 //J
Medical
Nebraska
Center
68198
and K. Sayood Department Center
of Electrical and
for Communication University
Engineering
and Information
Science
of Nebraska-Lincoln
Lincoln,
NE 68588
Telephone: (402) 472-6688 FAX: (402) 472--4732 email:
COMSOC
Technical
[email protected]
Committee:
Signal Processing
Hot topic session
and Communication
number:
Electronics
HT28
Abstract
In a standard data, especially schemes
image coding
scenario,
if the image is a natural
(e.g., DPCM)
to perform
pixel-to-pixel
correlation
scene. This correlation
efficient
compression.
In a color-mapped
in the pixel array are no longer
directly
which
are numerically
may point to two very different
(close)
still exists,
but only via the colormap.
reintroduce
the structure.
resulting
structure
In this paper
to the pixel intensity.
This fact can be exploited we study the sorting
can be used in both lossless
always
exists in the
is what allows predictive
stored
adjacent
related
nearly
image,
coding
the values
Two color indices
colors.
The correlation
by sorting the color map to
of colormaps
and lossy compression
and show how the
of images.
" This work was supported by the NASA Goddard Space Flight Center (NAG 5-1612) and the NASA Lewis Research Center (NAG 3-806).
3
Compression
of Color-mapped
Images _ A.C. Hadenfeldt
°
1 Introduction
Many lower-cost
image display systems
there has been considerable images, the compression The human depending
attention
on viewing
a means of displaying
devoted
of color-mapped
eye can distinguish conditions
use color-mapped
to the compression
images
hundreds
of thousands
similar
of different
is illustrated
While
and fuLl-color
attention.
colors
(also called true-color)
Such a system
displays.
of monochrome
has not received
[1]. A fuU-color
this wide range.
(or pseudo-color)
in a color space,
frame buffer
in Figure
provides
1.
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t ! I |
t I _ I 1 I I ! I I I _ _
CRT _reen Figure Many applications If a fuU-color display
of digital images benefit is used, an application
Also, the images
involved
on a mass-storage
device.
These applications
1 Full-Color
naturally
A smaller amount of image memory The values stored
may become
in memory
of storage
solution
too costly to implement space, whether
or color-mapped
of those found on personal
is required,
are used as indices
to be effective. practically.
in display memory
or
is needed.
lead to the pseudo-color
Figure 2. This type of display is typical
Buffer
from, or require, color capabilities
require large amounts A less expensive
Frame
one-third
frame buffer, shown in
computers
that of the fuU-color
into a 24-bit
and workstations. system example.
table, the colormap.
Compression
of Color-mapped
hnages
_ A.C. Hadenfeldt
°
Each entry in the colormap the pixel. These three values
consists
are then passed
guns of the CRT, as with fuU-color small number of colors
colormap,
a large variety
full-color
display
of images
of
DACs to the red, green, and blue electron
The color-mapped
(224 for this example).
system
in the figure,
By careful
can be displayed,
allows the display
which can be selected
selection
often with quality
of a
of the colors approaching
in the
that of a
system.
q_
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system.
through
for the red, green, and blue portions
at a time, 2 s for the system shown
from a larger set of colors
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of 8-bit vaIues
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CRT_reon
Figure
2 Color-Mapped
The use of the colormap, however, of this can be obtained quantities
were computed
are listed in Table
1.
by calculating
disguises the zero-th
using the index arrays
Frame
Buffer
the spatial structure
in the image.
and first order entropies
An indication
of the image.
These
for the four test images shown in Figure 3, and
Compression
of Color-mapped
hnages-
Table
The large values values
1 Entropies
of the Source Ito
IIt
Lena
7.617
7.413
Park
7.470
7.797
Omaha
7.242
7.165
Lincoln
5.916
6.674
of H1 in the table verify
of selecting
quantization
implies
that the color
in, for example,
index
an achromatic
in which
image,
to what might be encountered
resultant Since
errors
were introduced
images were of poor subjective
quantization
schemes
2 Colormap
Sorting
In the previous
become
section,
The root of these problems
several
Colon'nap as vectors, dimensional
problems
coding
is defined
function
at best,
in the image are more
To further
index
critical
verify this, an experiment bit of a color-mapped
values
were quantized.
and often completely
coding
unique
schemes,
The
unrecognizable.
the available
to color-mapped
choices
for
images
were discussed.
indices stored in the image have little relationship
for progressive
optimization
as foUows.
vector space and a distance
a.{, find an ordering
image.
due to the color
transmission.
