This document discusses intra-frame compression using Huffman and arithmetic entropy coding techniques. It implements both techniques in MATLAB on two test images and evaluates the results. Huffman coding performed slightly better, with lower bit rates and higher compression rates. However, arithmetic coding would likely perform better on images with larger symbol dictionaries or alphabets due to its adaptive modeling capabilities. Both techniques achieved high efficiency for the test images.

1. INTRA-FRAME COMPRESSION USING ENTROPY CODING
ANIL ULAS KOCAK
Ankara University
Golbasi ANKARA
anilulaskocak@gmail.com
Abstract
Abstract- In this project, I only focus on intra-frame compression using two old entropy coding
techniques huffman and arithmetic coding. While implementing them separately, i also get good experi-
mental results. Thanks to these results it is easy to compare these two techniques and decide which one
is better.
1. Introduction
Compression is very old and common problem in data storage and data transfer field. While technol-
ogy is evolving, big data plays important role and triggers bandwidth and data storage problem. To solve
this problem, main contributions are made in bandwidth and the data which is used over bandwidth to
be sent or received. Approximately 35 years ago, researchers and scientist noticed that data compression
would play key role in data transferring field. Since that time, a lot of method have been come up and
been tried to find most effective solution.
In this work, My main aim is to implement two old and counted as a milestone of data compression
field method huffman and arithmetic coding. By implementing, i analyze both methods. After imple-
mentation, i focus on their results in compression capacities and compression performance. To evaluate
results, codeword length, entropy, bit per pixel rate and compression rate metrics are used for realizing
their strong side.
2. Experimental Works
Experimental works are implemented in MATLAB for this project. The project occurs from some
parts such as Discrete Cosine Transform, Quantisation, sending/receiving binary data and reversing
previous transformation. In this work, my first step is to transform image by DCT and quantize it with
constant quantization table in 8x8 blocks. After zigzag scanning, i extract symbols which are consisting
Dc, Ac value and zeros. I use both huffman and arithmetic entropy coding separately to transform these
values to bitstream. I also used codeword length, entropy bit per pixel rate and compression rate as
metrics for deciding which is better.
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2. Figure 1: Schema of image compression.
For more detailed, In this work, i extract symbols which has zero counts and AC values (zeros, | AC |)
from array which is occurred after zigzag scanning. This procedure is applied iteratively to get symbol
dictionary ans source symbols which will be transformed to bitstream from entire image. With these
dictionary symbols and source symbols, frequency of occurrences of each symbol is calculated so that
probabilities of each symbol is assigned as probability array. After getting probabilities, next step is to
use a entropy coding algorithm to transform source symbols to bitstream.
Huffman entropy coding is very old but important lossless entropy coding in data compression field.
By huffman, symbol probabilities is sorted and bottom two probabilities is added. This loop is made
iteratively until get probability value 1 at top. Then, by putting binary ’0’ to top and binary ’1’ to bottom,
codewords are extracted.
Figure 2: Example of Huffman tree and extracting codewords
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3. Arithmetic coding is also old but one of the lossless universal consent entropy coding algorithms. Un-
like huffman, in this algorithm bitstream is generated from probability differences of each symbols which
is called range. After transforming symbol to bitstream, new range is rescaled by assigning symbol’s
range. This continues until each source symbol are transformed and added to bitstream iteratively. After
getting bitstream, last step to evaluate finally is computing entropy value by H = i=1 p(si)log2(si)
equation, mean codeword length by L = i=1 p(si)codeword(si) equation, count of total bits, bit rate
by NumberofBitstotal
Heightimage×Widthimage
in bit per pixel, compression rate by
uncompressedimage
compressedimage
, efficiency by E = H
L
and
lastly redundancy by R = 1 − E.
Figure 3: Example of Arithmetic Coding
3. Experimental Results
In this section, i will mention about this work’s steps and evaluate them separately. As first step of
project, after getting probabilities from symbols and their number of occurrence, entropy which consists
of dictionary symbols (zeros, | AC |) and EOB symbol is computed by H = i=1 p(si)log2(si) equa-
tion (zigzag.m, make − symbols.m and make − p.m in attached matlab file). Entropy(H) = 3.847
bit pixel for image (a) 4a and Entropy(H) = 3.939 bit pixel for image (b) 4b
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4. (a) Lena.tif size(512X512) (b) ice cif 001.tif size(288X352)
Figure 4: Test images
To find required maximum bitsize for quantized DC values of each 8x8 block finding biggest DC value
is needed. After scanning all DC values in image, biggest one’s bitsize assign to other ones so each DC
values consist of completely same size with biggest DC value. Furthermore, one sign bit for each DC
value is added to these bitsizes and sum of these bitsizes is assigned as maximum bitsize for quantized
DC values (DCbits.m in attached matlab file). 28672 bits are computed for picture (a) 4a and 11088 bits
for picture (b) 4b
First entropy coding implementation of this work is huffman coding (encoder-huffman.m. After im-
plementing huffman coding and getting codewords by symbols’ probabilities for each image separately,
bitstream which can transfer is generated by source symbol. With bitstream and other variables mean
codewords length-bit rate-compression rate-efficiency-redundancy are computed by their formula.
