This document summarizes work done to properly apply the discrete cosine transform (DCT) to an image. It was found that shifting pixel values by 128 before applying the DCT was necessary to avoid negative DC coefficient values after reconstruction. Using integer 16 data type for the image instead of uint8 helped preserve pixel values below 128 after reconstruction. Not shifting the data led to incorrect pixel values being lost during reconstruction.

2. clear all,clc
I = imread('Market.png');
I=rgb2gray(I);
fun = @dct2;
mn=max(max(I))
% I1=im2double(I);
I2=I-128;
mn2=max(max(I2))
J = blkproc(I2,[8 8],fun);
imagesc(J), colormap(hot)
fun2=@idct2;
J1 = blkproc(J,[8 8],fun2);
figure(2),imshow(uint8(J1));

3. Original Image

4. After restoring the image without
shifting back

5. After shifting back some values are
lost

6. We see a problem with the values
under 127
• I(9:16,9:16)
• ans =
• 45 56 81 129 148 151 152 145
• 43 46 69 115 147 154 152 147
• 41 44 59 96 138 151 153 152
• 38 40 52 78 124 148 152 151
• 39 42 47 59 104 139 151 148
• 46 43 45 47 89 130 150 152
• 56 46 44 44 71 114 146 152
• 61 45 40 47 50 95 138 149
• >> I3(9:16,9:16)
• ans =
• 128 128 128 129 148 151 152 145
• 128 128 128 128 147 154 152 147
• 128 128 128 128 138 151 153 152
• 128 128 128 128 128 148 152 151
• 128 128 128 128 128 139 151 148
• 128 128 128 128 128 130 150 152
• 128 128 128 128 128 128 146 152
• 128 128 128 128 128 128 138 149

7. Now fixing the values using int16
clear all,clc
I = imread('Market.png');
I=rgb2gray(I);
I=int16(I);
fun = @dct2;
mn=max(max(I))
% I1=im2double(I);
I2=I-128;
mn2=max(max(I2))
J = blkproc(I2,[8 8],fun);
imagesc(J), colormap(hot)
title('Color map of DCT of image');
fun2=@idct2;
J1 = blkproc(J,[8 8],fun2);
J1=J1+128;
figure(2),imshow(uint8(J1));
title('Restoring the image after shifting back');

10. Shifting before the DCT
J(1:8,1:8)
ans =
-638.8750 -29.3727 4.1062 -16.0527 0.1250 -7.0450 -0.2126 -4.8849
85.6364 65.3826 33.3134 -17.2827 8.4377 -6.4442 -0.1027 -3.7384
-39.9354 -15.6904 30.2180 9.2577 4.1787 -2.2267 2.9812 2.4027
2.5924 -13.6623 -5.4940 -7.8364 3.6032 2.0654 1.9024 -1.0399
2.3750 4.8097 8.6232 -5.1047 -5.6250 0.9445 0.8930 0.3232
2.9945 -3.2131 -0.8315 6.9848 4.8601 0.3043 -1.6718 1.9621
1.2530 -0.0697 0.7312 -0.2981 1.9222 2.9600 -0.7180 -2.5778
2.9642 0.1810 -0.3388 -2.6802 0.8737 -1.0296 -2.5350 -1.8505
>> J(9:16,9:16)
ans =
-258.2500 -337.8517 15.4053 44.1097 -3.0000 3.1585 -0.6985 2.7715
93.1163 -12.7923 -105.2548 6.4951 21.3826 -6.4124 -5.7623 -1.4411
3.9429 29.0674 5.5570 -20.2654 8.7304 15.5683 -4.3624 -6.9161
6.3625 -0.4752 -3.8135 1.3667 -8.5273 -3.2138 1.0229 2.4084
-4.7500 5.8781 -0.2078 -5.6255 -0.5000 3.6717 -1.8081 -3.3458
5.5093 -0.4890 -0.0957 -0.4586 -0.3461 -1.6795 0.1375 1.1942
-0.2802 0.6512 1.8876 -2.0940 0.1721 -0.2394 -0.8070 -0.5737
0.1225 1.8141 1.4955 -0.7570 -1.3032 -0.4243 -1.4556 0.1052