In this section,
methods
are discussed.
sorting is a combinatorial
the problem
stored
pixel
are also relatively
The data compression
values
source
is that the colormap
this relationship
of/'/o
in the least significant
quality
in the image
limited.
with each other, which complicates of restoring
correlation
The values
if the color
is a part of many popular
compression
process.
an 8-bit colon-nap.
was conducted similar
that the spatial
by the color-mapping
large, a direct result process
Images
Image
has been reduced
than similar values
A.C. Hadenfeldt
L(k)
Given
measure
which
problem.
a set of vectors
d(i,j)
minimizes
Treating
defined
{ax,a2,
between
the total distance
the K colormap ...,a_c}
entries
in a three-
any two vectors
ai and
D:
K-1
D=
,IC Ck),
+ i))
O)
-
Compression
of Color-mapped
The ordering function Another possibility is, the colormap by/_,(1).
Images
L is constrained
by L(K)
In this case, an additional
The sorting problem
complete
of possible
only for K no greater
exactly
second
were tested.
is an algorithm annealing
described
in more detail
has performed
was chosen
L*a*b*
were investigated.
Three
to the color primaries
would
selected
since they provide
2.1 Sorting
give similar
results a means
Using Simulated
Simulated annealing basis for the technique of materials implementation
as they
images.
were not considered,
measure
[3][6]
discussed
is necessary.
in Section
as simulated
2.1. The annealing.
in this work,
and is
Euclidean
were selected:
distance,
and different
the NTSC RGB space,
color
the CIE
space. The NTSC RGB space was chosen since it corresponds
of the original
the NTSC RGB space
a local minimum
approach
for the colormaps
to be (unweighted)
color spaces
space, and the CIE L*u*v*
method
are computationally
for locating
known
to be NP-
2.2.
The distance metric d was chosen spaces
technique,
D.
- 1)!][4]. Algorithms
these algorithms
well in practice,
as the sorting
in Section
is known
such as K = 256, another
first is a "greedy"
formula
problem, and is identical
is 1/2[(I(
algorithms
That
to the entry specified
the problem
to consider
[4][5]; however,
colormaps
The
which
Simulated
As such,
than about 20. Efficient
exist [4] for K < 145. For large
salesman
{1, ..., K}.
as a ring structure.
is added to the distance
travelling
orderings
of integers
to be adjacent
L(1))
as a ring structure.
[3], and the number
Two techniques
is now considered
is similar to the well-known
is considered
of the sequence
entries is considered
term of d(L(K),
exist which can solve the problem feasible
to be a permutation
results when the list of colormap
entry specified
if the colormap
_ A.C. Hadenfeldt
Color spaces
which can be linearly transformed
since the use of an unweighted
for such a color to measure
space.
perceptual
The
Euclidean
to
distance
two CIE color spaces
were
color differences.
Annealing
is a stochastic
technique
comes from thermodynamics are cooled.
The
technique
in [6].
4
for combinatorial
and observations described
minimization.
concerning
in this section
The
the properties
is based
on the
Compression
of Color-mapped
To illustrate move
freely
molecules
this concept,
with respect
(cooled
consider
A.C. Hadenfeldt
an iron block.
to each other.
will be locked
annealed
Images _
together
very slowly),
If the block
in a high-energy
the molecules
energies
is characterized
is quenched
state.
lattice
which
by the Boltzmann
the iron molecules
(cooled
very quickly),
On the other hand,
will tend to redistribute
energy, with the result being a lower energy molecular
At high temperatures,
if the block
themselves
is much stronger.
the
as they lose
The distribution
of this distribution
will have a high energy.
In a combinatorial
can be used to temporarily
allow increases
achieve
(2)
and k is Boltzmann's
is that even at low temperatures,
of
distribution:
P( E) ~ e-z/kr where E is the energy state, T is the temperature,
is
there
optimization
constant.
The significance
is some probability
that a molecule
situation,
in the cost function,
the Boltzmann
distribution
while still generally
striving
to
a minimum.
Solving minimizing annealing, algorithm
the
colorrnap
sorting
problem
the sum of the distances an initial
path through
then proceeds
1. Select
3. If AN
the colors. (colors)
T and a cooling
new path by perturbing
in path cost, AE
> 0, randomly
number
selecting
each
color
To find a solution
is chosen,
only
once
while
using simulated
and its cost computed.
as follows:
a temporary
the change
between
the nodes
an initial temperature
2. Choose
involves
decide
r from a uniform
= E_w whether
factor
a.
the current path (see below),
- Eola. If AE
< 0,
or not to accept
distribution
in the range
and compute
accept the new path.
the path.
