Bivariatealgebraic integerencoded arai algorithm for

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 03 Special Issue: 01 | NC-WiCOMET-2014 | Mar-2014, Available @ http://www.ijret.org 37
BIVARIATEALGEBRAIC INTEGERENCODED ARAI ALGORITHM FOR
EXACT COMPUTATION OF DCT
Sumi Thomas1
, M. Mathurakani2
1
P.G Student, 2
Formerly Scientist in DRDO, Professor, Electronics and Communication Engineering, Toc H Institute of
Science And Technology, Kerala, India
Abstract
Discrete Cosine Transform (DCT) function is widely used in many standard image and signal processing applications. Error free
architectures with high throughput and reduced hardware area is a major concern.This paper surveys and particularly collates all the
information available on arai algorithm that uses algebraic integer coded computation.Bivariate Algebraic Integer(AI) encoded 2-D
DCT algorithm ensuresquantization noise free implementation of2-D DCT. This algorithm realizes an error-free 2-D DCT without
using Final Reconstruction Steps(FRS) between row–column transforms, leading to an 8×8 2-D DCT that is entirely free of
quantization errors in AI basis,thereby avoiding the leakage of quantization noise between DCT channels.This architecture has
infinite accuracy till the reconstruction step. The architecture enables low-noise high-dynamic range applications in image
processing that requires full control of the finite-precision computation of the 2-D DCT. The 2D DCT algorithm is simulated under
fourdifferent conditions, on a gray coded 512X512 image, and PSNR value is measured and compared using MATLAB.
Keywords: Algebraic Integer quantization, Discrete Cosine Transform (DCT), Scaled DCT/IDCT, Quantization Noise,
Fixed point representation.
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1. INTRODUCTION
High Dynamic Range image processing are under exponential
growth and, therefore, the demand for applications capable of
high dynamic range (HDR)imagingis accordingly increasing.
Such HDR systems operating at high resolutions require an
associate hardware capable of significant throughput at
allowable area-power complexity. Efficient codec circuits
capable of both high speeds of operation and high numerical
accuracy are needed for next generation systems. Such
systems may process massive amounts of input feeds, each at
high resolution, with minimal noise and distortion while
consuming as little energy as possible.
Straightforward techniques for spatial domain processingof
images and videos are expensive and time consuming. An
alternate approach is to process the image in compressed DCT
domain, so that the output compressed stream corresponds to
the output image and it confirms the syntax
of 8 X 8 block.
The major difficulties encountered are: the computational
complexity of image compression and decompression
algorithms, the high rate of data to be manipulated and
increase in hardware area. For this reason there has been a
great effort in recent years to develop fast algorithms that
perform these tasks directly in the transform domain and
thereby avoid the need of direct compression – the DCT
which requires 38.7% of the execution time[14]. Usually, the
2-D DCT is computed by successive calls of the 1-D DCT
applied to the columns of an 8×8 sub image, then to the rows
of the transposed, resulting in intermediate calculations [18].
The very large-scale integration (VLSI) implementation of
trigonometric transforms such as DCT and discrete Fourier
transform is indeed an active research area [14][15].
An ideal eight-point 1-D DCT requires multiplications
bynumbers in the form c[n] = cos (nπ/16), n = 0, 1. . .7.
Theseconstants impose computational difficulties in terms of
number binary representation since they are not rational. For
fast implementation we replace floating point numbers by
appropriate dyadic rationalsby truncating or rounding offto
approximate values. Both, rounding and approximation leads
to truncation errors. Thus, instead of employing the exact
value c[n], a quantized value is considered. Clearly, this
operation introduces errors.
In order to overcome this disadvantage Algebraic Integer
Encoding scheme is adopted for DCT calculation[16]. With
this scheme an error free representation of cosine functions
become possible. Algebraic integers are roots of
monicpolynomials that have integer coefficients with leading
coefficient equal to unity. Here the cosine valuesarereplaced
by a set of array of integers. After performing thecomputation
in AI domain,a reconstruction algorithm is required to map the
array of AI encoded functions back into their fixed point
representation at a given precision [4].

_______________________________________________________________________________________
In [11], AI-based procedures for the 2-D DCT are proposed.
This architecture was based on the low-complexityArai
algorithm [2], which formed the building-block of each1-D
DCT using AI number representation. The Arai algorithm is a
popular algorithm for video and image-processing applications
because of its relatively low computational complexity. The
eight-point Arai algorithm proposed in [10] needs only five
multiplications to generate the eight output coefficients.
