Term Project report for EE5302

Transcription

1 Term Project report for EE5302 Submitted by: Vidhya N.S Murthy Student ID:

2 Project statement To study the statistical properties of a video signal and remove spatial redundancy using different differential pulse code modulation techniques. Approach The project is broken into a set of smaller problems. 1. In the first step a routine for getting the histogram was written and was tested by generating a set of uniform random variables. 2. In this step a set of images where taken and the central limit theorem was observed and the entropy of the images were calculated leading us to conclude about the redundancy in information content in an image. 3. In this step different DPCM techniques were carried out to remove spatial redundancy from an image. 4. In this step the different predictors in step 4 are compared. 5. In this step the detrimental effect of errors is determined. Problem statement 1 Use MATLAB to generate random numbers (uniform distribution) between a and b (you choose your own a and b) and plot their histogram. Details: a ) First generate 1000 values for this distribution and write C code to perform the histogram function. You can display your histogram using MATLAB (no need to write your own graphical display). b) Now repeat step for 10,000, 100,000 and a million samples. c) Observe how the histogram changes and comment. Approach 1. The random variables were generated in MATLAB using the rand function. The interval (a,b) chosen for the generating the uniform random variable was (0,1). 2. Histogram routine was written in C and interfaced to MATLAB using MEX files concept. 3. The histogram routine mimics the MATLAB hist function with the prototype N = hist(m) Here M is a 1-D array and N is the histogram. The routine bins the array elements into 10 equally spaced containers. N is an array of elements denoting the number of elements in each container. 4. A MATLAB.m file was written to generate the histograms for 1000,10000, and a million samples of the uniform random variable. 5. The graphical display was achieved using the bar function.

3 Results Frequency Sample values Fig 1: Histogram for 1000 samples of Uniform random variables in the interval (0,1) Frequency Sample values Fig 2: Histogram for samples of Uniform random variables in the interval (0,1)

4 Frequency Sample values Fig 3: Histogram for samples of Uniform random variables in the interval (0,1)

5 Frequency Sample values Fig 4: Histogram for a million samples of Uniform random variables in the interval (0,1) Observations 1. The histogram approaches the distribution of the uniform random variable as the number of samples are increased Comments 1. The observations are consistent with the principle of statistical regularity that averages obtained in long sequences of repetitions of random experiments yield approximately the same value. 2. Hence as the number of samples increases the histogram approaches the distribution of uniform random variable

6 Source Code M file k = 0.05:0.1:0.95; m = rand(1000,1); n = histogram(m,10); bar(k,n,1); figure; m = rand(10000,1); n = histogram(m,10); bar(k,n,1); figure; m = rand(100000,1); n = histogram(m); bar(k,n,1); figure; m = rand( ,1); n = histogram(m); bar(k,n,1);

7 Problem Statement 2 Collect 5 images of size greater or atleast equal to 512 x 512 pixels. Try to select pictures which are diverse with lot of different objects. Now plot the pdf of gray value of all pixels accumulated from the 5 images using Matlab functions. What do you conclude from the result? Show the plot and explain your observation. Approach 1. 6 grayscale images were selected at random. Image dimensions were 512x512,1976x1978,512x512,378x504,800x701 and 2245x They were read into the Matlab workspace using imread(). 3. All images chosen contained only luminance values. The matrices were then reshaped into single dimension array using the MATLAB function reshape(). 4. The histogram function created as a mex file for step 1 was enhanced to take in inputs of the form shown below N = histogram(m) N = histogram(m,numbins) When used as prototype 1 histogram bins the elements of a 1 D array into 10 containers and when used as prototype 2 histogram bins the elements of a 1-D array into NumBins number of containers. This follows the design of the MATLAB hist function. 5. The histograms of the 6 images were calculated and summed up. 6. The pmf was calculated and plotted. 7. Mean and Entropy were calculated. Thumbnails of Images used boat mary carter nebula lena statue

8 Results PMF plot Fig 5: Histogram plot for the pixels in the image Mean Mean was calculated to be Entropy Entropy was calculated to be Observations 1. Entropy defines the upper limit of compression that can be achieved. Here it can be inferred that the image data of above images can be represented by a min of bits per pixel before we start losing information.