11. Now without shifting

12. DCT without shifting
J(1:8,1:8)
ans =
385.1250 -29.3727 4.1062 -16.0527 0.1250 -7.0450 -0.2126 -4.8849
85.6364 65.3826 33.3134 -17.2827 8.4377 -6.4442 -0.1027 -3.7384
-39.9354 -15.6904 30.2180 9.2577 4.1787 -2.2267 2.9812 2.4027
2.5924 -13.6623 -5.4940 -7.8364 3.6032 2.0654 1.9024 -1.0399
2.3750 4.8097 8.6232 -5.1047 -5.6250 0.9445 0.8930 0.3232
2.9945 -3.2131 -0.8315 6.9848 4.8601 0.3043 -1.6718 1.9621
1.2530 -0.0697 0.7312 -0.2981 1.9222 2.9600 -0.7180 -2.5778
2.9642 0.1810 -0.3388 -2.6802 0.8737 -1.0296 -2.5350 -1.8505
>> J(9:16,9:16)
ans =
765.7500 -337.8517 15.4053 44.1097 -3.0000 3.1585 -0.6985 2.7715
93.1163 -12.7923 -105.2548 6.4951 21.3826 -6.4124 -5.7623 -1.4411
3.9429 29.0674 5.5570 -20.2654 8.7304 15.5683 -4.3624 -6.9161
6.3625 -0.4752 -3.8135 1.3667 -8.5273 -3.2138 1.0229 2.4084
-4.7500 5.8781 -0.2078 -5.6255 -0.5000 3.6717 -1.8081 -3.3458
5.5093 -0.4890 -0.0957 -0.4586 -0.3461 -1.6795 0.1375 1.1942
-0.2802 0.6512 1.8876 -2.0940 0.1721 -0.2394 -0.8070 -0.5737
0.1225 1.8141 1.4955 -0.7570 -1.3032 -0.4243 -1.4556 0.1052

13. Why Shift data?
• Because the DCT is designed to work on pixel
values ranging from -128 to 127, the original
block is “leveled off” by subtracting 128 from
each entry.

15. With shifting DC component
becomes negative
dct2(int16(I(17:24,17:24))-128)
ans =
-257.2500 113.9048 190.8229 -14.2258 -31.7500 -26.0361 -7.0623 2.5148
-2.2054 -19.3567 26.7378 49.5825 18.4131 11.2723 -8.1334 -1.2940
-3.0683 -6.7897 -8.6176 -2.1773 0.6929 2.1958 0.9660 -1.2293
2.3132 3.9376 3.2723 8.6746 4.5256 2.9021 1.4571 1.7553
0 1.4061 -3.0190 -1.6145 1.5000 1.3845 -0.1025 -2.3183
-0.8553 -1.8905 2.8431 0.5789 3.0710 -1.4592 0.0228 -1.5351
-4.1410 2.3969 2.2160 2.8282 -0.2870 -0.1222 -0.1324 1.9538
1.8516 1.3828 -0.3954 2.7004 0.8187 1.7177 -0.6069 -1.3586
K>> dct2(int16(I(17:24,17:24)))
ans =
766.7500 113.9048 190.8229 -14.2258 -31.7500 -26.0361 -7.0623 2.5148
-2.2054 -19.3567 26.7378 49.5825 18.4131 11.2723 -8.1334 -1.2940
-3.0683 -6.7897 -8.6176 -2.1773 0.6929 2.1958 0.9660 -1.2293
2.3132 3.9376 3.2723 8.6746 4.5256 2.9021 1.4571 1.7553
0 1.4061 -3.0190 -1.6145 1.5000 1.3845 -0.1025 -2.3183
-0.8553 -1.8905 2.8431 0.5789 3.0710 -1.4592 0.0228 -1.5351
-4.1410 2.3969 2.2160 2.8282 -0.2870 -0.1222 -0.1324 1.9538
1.8516 1.3828 -0.3954 2.7004 0.8187 1.7177 -0.6069 -1.3586