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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 20, NO. 8, AUGUST 2010
Comments
Comments on “An Analog 2-D DCT Processor”
Surya Prakash Noolu and Maryam Shojaei Baghini, Senior Member, IEEE
Abstract—In the paper “An analog 2-D DCT processor,”
authors have presented a row–column method for computing 2-D
discrete cosine transform (DCT). They have reported minimum
peak signal-to-noise ratios (PSNR) 40.6 dB and 31.4 dB for 4point and 8-point DCT, respectively. The main objective of this
comment letter is to point out that those PSNR values are
not correctly calculated. The actual minimum PSNR values are
shown to be 24.7 dB and 22.7 dB for 4-point and 8-point DCT,
respectively. Similarly, maximum PSNR values are corrected in
this letter.
TABLE I
2-D DCT Coefficient Errors Ek1 ,k2 in Terms of LSBs for 4-Point
2-D DCT Processor [1]
k2
0
1
2
3
Index Terms—Discrete cosine transform (DCT), peak signalto-noise ratio (PSNR).
All 1µA
0
1
k1
−1
−1
0
0
1
0
0
0
2
3
0
0
−1
0
0
−1
0
0
k1
1
2
3
4
k2
All 5µA
0
1
−3
−5
0
−2
0
−1
0
−1
2
3
0
1
−2
−1
−2
−2
−1
−1
as
I. Introduction
MSE =
P
EAK signal-to-noise ratio (PSNR) is the most commonly
used figure of merit to measure the quality of a reconstructed image, compared with the original image in lossy
compression codecs. PSNR is defined as ratio of the maximum
possible power of a signal and the power of corrupting noise.
In the context of 2-D discrete cosine transform (2-D DCT),
signal is the original data and noise is the error introduced by
the processor. DCT transforms the signal or image from spatial
domain to frequency domain. N × N matrix I, 2-D DCT of
an N × N real signal matrix Ki,j (i, j = 0, 1, ..., N − 1) is
defined as
2
(2i + 1)kπ
= c(k)c(l)
Ki,j × cos
N
2N
i=0 j=0
N−1 N−1
Ik,l
× cos
(2j + 1)lπ
k, l = 0, 1, ..., N − 1
2N
In (3) K is the original image data, i.e., input to the DCT
processor and L is the reconstructed image data obtained by
calculating ideal inverse discrete cosine transform (IDCT) of
the DCT processor output, denoted by Y
Li,j
N−1 N−1
2 (2i + 1)kπ
=
c(k)c(l)Yk,l × cos
N k=0 l=0
2N
× cos
(2j + 1)lπ
i, j = 0, 1, ..., N − 1
2N
(4)
where c(0) = √12 and c(k), c(l) = 1 for k, l = 0. In [1], MAX
is 5 µA, since the DCT processor is designed for the input
current range from 1 µA to 5 µA.
II. Comment on PSNR
and c(k), c(l) = 1 for k, l = 0.
where c(0) =
Accordingly, PSNR is given as
MAX2
MSE
(3)
(1)
√1
2
PSNR = 10. log
N−1 N−1
1 K(i, j) − L(i, j)2 .
N 2 i=0 j=0
In [1], authors have introduced 2-D DCT coefficient error
matrix E for different data inputs which is
Ek1 ,k2 = Ik1 ,k2 − Yk1 ,k2
k1 , k2 = 0, 1, 2, ..., N − 1.
(2)
where MAX is the maximum possible pixel value of the
image. In (2), MSE is the mean squared error and is defined
Manuscript received June 24, 2009; revised March 8, 2010; accepted March
8, 2010. Date of publication May 20, 2010; date of current version August
4, 2010. This paper was recommended by Associate Editor Y.-K. Chen.
The authors are with the Department of Electrical Engineering,
Indian Institute of Technology Bombay, Mumbai 400076, India (e-mail:
noolusurya@iitb.ac.in; mshojaei@ee.iitb.ac.in).
Digital Object Identifier 10.1109/TCSVT.2010.2051281
(5)
In (5) I is the ideal DCT coefficient matrix. Y is the actual
DCT coefficient matrix which is output of the processor in [1].