Generate
a random
[0, 1), and accept
the new path
for I iterations.
Then, "coor'
if r < exp(-AE/T). 4. Continue
to perturb
the system
the path at the current
by the cooling
factor:
Tn_,
temperature
= c_Told. Continue
iterating
using
the new
at a particular
temper-
temperature. 5. Terminate ature.
the algorithm
when no path changes
are accepted
The
Compression
of Color-mapped
The decision-making process
process
function.
annealing
to the path which
method
to avoid easily
Hence, the algorithm
For the images space used.
algorithm
of this work,
to improve
to the choice
was chosen
the execution
path changes
Also, a method
to occur too slowly
speed of the algorithm
were made using the suggestions to the path are made, chosen
of Lin [4][6].
at random.
removes
The simulated
as values outside
or too quickly.
The number
of nodes (colors),
the range
of iterations
or 25,600.
suggested
on the
annealing
of this value,
However,
by [6] was added,
if (10)(number
of nodes) = 2560
In this work, the perturbations one of two possible
The first is a path transport,
a segment
in the path, but with the nodes
point in the path. at random.
of the current
in reverse
order.
which removes
The location
The second
path and reinserts
The location
and length
changes a segment
of the segment,
perturbation
method,
it at the same point of the segment
are
chosen.
The algorithm outlined problem.
structure.
the case,
allowing
in the previous
This type of problem
desires
ring-like
both structures
as 0.9.
At each iteration,
point are chosen
and the new insertion
the salesman
chosen
the path must be selected.
its length,
salesman
for
of the cost
from 80 to 500, depending
to the next temperature
it at another
randomly
in a local minimum
an improvement
of the current path and reinserts
again
it possible
are made at a given temperature.
for perturbing
called path reversal,
trapped
as 100 times the number
to proceed
Note that the decision
its cost. This makes
of T ranged
to be most sensitive
which causes the algorithm successful
initial values
_x was usually
the cooling
algorithm.
to the initial path choice.
factor
[0.85, 0.95] caused per temperature/"
increase
being
is less sensitive
The cooling
seemed
-- A.C. Hadenfeldt
is known as the Metropolis
will allow some changes
the simulated
color
Images
paragraphs usually
to return to the original However,
the simulated
the colormap
were
formulates
assumes
city).
Hence,
annealing
to be considered
conducted.
6
colormap
a complete
tour will be made (i.e.,
the colormap
technique
sorting as a travelling
is assumed
to have a
can also be used if this is not
as a linear list structure.
Experiments
using
-
Compression
of Color-mapped
3 Colormap
Sorting
and
Images
Lossless
_ A.C. Hadenfeldt
Compression
The results of sorting the colormaps in the following
of the test images using simulated
tables. Table 2 shows results
for sorting the colormap
while Table 3 shows the results of sorting the colormap are values for the resulting
first-order
Table 2 Resultant Image
entropy Images
as a circular
as a linear structure.
With Circularly
Sorted
are shown
ring structure,
Given in the tables
and the final path cost (the distance
RGB Space
measure
D).
Colormaps
L*a*b Space
L*u*v*
Space
Name
Cost
Ht
Cost
Ht
Lena
13.88
5.641
857.80
5.627
208.49
5.480
Park
19.32
6.325
1609.46
6.330
310.41
6.218
Omaha
11.04
6,209
1081.82
6.303
363.21
6.178
Lincoln
10.62
5.513
1193.88
5.831
224.06
5.478
Table 3 Resultant
Images
L*a*b
Cost
I/1
Lena
II.68
5.575
Park
15.66
Omaha Lincoln
the colormap first-order
847.29
5.933
200.3I
5.512
6.260
1509.25
6.775
292.29
6.546
10.81
6.532
1004.69
6.554
283.66
6.199
10.61
5.774
1177.80
6.120
204.64
5.735
entropy//o
the frequency
of the resultant
images
results,
space
between
sorting
colormap
by the sorting
of occurrence
indicate
in each case.
the L*u*v*
differences
is not changed
The sorting if entropy
quality, the second
In terms of lossless
goal stated
compression,
previously.
the sorting
reduction
color.
The lower
correlation
between
(the first goal stated above) is with the added advantage
has been considered.