Thus, low complexity Arai algorithm is chosen as a
foundation for proposing optimized architectures having lower
complexity and lower noise. This paper extends the eight-
point 1-D AI-based DCT architecture into a fully parallel
time-multiplexed 2-D architecture for 8×8 data blocks[4]
which is further extended to 512 x 512 image and its inverse
DCT calculation.
2. PRELIMINARIES AND PAPER DESCRIPTION
The 2D-DCT transformation a block {x (m,n) }m,n=0
N-1
in the
spatial domain into a matrix of frequency components { X(i,j)
}i,j=0
N-1
according to the following equation:
(1)
Where
C (u) = if u= 0
1 if u ≥1
(2)
And N is the length of the sequence.
Inverse DCT is calculated by:
(3)
DCT has the following properties:DCT has energy packing
capability. DCT is orthogonal and separable, it leads to the
reduction of spatial redundancy forthe input signal and has
found wide applications in speech and image processing. The
2-Dimensional DCT, over a small block of pixels, has been
widely used as afrequency analysis and compression algorithm
in image processing standard likeMPEG-2.
2.1 Theoretical Background
The AI encoding for image processing was originally
proposed by Cozzens and Finkelstein[5].They introduced a
new approach for computing the DCT that uses the residue
number system (RNS) in a ring of algebraic integers. In their
work, the signal sample is represented by a set of typically
four to eight small integers, combines, with the Residue
Number System (RNS) to produce processors composed of
simple parallel channels. This leads to a 1-D bivariate-encoded
Arai DCT algorithm by Wahid and Dimitrov [11], [16].
Recently, subsequent contributions by Wahid using bivariate-
encoded 1-D Arai DCT blocks for row and column transforms
of the 2-D DCT has led to practical area-efficient VLSI
image-processing circuits with low-power consumption. This
paper summarizes and compares the accuracy of both 1-D and
2-D DCT based on conventional fixed-point arithmetic as well
as on AI encoding. The use of AI encoded array has the
following advantages:
• The architecture is made error free by using exact
integer representation instead of quantized value.
• Eliminate the need of intermediate reconstruction
there by eliminate error propagation.
• Small array of integer values are used instead of
truncated fixed point representation.
• Infinite precision and accuracy till the last
reconstruction step.
• Saves hardware area by completely replacing
multiplications by shift and negation operations
thereby increasing the speed of computations.
• Reduces the total hardware area.
2.2 AI Encoding And Decoding
Algebraic integer encoded DCT is adopted to eliminate the
quantization noise. Such a representation is based on a
mapping function that links input numbers to integer arrays.
AIs are root of monic polynomials with leading coefficient
equal to unity [2]. The set of AIs are commutative, that is
addition and multiplication operations are commutative and
also satisfy distribution over addition. A general AI encoding
mapping has the following format[4]:
fenc(x; z) =a = [a0 a1]T
(4)
Wherea is a multidimensional array of integers and z is afixed
multidimensional array of AIs. It can be shown thatthere
always exists an integer array such that any real number can
berepresented with arbitrary precision [18]. For eg.,the
irrational number 1-3 can be encoded as:
fenc
This forms an exact representation.Decoding operation is
furnished by
fdec(a; z) = a •z=a0+a1.z1 (5)
An exact AI array representation for all the DCT constants can
be derived as shown in [11]. The Al representation has the
following advantages:

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• Represents the required constants without error.
• Small integers are used andhence adopt a simple
architecture and fast signal processing at an
increased speed.
• Simple encoding and decoding operation.
2.3 AI Encoded 1-D DCT
Based on theseparability property of DCT and applying row
column decomposition [7], the 2-D 8 x 8 DCT/IDCT can be
realized by computing one-dimensional (1-D) 8-
pointDCT/IDCT on all the rows followed by columns of
thetotal 8 x 8 block. The computational complexity can be
reduced by simplifying the DCT/IDCT coefficients by
eliminating redundant coefficients. Thus for computation of
DCT based on arai algorithm [4] we require the following
coefficients:
C[4]= (6)
C[6] = (7)
C[2] – C[6] = (8)
C[2] + C[6] = (9)
Byapplying 2D AI Encoding [4]
Let z1 = (10)
And z2 = (11)
The 2-D array for AI encoding is:
z= 1 z1
z2 z1z2 (12)
This leads to a 2-D-encoded coefficient of the form
fenc(x; z) = a = a 0,0 a 1,0
a0,1 a 1,1 (13)
2D error free integer representation of the cosine values can be
represented as shown in Table 1[4]. The employed integers
are powers of two, and hence can be implemented by simple
shift registers as shown in fig-1.For a given encoded number
a, the decoding operation is expressed by:
fdec(a; z) = a • z = a0,0 + a1,0z1 + a 0,1z2 + a 1,1z1z2
(14)
Table -1: 2 D AI Encoded DCT constants
C[4] C[6] C[2] – C[6] C[2] + C[6]
0 0
0 1
0 1
-1 0
0 0
2 0
0 2
0 0
3. AI ENCODED ARAI DCT ARCHITECTURE
The image to be compressed is the input to the architecture in
a row-wise order at the rate of one pixel (8 bits per pixel) per
clock cycle as shown in the fig.1.The AI array of
DCTcoefficients is multiplied with the input. Thus each input
pixel x0 after AI multiplication will be represented as an array
as shown below:
x0
(a)
x0
(b)
x0
(c)
x0
(d)
= x = x0
(a)
+ x0
(b)
z1 + x0
(c)
z2 + x0
(d)
z1z2
(15)
Where, (a),(b),(c) and (d) shows the position. Thus for each
input pixel there will be 4 output channels, resulting in a total
of 32 channels. Redundant bits can be eliminated, thereby
resulting in only 22 channels as shown in the architecture in
fig.1.