Table 1: Huffman coding performance
HUFFMAN CODING
Image (a) (b)
Total Number of Bits 164250 57383
Mean Codeword Length (bit/pixel) 3.876 3.967
Bit rate(bit/pixel) 0.627 0.566
Compression Rate (times) 12.8 14.1
Efficiency 0.993 0.993
Redundancy 0.007 0.007
Total Number of bits metric is related with image size, unique symbols which symbol dictionary has
and symbols which have low frequency of occurrence. Although image 4b has smaller size, it has almost
same amount of unique symbols but it still costs less size of bitstream. Huffman coding is very useful for
these test images because of that mean Codeword Length is almost same entropy. If we look efficiency,
we can easily realize that it is almost 1 (top value) and redundancy is 0 (bottom value). Bit rate and
related compression rate is also at acceptable level. We can notice that huffman is very appropriate
entropy coding algorithm for both images.
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5. After analyzing results and metrics, dekodlama.m is called and then reconstructed image is got. Last
metric of analysis is PSNR. Because of huffman is lossless entropy coding algorithm, only loss has been
occurred in quantization stage of image. Expected value of PSNR is that image which has smaller size
has bigger PSNR value. PSNRs are PSNR = 35.78 for image (a) and PSNR = 36.76 for image (b)
Another entropy coding algorithms for this work is arithmetic coding. encoder − arithmetic.m is
called for implementation and getting results from test images. By implementing arithmetic coding fixed
model is used, so probabilities do not update itself iteratively. It is expected that because of arithmetic
coding has less dependencies to extended sources, by enlarging symbol dictionary, arithmetic coding
does not lose too much its efficiency relatively according to huffman coding.
Table 2: Arithmetic coding performance
ARITHMETIC CODING
Image (a) (b)
Total Number of Bits 167528 58695
Bit rate(bit/pixel) 0.639 0.579
Compression Rate (times) 12.5 13.8
After arithmetic coding implementation, we notice that total number of bits is bigger in image 4a than
image 4b like it is expected. Because, like in huffman impelementation, Total Number of bits metric is
related with image size. Also Bit Rate and Compression Rate which corresponds to Total Number of
bits is bigger in image 4a due to image size.
4. Evaluation
Huffman and Arithmetic coding are two old entropy coding techniques. After implementation of these
algorithms in matlab, results are occurred at table.
Table 3: Comparison of Arithmetic and huffman coding for test images
ARITHMETIC CODING HUFFMAN COD˙ING
Image (a) (b) (a) (b)
Total Number of Bits 167528 58695 164250 57383
Bit rate(bit/pixel) 0.639 0.579 0.627 0.566
Compression Rate (times) 12.5 13.8 12.8 14.1
As a conclusion, Is is expected before implementation that image that has bigger image size is im-
plemented with more bits theoretically and it is occurred experimentally like it is expected. But it is
also expected that arithmetic coding has less bits than huffman coding in same test image due to being
newer according to huffman. It is seems that the reason why arithmetic coding has more total number of
bits than huffman coding is that huffman coding is working well because of images that does not have
extended sources and too much unique symbols. In other word, huffman coding has good compression
rate, bit rate and performance on images that does not contain large alphabets or symbol dictionary for
extended source. But if i used test images that has more unique symbols or large alphabets, arithmetic
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6. coding would be more useful and well performance against huffman coding. Another advantage of arith-
metic coding against huffman is the chance of choose of adaptive arithmetic model while in huffman it is
not possible. In other word, adaptive model of arithmetic coding provides varying symbol probabilities
by time while huffman provides only stable symbol probabilities.
References
[1] IAN H. WITTEN, RADFORD M. NEAL, and JOHN G. CLEARY ARITHMETIC CODING FOR
DATA COMPRESSION 30.6.520. 1987.
[2] Rafael C. Gonzalez, Richard E. Woods and S.Eddins Digital Image Processing Using MATLAB 2nd
Edition 2009: Gatesmark Publishing
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