9 Source Code M file image = double(imread('c:\bw\carter.jpg')); [m,n] = size(image); hist_input = reshape(image,m*n,1); num_inputs = m*n; acc1 = histogram(hist_input,255); image = double(imread('c:\bw\statue.jpg')); [m,n] = size(image); hist_input = reshape(image,m*n,1); num_inputs = num_inputs + m*n; acc2 = histogram(hist_input,255); image = double(imread('c:\bw\mary.jpg')); [m,n] = size(image); hist_input = reshape(image,m*n,1); num_inputs = num_inputs + m*n; acc3 = histogram(hist_input,255); image = double(imread('c:\bw\lena.jpg')); [m,n] = size(image); hist_input = reshape(image,m*n,1); num_inputs = num_inputs + m*n; acc4 = histogram(hist_input,255); image = double(imread('c:\bw\boat.png')); [m,n] = size(image); hist_input = reshape(image,m*n,1); num_inputs = num_inputs + m*n; acc5 = histogram(hist_input,255); image = double(imread('c:\bw\nebula.jpg')); [m,n] = size(image); hist_input = reshape(image,m*n,1); num_inputs = num_inputs + m*n; acc6 = histogram(hist_input,255); final_hist = acc1+acc2+acc3+acc4+acc5+acc6; pmf = final_hist./num_inputs; pixel_val = 1:255; stem(pixel_val,pmf); xlabel('pixel values'); ylabel('pmf'); title('probability mass function plot using data from 6 images'); mean = sum(pixel_val.*pmf); entropy = sum(pmf.*log(1./pmf));

10 MEX file implementation of the function Histogram() #include "mex.h" #include <math.h> #include <stdlib.h> double mymax(double *array_in,int array_size) double maximum; int i; maximum = array_in[0]; for (i = 1; i < array_size; i++) if (array_in[i] > maximum) maximum = array_in[i]; return maximum; double mymin(double *array_in,int array_size) double minimum; int i; minimum = array_in[0]; for (i = 1; i < array_size; i++) if (array_in[i] < minimum) minimum = array_in[i]; return minimum; void SplitIntoIntervals (double *bin_upper_limits,double data_min, double data_max,int num_bins) int i; double bin_size; bin_size = (data_max - data_min)/num_bins; bin_upper_limits[0] = data_min+bin_size; for (i = 1; i < num_bins; i++) bin_upper_limits[i] = bin_upper_limits[i-1] + bin_size; void myhist(double *data_in,double *hist_out,int m_in, int num_bins) int i,j; double data_in_max,data_in_min; double data_max,data_min; double *bin_upper_limits; data_in_max = mymax(data_in,m_in); data_in_min = mymin(data_in,m_in); data_min = floor(data_in_min); data_max = ceil(data_in_max); bin_upper_limits = (double*)malloc(num_bins*sizeof(double)); /* split data range into equal intervals */ SplitIntoIntervals(bin_upper_limits,data_min,data_max,num_bins); /* set histogram to zero initially */ for (i = 0; i < num_bins; i++) hist_out[i] = 0;

11 /* bin data */ for (i = 0 ; i < m_in; i++) for (j = 0 ; j < num_bins; j++) if ((data_in[i] < bin_upper_limits[j])) hist_out[j]++; break; free(bin_upper_limits); /* * * * */ This routine bins the elements of Y into 10 equally spaced containers and returns the number of elements in each container. Currently Y can only be an array void mexfunction(int nlhs, mxarray *plhs[],int nrhs, const mxarray *prhs[]) int m_in,n_in; int num_bins; double *data_in, *hist_out; if (nrhs > 2) mexerrmsgtxt("must have one/two input argument"); if (nlhs!=1) mexerrmsgtxt("must have one output argument"); if (nrhs == 1) num_bins = 10; else if (nrhs == 2) double *bin; bin = mxgetpr(prhs[1]); num_bins = bin[0]; /* Find the dimensions of the data */ m_in = mxgetm(prhs[0]); n_in = mxgetn(prhs[0]); if (n_in!= 1 ) mexerrmsgtxt("input must be 1 -D array"); plhs[0] = mxcreatedoublematrix(1, num_bins, mxreal); data_in = mxgetpr(prhs[0]); hist_out = mxgetpr(plhs[0]); myhist(data_in,hist_out,m_in,num_bins);