Fig -1: 1-D AI Arai DCT block [4]
Since the 2-D DCT is a separable transform, the row-column
decomposition technique may be used to calculate the 2-D
DCT. As a result, the functional units used for the first 1-D
stage, can be re-used for the second 1-D DCT stage. In this
case, the 2-D DCT of a 512 x 512 image hasbeen performed
first by grouping it into blocks of 8 x 8 data. Then scaled arai
1-D DCT algorithm is applied to each row.This paper adopts
the work of Agostini[11] that implemented Arai scaled 1-D

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DCT algorithm. The DCT coefficients produced by this
algorithm are scaled values. Hence, we should multiply the
scaled ones with a postscalar matrix to get the actual values
[1].
4. FINAL RECONSTRUCTION STEP
The 22 channel representation corresponding to 8 pixel
inputs must requires a reconstruction step to convert it back
into corresponding 8 pixel DCT output. Intermediate
reconstruction after each 1 D DCT is eliminated inorder to
prevent quantization noise cross coupling between DCT
channels. The quantization noise is injected only at the final
output. Therefore, noise signals are uncorrelated.
ArjunaMadanayake proposes two FRS based on1) optimized
Dempster–Macleod multipliers, and 2) expansion
factor scaling[4]. DCT of an image is obtained in MATLAB
using both the FRS methods and their PSNR values are shown
in table 2.
4.1 Dempster- Macleod Method Based FRS
In this method, the encoded elements aredecoded according to
equation(15)as:
X i,k
(q)
= X i,k
(q)(a)
+ X i,k
(q)(b)
z1 + X i,k
(q)(c)
z2+X i,k
(q)(d)
z1z2 , q
{a, b, c, d} (16)
Values of z1and z2are given by equ.(10) and (11).The result is
the kth
row of the final DCT data Xi,k, i = 0, 1,…7 .
For each value of q, equation (16)can beunfolded as:
X i,k
(a)
= X i,k
(a)(a)
+ X i,k
(a)(b)
z1
+ X i,k
(a)(c)
z2+X i,k
(a)(d)
z1z2 (17)
X i,k
(b)
z1 =X i,k
(b)(a)
z1 + X i,k
(b)(b)
z1
2
+ X i,k
(b)(c)
z1z2+X i,k
(b)(d)
z1
2
z2 (18)
X i,k
(c)
z2 = X i,k
(c)(a)
z2 + X i,k
(c)(b)
z1z2
+ X i,k
(c)(c)
z2
2
+X i,k
(c)(d)
z1z2
2
(19)
X i,k
(d)
z1z2 = X i,k
(d)(a)
z1z2 + X i,k
(d)(b)
z1
2
z2
+ X i,k
(d)(c)
z1z2
2
+X i,k
(d)(d)
z1
2
z2
2
(20)
Thesum ofthe above given equ. returns X i,k.The values z1, z2,
z1z2 used in the above given equations are irrational numbers.
So the values recovered after this decoding are approximated
fixed point values.
4.2 FRS Based on Expansion Factor Scaling
In this method, the quantities z1,z2,z1z2are scaledinto integer
values, thereby eliminating quantization error since we use
integer values instead of fixed point values. This would
facilitate the usage of simple integer arithmetic.