2-D DCT coefficient error matrices E of 4-point and 8-point
transforms for two different data input sets were taken from
[1] and shown in Tables I and II. In Tables I and II, elements of
E (Ek1 ,k2 for k1 , k2 = 0, 1, 2, ..., N −1) are mentioned in terms
of least significant bits (LSBs) and according to [1] each LSB
is equivalent to 156 nA. Table III shows ideal dc coefficients
c 2010 IEEE
1051-8215/$26.00 NOOLU et al.: COMMENTS ON “AN ANALOG 2-D DCT PROCESSOR”
1163
TABLE II
TABLE IV
2-D DCT Coefficient Errors Ek1 ,k2 in Terms of LSBs for 8-Point
2-D DCT Processor [1]
The Reconstructed Image L of 8-Point Transform for Input
Data Set All 1µA
k2
0
1
2
3
4
5
6
7
k2
0
1
2
3
4
5
6
7
k1
k1
0
1
−11
−2
−1
−1
−1
0
0
−1
0
1
0
0
0
0
0
1
0
1
−9
−6
−5
−6
−3
−2
−2
−4
−4
0
1
0
1
0
0
0
All 1µA
2
3
4
2
1
0
0
0
0
0
0
0
1
0
0
1
1
−1
−1
All 5µA
2
3
−2
0
−1
0
−1
0
2
−1
−4
0
−1
−1
2
1
3
−3
5
6
7
3
0
0
0
2
1
0
1
2
0
1
−1
0
1
0
0
1
0
0
0
−1
−1
1
0
1
0
0
−1
1
−1
0
0
4
5
6
7
4
1
−1
0
4
1
−2
0
0
0
1
−1
1
1
0
0
−2
0
0
0
−2
−2
2
1
−2
−1
1
−2
1
−2
0
−1
k2
1
2
3
4
5
6
7
8
All 1µA
4
1
2
3
1.027
1.164
0.967
0.945
1.105
1.020
1.140
0.917
1.483
1.298
1.371
1.334
1.334
1.229
1.297
1.472
1.509
1.105
1.135
1.387
1.263
1.229
1.227
1.241
k1
1.239
1.315
1.199
1.170
1.236
1.342
0.950
1.260
5
6
7
8
1.319
1.311
1.299
1.106
1.058
1.295
1.267
1.112
1.471
1.236
1.273
1.301
1.389
0.981
1.291
1.259
1.376
1.193
1.256
1.059
1.380
1.157
1.155
1.044
1.310
1.223
1.187
1.114
1.155
1.009
1.117
1.112
TABLE V
Calculated PSNRs and Given PSNRs (DB) in [1]
PSNR
4-point transform
8-point transform
Calculated PSNR values
All 1 µA
All 5 µA
32.56
24.67
25.84
22.65
PSNR values in [1]
All 1 µA
All 5 µA
63.0
40.6
32.7
31.4
TABLE III
Ideal DC Coefficients I0,0 for Different Input Data Sets
4-point transform
8-point transform
All 1 µA
4.0
8.0
All 5 µA
20.0
40.0
(i.e., ideal DCT coefficient Ik1 ,k2 with k1 , k2 = 0) of 4-point
and 8-point 2-D DCT, for two dc input data sets, all 1 µA and
5 µA. Since input data is dc, all ideal ac coefficients are zero.
In this letter, ultimate PSNR values are derived from reported matrices I and E in [1]. By rearranging (5) actual DCT
coefficients, Yk1 ,k2 , are computed from ideal DCT coefficients
Ik1 ,k2 and DCT coefficient errors Ek1 ,k2 (k1 , k2 = 0, 1, 2, ...,
N − 1) as given by (5)
Yk1 ,k2 = Ik1 ,k2 − Ek1 ,k2
k1 , k2 = 0, 1, 2, ..., N − 1.
As an example, PSNR of 8-point DCT transform for input
data set all 1 µA is calculated as follows:
1) matrix Y is calculated by (6) using the data corresponding to 1 µA in Tables II and III;
2) matrix L is computed by applying ideal IDCT on Y . L
is given in Table IV. All elements of K are 1 µA;
3) value of MSE is obtained as 0.0650 µA2 by solving (3).
Finally, PSNR value of 25.84 dB results from (2) with
MAX2 = 25 µA2 .
The basic concept of PSNR calculation is the same for both
4-point and 8-point transforms. Table V shows PSNR values
obtained from (2) and corresponding values reported in [1].
As Table V shows error in reported values of PSNR varies
from 6.86 dB to 30.44 dB. Accordingly, we informed the first
author of [1] regarding the same [2].
(6)
Thereby, reconstructed image matrix L is computed by
performing ideal IDCT on Y using the MATLAB program.
The actual PSNR values are then obtained using (2) and (3).
PSNR values are calculated for two input data sets K, all 1 µA
and all 5 µA.
References
[1] M. Pänkäälä, K. Virtanen, and A. Paasio, “An analog 2-D DCT processor,”
IEEE Trans. Circuits Syst. Video Technol., vol. 16, no. 10, pp. 1209–1216,
Oct. 2006.
[2] Mikko Pänkäälä, private communication, Jan. 11, 2009.