12 Problem statement 3 To perform various Differential Pulse Code Modulation (DPCM) techniques on a video signal to remove spatial redundancy. Techniques include both non adaptive and adaptive methods. + Quantizer predictor Fig 6: DPCM encoder + + predictor Fig 7: DPCM decoder The generalized DPCM encoder and decoder are shown in figures 6 and 7. In the following experiments the quantizer is not used. Input Video Sequence The video sequence used is the hall monitor sequence. Only the luminance components of the pictures were used in all the experiments. The 54th and the 55th frames were extracted for this purpose from the raw video file. The video format is the planar YUV 4:2:0 format. Frame 54 of Hall monitor sequence Frame 55 of Hall monitor sequence Classes of predictors used. Two classes of predictors were used. 1. Linear 2. Adaptive. Linear Predictors Predictors of the form X' = a1x1 + a2x2 + a3x3...anxn are called linear predictors. Here a1... an are calculated by minimizing the mean square error between the predicted value and the input pixel. Thus the a1... an which minimize the mean square error between the predicted pixel and the input pixel form the linear predictors.

13 i.e. E [(X X')2] = minimum. Therefore d/dai(e [(X X')2] ) = 0 for i = 1...n. Therefore we have n equations and n unknowns. We solve them to get the values of a1... an. Predictor 1... X2 X3 X X1 X... Fig 8: Arrangement of neighbours inside image X is the pixel to be predicted Predictor 1 is defined as X' = X1 if X is not a border pixel 0 if X is a border pixel Fig 9: Histogram of error values for predictor 1 Hence predictor 1 uses the previous pixel as prediction for the current pixel. This is the simplest possible predictor. The histogram plot for the error generated through predictor 1 is shown in the plot above. Predictor 2 Referring figure 8 predictor 2 is a third order linear predictor which is defined as below

14 X' = a1x1 + a2x2 + a3x3 0 if X is not a border pixel if X is a border pixel Predictor 2 uses the three neighbours left, top and topleft pixels to generate the prediction. Fig 10: Histogram of error values for predictor 2 The histogram plot for the error generated through predictor 2 is shown in figure 10. Predictor 3 Xp..... X2 X3 X X1 X... Fig 11: The neighbours in the current frame and the colocated pixel in the past picture is denoted by Xp Predictor 3 follows the same third order predictor method however instead of using the top left pixel it uses the pixel co located with the pixel to be predicted in the previous frame. X' = a1x1 + a2x3 + a3xp if X is not a border pixel 0 if X is a border pixel The histogram plot for the error generated through predictor 3 is shown in figure 12

15 Fig 12: Histogram of error values for predictor 3 Adaptive Predictors Linear predictors work well in most scenarios except when there are edges. Since the linear operation is a weighted average with global constants the predictor does not work well in the uncorrelated areas of the image, mainly the edges. This results in higher values of errors. It is well known in image processing that edges are preserved when median filters are used. In the following predictors a median predictor with linear sub-predictors is used. The predictors 4 and 5 are from [1]. Predictor 4 a2 a3 a4 a1 p Fig 13: spatial neighbours for predictor 4. The pixel to be predicted is pixel p The predictor is given by p = MED5 (a1,a3,l,r,a1) here l = a1/2 + (a3 + a4)/2 (overall level predictor); d = (a1 + a3)/2 (upper nearest neighbour level predictor) r = 2d a2 (-45o ramp predictor)

16 The histogram plot for the error generated through predictor 4 is shown in figure 14 Fig 14: Histogram of error values for predictor 4 Predictor 5 This predictor takes motion/moving regions of a picture into account. In this predictor which is an enhancement to the previous predictor the prediction is such that when there is a moving part in an image it is predicted from the spatial neighbours (neighbours in the current frame) and when there is a still portion in the picture it is predicted from the temporal neighbours (i.e from the past frame) a2 a3 a4 a1 p a7 a8 a9 a6 a5 a10 Frame n Frame n-1 Fig 15: Spatial and Temporal neighbours for predictor 5. The pixel to be predicted is pixel p The predictor is defined as follows p = MED5 (a1,a3 a8 + a5, a5, r, a1 a6 + a5) where as before d = (a1 + a3)/2 (upper nearest neighbour level predictor) r = 2d a2 (-45o ramp predictor)