We examine
has resulted 7
of a particular
since permuting
results for the NTSC RGB space show
gives better results, entries
process,
that some of the spatial
from this sort should also be able to accept quantization
subjective
Space //1
that sorting in this space yields good
the perceptual
L*u*v* Cost
color indices has been restored
the goal. However,
H1
Colormaps
Space
Cost
entries does not change
entropies
Sorted
Cost
//1
Note that the zero-order
images
With Linearly
RGB Space
Image Name
-
annealing
errors
Hence, while
that
the resultant
maintaining
good
this further in the next section.
in a drop of 2 bits per pixel for the
Compression
of Color-mapped
Images _
A.C. Hadenfeldt
Lena image and 1 to 1.5 bits per pixel for the other images.
For a 512x512
to a savings of between
32,768 to 65,536
For a large database
could
saving.
be a considerable
4 Coiormap
Sorting
and Lossy
The sorting of the colormap
bytes
per image.
restores
some perceptual
subjective
the three least significant
results were obtained
8-bit original.
quantizing
the Park image to 5 bits/pixel,
list in L*u*v*
before
the sorted
not be assumed
(compressed between distances
image.
In some cases,
and decompressed)
those pixels
would correspond
compress
them using particular
the Discrete
Cosine Transform
4.1 DCT Coding
in shading
of Color-mapped
and after sorting.
The
5 shows the result of
properties
the original
image.
A caveat meaning,
as an eight-
and reconstructed
In the monochrome
might be overlooked
lend themselves
Pulse
case large
by the viewer.
to lossy compression
lossy compression
Code Modulation
we
techniques,
(DPCM).
Images
with the unsorted
colormap
color map. As can be seen from the figure, the original
that remains is seemingly
Good
there might be a drastic change in color
of two popular
and Differential
it
from the
has been sorted.
the same
between
which
images
implementations
In Figure 6 we coded the Lena image the unsorted
to have
if the distance
color-mapped
(DCT)
Park images.
indices have more perceptual
and reconstructed
to changes
To see how well the sorted
the eight-bit
indices is large enough,
in the original
sense. Therefore
the image. To verify this
space. Figure
and after the color'map
between
bit monochrome
destroying
indices in the
levels down to as low as 5 bits/pixel
is in order here. While the distance image should
without
for the Park image, before
shown was sorted as a linear
colonnap
to the colormap
bits of the L*u*v*-sorted
using quantization
Figure 4 shows the colormap
sorted colormap
structure
value are also close in some perceptual,
should be possible to introduce errors into the indices we dropped
of images this
Compression
sense that indices close in numerical
hypothesis,
image this translates
random colors.
at two bits per pixel using image is totally lost and all
It should be noted that for eight-bit
8
monochrome
images,
Compression
of Color-mapped
Images _
A .C. HadenfeMt
DCT coding at two bits per pixel generally
provides
a reconstruction
which is indistinguishable
from the original. In Figure 7 we show the same image, this time with the sorted color map, coded per pixel with the fixed bit allocation.
[7] The
images
in Figure
bit per pixel using the JPEG [8] algorithm, l Note that while allocation
shown in Figure 7 is far superior
annoying
artifacts.
minimizes
This is because
the average
the color-mapped
to the image
of the nonadapdve
sensitive
The JPEG algorithm
adapts
Therefore,
in Figure
is coded
4.2 DPCM
Coding
Standard
of Color-mapped
especially
edges,
In monochrome
which may be acceptable
the prediction images
of large errors in individual
its bit allocation
for certain
while it
blocks.
As
for the low
on a block-by-block
at half the rate of the image
of the quantization
error is always
output alphabet
to entropy
bounded
images.
is problematic
is generally values
because
large,
result
However,
colors.
by the use of a recursively
the quantizer
error
application.
avoids t.his problem
to the coding of color-mapped
images
these noise
values will result in splotches of different
indexed by _.
Another
in color-mapped
quantizer
This attribute advantage
without
leading
in a blurred
The Edge Preserving
can be kept small
basis.
in Figure
7
to large overload look around edges, images these noise
DPCM (EPDPCM)
[9,10], makes
system
in which the magnitude it ideal for application
of the EPDPCM
incurring
in the busy regions
overload
system is that, as error, the output
is
coding.
