it talks about shifting values after Discrete Cosine Transform

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