17 The histogram plot for the error generated through predictor 5 is shown in figure 16 Fig 16: Histogram of error values for predictor 5 Notes 1. The spatial top,top-left and top-right neighbours are at the immediate row above the pixel to be predicted. The implementation in [1] assumes deinterlaced field pictures however since in all of the above experiments a progressive picture source was used, the neighbour pixels were not picked depending on the field polarity

18 Problem Statement 4 To compare predictors 1 through 5. Comparison Metric The comparison metric chosen is the mean square error. The error values obtained are squared and their mean is obtained for each predictor. Comparison Result Fig 17: Mean square errors of predictors 1 through 5 when using the Hall monitor sequence Conclusions As can be seen predictor 5 yields the minimum mean square error. This also means that the energy of the error signal is the minimum in predictor 5. Hence fewer bits will be required to code the error and hence more compression is achieved.

19 Predictor Source Code Linear Predictors M File for Predictors 1 to 3 % XP - pixel from past frame X1 - previous pixel X2 - immediate top pixel X3 - top left pixel TBD - how to find which is the best predictor % height = 144; width = 176; fileid = fopen('c:\spring08\multimdediaprocessing\code\mpeg2vidcodec_v12\mpeg2\par\ hall0.yuv', 'r'); buf1= fread(fileid, width * height, 'uchar=>uchar'); fclose(fileid); fileid = fopen('c:\spring08\multimdediaprocessing\code\mpeg2vidcodec_v12\mpeg2\par\ hall1.yuv', 'r'); buf2 = fread(fileid, width * height, 'uchar=>uchar'); fclose(fileid); cur_frame = double(reshape(buf2,width,height)); cur_frame = transpose(cur_frame); past_frame = double(reshape(buf1,width,height)); past_frame = transpose(past_frame); X1 = [zeros(height,1) cur_frame(:,1:width-1)]; X2 = [zeros(1,width); cur_frame(1:height-1,:)]; X3 = [zeros(height,1) X2(:,1:width-1)]; EX4X2 = mean(mean(cur_frame.*x2)); EX4X1 = mean(mean(cur_frame.*x1)); EX4X3 = mean(mean(cur_frame.*x3)); EX1X2 = mean(mean(x1.*x2)); EX1X3 = mean(mean(x1.*x3)); EX2X3 = mean(mean(x2.*x3)); EX1X1 = mean(mean(x1.*x1)); EX2X2 = mean(mean(x2.*x2)); EX3X3 = mean(mean(x3.*x3)); CoeffMatrix = [EX1X1 EX1X2 EX1X3;EX1X2 EX2X2 EX2X3;EX1X3 EX2X3 EX3X3]; InvCoeffMatrix = inv(coeffmatrix); parms1 = InvCoeffMatrix*[EX4X1;EX4X2;EX4X3]; X1 = [zeros(height,1) cur_frame(:,1:width-1)]; X2 = [zeros(1,width); cur_frame(1:height-1,:)]; X3 = [zeros(height,1) X2(:,1:width-1)]; EX4XP = mean(mean(cur_frame.*past_frame)); EX1XP = mean(mean(x1.*past_frame)); EX2XP = mean(mean(x2.*past_frame));

20 EXPXP = mean(mean(past_frame.*past_frame)); CoeffMatrix = [EX1X1 EX1X2 EX1XP;EX1X2 EX2X2 EX2XP;EX1XP EX2XP EXPXP]; InvCoeffMatrix = inv(coeffmatrix); parms2 = InvCoeffMatrix*[EX4X1;EX4X2;EX4XP]; dpcm_error = double(cur_frame) - double(x1); third_order_predx1x2x3 = double(cur_frame) - double((parms1(1)*x1 + parms1(2)*x2 + parms1(3)*x3)); third_order_predx1x2xp = double(cur_frame) - double((parms2(1)*x1 + parms2(2)*x2 + parms2(3)*past_frame)); dpcm_error = reshape(dpcm_error,height*width,1); figure hist(dpcm_error,512); mse_pred(1) = mean(dpcm_error.^2); figure third_order_predx1x2x3 = reshape(third_order_predx1x2x3,height*width,1); hist(third_order_predx1x2x3,512); mse_pred(2) = mean(third_order_predx1x2x3.^2); figure third_order_predx1x2xp = reshape(third_order_predx1x2xp,height*width,1); hist(third_order_predx1x2xp,512); mse_pred(3) = mean(third_order_predx1x2xp.^2); figure; stem(mse_pred); Adaptive Predictors M file for Predictor 4 % Trying to implement a 2D median predictor X1 - previous pixel X3-1 row up pixel X2-1 row up lefts pixel X4-1 row up right pixel X6 X8 % height = 144; width = 176; fileid = fopen('c:\spring08\multimdediaprocessing\code\mpeg2vidcodec_v12\mpeg2\par\ hall1.yuv', 'r'); buf1= fread(fileid, width * height, 'uchar=>uchar'); fclose(fileid); cur_frame = double(reshape(buf1,width,height)); cur_frame = transpose(cur_frame); % neighbour pixel frames X1 = [zeros(height,1) cur_frame(:,1:width-1)]; X3 = [zeros(1,width); cur_frame(1:height-1,:)]; X2 = [zeros(height,1) X3(:,1:width-1)]; X4 = [X3(:,1:width-1) zeros(height,1)]; median = Adaptive2D(X1,X2,X3,X4); error_adpcm = cur_frame - median; error_adpcm = reshape(error_adpcm, 1, height*width); hist(error_adpcm,512);