Results using the EPDPCM system
are shown
coded at a rate of 2 bits per pixel, while the image
1
which,
Images
DPCM coding of color-mapped
noise values.
amenable
nature of the algorithm
superior quality.
still provides
of images,
8(b) which
the image coded using the fixed bit
to large errors, this could account
quality reproduction. the image
8 were coded at two and one
in Figure 6 there are still quite a few
error, may permit the introduction
images are particularly
at two bits
in Figure in Figure
10. The image
in Figure
10(a) was
10(b) was coded with 1.35 bpp.
The IPEG coded images were coded using software from the independent JPEG foundation.
Compression
of Color-mapped
The advantage and higher speed. is generaUy
Images _ A.C. Hadenfeldt
of DPCM However,
significantly
systems
over transform
the reconstruction
higher
coded image is actually
coded image. Thus using the EPDPCM and speed,
and reconstruction
systems
not the case for the sorted
quality of the two-bit EPDPCM
systems
quality obtained
than that of DPCM
10(a) and 8(a), this is obviously
coding
system provides
is their low complexity
using transform
at a given
rate.
colormapped
somewhat
higher
advantages
coding systems
Comparing
images.
Figure
In fact, the
than the two-bit DCT
both in terms of complexity
quality.
5 Conclusion
In this paper amenable
we have shown
to both lossless
that use of sorted
and lossy compression.
dictates
the use of DCT coding
DPCM
coding
colormaps
makes
color-mapped
images
For lossy compression
conventional
wisdom
However,
for color-mapped
images
Computer
Graphics:
for most types of images.
might be more advantageous.
6 References
[1] Foley, J.D., van Dam, A., S.K. Feiner, and Practice [2] Graphics June
(Second
Interchange
Edition), Format
Reading,
MA: Addison-Wesley,
(GIF) Specification,
CompuServe,
Principles
1990. Inc.,
Columbus,
OH,
1987.
[3] Aarts, E., and J. Korst, Simulated Wiley
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pp.
Solutions
2245-2269,
R.E.,
Annealing
and Boltzmann
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New York: John
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-
and J.F. Hughes,
and
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S.E.
Dreyfus,
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Salesman
Problem,"
Bell System
Technical
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10
Dynamic
Programming,
Princeton,
NJ:
Compression
of Color-mapped
Images u
[6] Press, W.H., B.P. Flannery, C, New York: [7] Iayant,
ACM, [9] Rost,
Cambridge
G.K., '_'he
34(4):31--44,
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April
Sayood,
IEEE Transactions
Coding
Numerical
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in
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of Waveforms, compression
Prentice-Hall,
standard,"
1984.
Communications
of the
1991. "An Edge Preserving
on Image Processing,
K. and S. Na.
and W.T. Vetterling,
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M.C. and K. Sayood,
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S.A. Teukolsky,
N.S. and P. Noll, Digital
[8] Wallace,
A.C. Hadenfeldt
"Recursively
on Information
Differential
1:250-256, Indexed
Theory,
Coding
Scheme,"
1992.
Quantization
IT-38,
11
April
Image
November
of Memoryless 1992.
Sources,"
Compression
of Color-mapped
Images -- A.C. Hadenfeldt
List of Figures
Figure
1. Full-Color
Frame
Figure 2. Color-Mapped Figure
3. Test Images
Figure
4. Colorrnap
Buffer
Frame
Buffer
for Park Image
Figure 5. Park image
quanfized
(a) before
and (b) after sorting
to five bits per pixel
with (a) unsorted
and (b) sorted
colormaps Figure
6. Lena image
with unsorted
colormap
coded
at two bits per pixel using
IPEG
DCI _ algorithm Figure 7. Lena image
coded at two bits per pixel using DCT with fixed bit allocation
Figure
coded
8. Lena JPEG
Figure Figure
9. DPCM
image
at (a) two bits per pixel
and (b) one bit per pixel
using
and (b) 1.35 bits per pixel
using
algorithm structure
10. Lena image EPDPCM
coded
at (a) 2 bits per pixel
12
F Lgure
3 .
Test
I mag
e'_
PI_EGEDING
PAGE
BLANK
NOT
FILMED
_, ..... ;: :i!:? :i_::ii:e:_!:_Lf!:
Figure
4.
Colormap
for
Park
image
(a)
before
and
(b)
after
sorting
Figure
5.
Park image quanitzed to five bits unsorted and (b) sorted colormaps
per
pixel
with
(a)
m
Figure
6.
Lena pixel
image using
with JPEG
unsorted colormap DCT algorithm
coded
at
two
bits
per