An expansion factor is defined as a real number α*
>1 that
satisfies the following minimization:
α*
= argmin||α· ζ − round(α· ζ)|| (21)
α>1
Where||· ||is a given error measure and round (·) is the
rounding function.In the range α∈[1, 256],taking an optimal
value α*
= 167.2309[4] , we have the following scaling:
*
However smaller values of α*
are more desirable because this
requires much more simple hardware design. Now let us take
α’
=4.5961 [4],then:
’
Considering a given expansion factor α, the decoding
operation given in equ. (16)can be generalized as:
X i,k=1/ α [ α · X i,k
(a)
+ n1· X i,k
(b)
+ n2· X i,k
(c)
+ n3· Xi,k
(d)
] (22)
Where n1, n2, and n3 are the integer constants implied by the
expansion factor α.Here these constants are {437, 181,473},
for α = α*
, and {12, 5, 13}, for α = α’
. The FRS is performed
using both values of αand the PSNRvaluesare tabulated in
Table -2.
5. COMPARISON AND RESULTS
We simulated the following four algorithms for DCT.
• Standard Arai Algorithm Based DCT
• AI Quantized DCT with α=4.5961
• AI Quantized DCT with α=167.2309
• Conventional Fixed Point Realization of DCT.
The image is decompressed back by applying conventional
floating point IDCT in MATLAB.The simulation is done in
MATLAB. The input is 512 x 512 lena.jpg image. The
proposed algorithms have been applied to the input. The
higher the value of PSNR, the better is the result. PSNR equal
to infinity means the input image is perfectly reconstructed
after IDCT. From Table-2 we conclude that, when we
represent the DCT coefficients interms of there fixed point
value and apply truncation with an accuracy of 8 bits,
quantization noise is injected resulting in lossy compression
and decompression. And hence the PSNR value is low.

_______________________________________________________________________________________
Table -2: Algorithms And PSNR values
Algorithm PSNR
Arai DCT INFINITY
Algebraic Integer DCT with
α=4.5961
41.3979
Algebraic Integer DCT with
α=167.2309
52.7532
Fixed Point DCT 24.6046
Algebraic integer encoded DCT is adopted to eliminate the
quantization noise. Such a representation is based on mapping
function that links input numbers toan integer arrays instead of
converting them into fixed point values.
Reconstruction step as discussed already is performed to map
the DCT array to its corresponding fixed point representation.
There is no intermediate reconstruction and hence prevents the
leakage of quantization noise between DCT channels. From
simulation results we can conclude that AIQ DCT has a better
PSNR compared with fixed point realization. The proposed
algorithms have been applied to the 512 x 512 image (fig.2
:(a)) and the input image is reconstructed back successfully
after performing both DCT (fig.2:(b)) and IDCT with floating
point MATLAB (fig.2:(c)) .
(a)
(b) (c)
Fig -2: (a) INPUT IMAGE LENA 512 x 512 (b)DCT
OUTPUT IMAGE (c) IDCT OUTPUT IMAGE
6. CONCLUSIONS
A time-multiplexed architecture proposed for the real-time
computation of the bivariate AI encoded 2-D Arai DCT was
simulated in MATLAB. The output of Arai AI DCT algorithm
is obtained and the image is reconstructed back. The algorithm
is not only completely multiplier-free but also quantization
free up to the final output channels. The architecture does not
have intermediate FRS sections between the column-wise and
row-wise DCT operations. Thus the quantizationnoise appears
only at the final output stage of the architecture. The location
of the FRS at the final output stage resulted in the complete
decoupling of quantization noise and provides the flexibility
of user selectable accuracy. Simple shift and negation
operations results in reduced hardware and ease of
computation.
In this paper we simulated five algorithms and concluded that
AI encoded DCT effectively reduces the overall arithmetic
operations and allows multiplication-free,parallel and very fast
implementation. With AI encoding DCT based image
compression can be made error free.The use of integers in the
encoding scheme also results exact reconstruction.
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BIOGRAPHIES
Sumi Thomas received the B.Tech degree in
Electronics and Communication Engineering
from Mar Baselios Christian College Of
Engineering And Technology, Mahatma
Gandhi University, Kerala, currently doing
M.Tech in VLSI and Embedded Systems at Toc H Institute of
Science and Technology, Cochin University.
Prof. M. Mathurakanihas graduated from
AlagappaChettiar College of Engineering and
Technology of Madurai University and
completed his masters from PSG college of
Technology, Madras University. He has
worked as a Scientist in Defence Research and Development
Organization (DRDO) in the area of signal processing and
embedded system design. He was honoured with the DRDO
Scientist of the year award in 2003.Currently he is a professor
in Toc H Institute of Science and Technology, Arakunnam.
His area of research interest includes signal processing
algorithms, embedded system implementations, reusable
architecture and communication systems.

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