21 mse_error_adpcm2 = mean(error_adpcm.^2); Mex file implementation of the function Adaptive2D used in predictor 4 #include "mex.h" #include <math.h> #include <stdlib.h> /* nrhs input argument nlhs output argument */ static void sort5(double *array) unsigned int i,j; double max,swap; for (i = 0; i < 5; i++) for (j = i; j < 5; j++) if (array[i] < array[j]) swap = array[j]; array[j] = array[i]; array[i] = swap; static double calculate_median5(double a, double b, double c, double d, double e) double array[5],median; array[0] = a; array[1] = b; array[2] = c; array[3] = d; array[4] = e; sort5(array); median = array[2]; return median; static void fill_median (double *a1,double *a2,double *a3, double *a4,double *median,unsigned int m,unsigned int n) unsigned int i; double l,r; for (i = 0; i < m*n; i++) l = a1[i]/2 + (a3[i] + a4[i])/4; r = (a1[i] + a3[i]) - a2[i]; median[i] =calculate_median5(a1[i],a3[i],l,r,a1[i]); void mexfunction(int nlhs, mxarray *plhs[],int nrhs, const mxarray *prhs[]) double *X1,*X2,*X3,*X4; double *median; unsigned int m,n; /* Find the dimensions of the data */ m = mxgetm(prhs[0]); n = mxgetn(prhs[0]); X1 = mxgetpr(prhs[0]);

22 X2 = mxgetpr(prhs[1]); X3 = mxgetpr(prhs[2]); X4 = mxgetpr(prhs[3]); plhs[0] = mxcreatedoublematrix(m,n,mxreal); median = mxgetpr(plhs[0]);; fill_median(x1,x2,x3,x4,median,m,n); M File for predictor 5 % Trying to implement a 2D median predictor X1 - left pixel X3 - current frame top pixel X2 - current frame top left pixel X4 - current frame top right pixel X6 - past frame left pixel X8 - past frame top pixel % height = 144; width = 176; fileid = fopen('c:\spring08\multimdediaprocessing\code\mpeg2vidcodec_v12\mpeg2\par\ hall1.yuv', 'r'); buf1= fread(fileid, width * height, 'uchar=>uchar'); fclose(fileid); fileid = fopen('c:\spring08\multimdediaprocessing\code\mpeg2vidcodec_v12\mpeg2\par\ hall0.yuv', 'r'); buf2= fread(fileid, width * height, 'uchar=>uchar'); fclose(fileid); cur_frame = double(reshape(buf1,width,height)); cur_frame = transpose(cur_frame); past_frame = double(reshape(buf2,width,height)); past_frame = transpose(past_frame); % neighbour pixel frames X1 = [zeros(height,1) cur_frame(:,1:width-1)]; X3 = [zeros(1,width); cur_frame(1:height-1,:)]; X2 = [zeros(height,1) X3(:,1:width-1)]; X4 = [X3(:,1:width-1) zeros(height,1)]; X5 = past_frame; X6 = [zeros(height,1) past_frame(:,1:width-1)]; X8 = [zeros(1,width); cur_frame(1:height-1,:)]; median = Adaptive3D(X1,X2,X3,X4,X5,X6,X8); error_adpcm = cur_frame - median; error_adpcm = reshape(error_adpcm, 1, height*width); hist(error_adpcm,512); mse_error_adpcm3 = mean(error_adpcm.^2); Mex file implementation of the function Adpative3D() used in Predictor 5 #include "mex.h" #include <math.h> #include <stdlib.h> /* nrhs input argument nlhs output argument */ static void sort5(double *array) unsigned int i,j;

23 double max,swap; for (i = 0; i < 5; i++) for (j = i; j < 5; j++) if (array[i] < array[j]) swap = array[j]; array[j] = array[i]; array[i] = swap; static double calculate_median5(double a, double b, double c, double d, double e) double array[5],median; array[0] = a; array[1] = b; array[2] = c; array[3] = d; array[4] = e; sort5(array); median = array[2]; return median; static void fill_median (double *a1,double *a2,double *a3, double *a4,double *a5,double *a6,double *a8, double *median,unsigned int m,unsigned int n) unsigned int i; double l,r,b,e; for (i = 0; i < m*n; i++) r = (a1[i] + a3[i]) - a2[i]; b = a3[i] - a8[i] + a5[i]; e = a1[i] - a6[i] + a5[i]; median[i] =calculate_median5(a1[i],b,a5[i],r,e); void mexfunction(int nlhs, mxarray *plhs[],int nrhs, const mxarray *prhs[]) double *X1,*X2,*X3,*X4,*X5,*X6,*X8; double *median; unsigned int m,n; /* Find the dimensions of the data */ m = mxgetm(prhs[0]); n = mxgetn(prhs[0]); X1 = mxgetpr(prhs[0]); X2 = mxgetpr(prhs[1]); X3 = mxgetpr(prhs[2]); X4 = mxgetpr(prhs[3]); X5 = mxgetpr(prhs[4]); X6 = mxgetpr(prhs[5]); X8 = mxgetpr(prhs[6]); plhs[0] = mxcreatedoublematrix(m,n,mxreal); median = mxgetpr(plhs[0]); fill_median(x1,x2,x3,x4,x5,x6,x8,median,m,n);

24 Problem Statement 5 To study the effect of error while reconstructing at the decoder end. Theory When the input image is decorrelated and the redundancy removed, the decoder needs to receive the error signal intact. When there is channel noise and the coded error is received with bit flips the reconstructed picture has artifacts. Types of errors introduced 1. Bit toggling. The sign bit was flipped 2. Error in the beginning of a scan line. Predictor 1 was used to see the effect of the errors. Fig 18 Effect of bit errors in the reconstructeed image As can be seen, error type 1 was introduced in the first row of the picture and type 2 error was introduced in rows The effect of type 1 error is not visible while the type 2 error has resulted in the line running through the image. This error is more pronounced since the first element of each row is the refresh element or the element which renews the prediction for each line. Source Code for seeing the effect of bitstream errors M file height = 144; width = 176; fileid = fopen('c:\spring08\multimdediaprocessing\code\mpeg2vidcodec_v12\mpeg2\par\ hall0.yuv', 'r'); buf1= fread(fileid, width * height, 'uchar=>uchar'); fclose(fileid); cur_frame = double(reshape(buf1,width,height)); cur_frame = transpose(cur_frame); X1 = [zeros(height,1) cur_frame(:,1:width-1)]; dpcm_error = double(cur_frame) - double(x1); %introduce error dpcm_error(1,40) = -dpcm_error(1,40); dpcm_error(70,1) = 0; dpcm_error(71,1) = 0; dpcm_error(72,1) = 0; %reconstruct recon(:,1) = dpcm_error(:,1); for i = 2:width; recon(:,i) = dpcm_error(:,i) + recon(:,i-1);

25 end; imshow(recon,[0,255]); Conclusions 1. The central limit theorem was observed. 2. The redundancy of information inside a picture was understood and exploited to achieve compression 3. 5 different predictors were implemented and compared. The adaptive predictor which made use of both temporal and spatial redundancy proved to be the best predictor. 4. The effect of bitstream errors in the coded signal was observed. References [1] T. Jarske and Y. Neuvo, Adaptive DPCM with median predictors, IEEE Trans. Comsumer Electronics,Vol. 37, No. 3, August 1991,pp [2] V.Devarajan and K. Rao, DPCM coders with adaptive prediction for NTSC composite TV signals, IEEE Trans. Communications. Vol. 28, No. 7, July 1980, pp