Editing JPEG (section)

===Decoding===
Decoding to display the image consists of doing all the above in reverse.

Taking the DCT coefficient matrix (after adding the difference of the DC coefficient back in)

:<math>
\left[
\begin{array}{rrrrrrrr}
-26 & -3 & -6 & 2 & 2 & -1 & 0 & 0 \\
0 & -2 & -4 & 1 & 1 & 0 & 0 & 0 \\
-3 & 1 & 5 & -1 & -1 & 0 & 0 & 0 \\
-3 & 1 & 2 & -1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}
\right]
</math>

and taking the [[Hadamard product (matrices)|entry-for-entry product]] with the quantization matrix from above results in

:<math>
\left[
\begin{array}{rrrrrrrr}
-416 & -33 & -60 & 32 & 48 & -40 & 0 & 0 \\
0 & -24 & -56 & 19 & 26 & 0 & 0 & 0 \\
-42 & 13 & 80 & -24 & -40 & 0 & 0 & 0 \\
-42 & 17 & 44 & -29 & 0 & 0 & 0 & 0 \\
18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}
\right]
</math>

which closely resembles the original DCT coefficient matrix for the top-left portion.

The next step is to take the two-dimensional inverse DCT (a 2D type-III DCT), which is given by:

<math>f_{x,y} =
 \frac{1}{4}
 \sum_{u=0}^7
 \sum_{v=0}^7
 \alpha(u) \alpha(v) F_{u,v}
 \cos \left[\frac{(2x+1)u\pi}{16} \right]
 \cos \left[\frac{(2y+1)v\pi}{16} \right]
</math>

where
* <math>\ x</math> is the pixel row, for the integers <math>\ 0 \leq x < 8</math>.
* <math>\ y</math> is the pixel column, for the integers <math>\ 0 \leq y < 8</math>.
* <math>\ \alpha(u)</math> is the normalizing scale factor defined earlier, for the integers <math>\ 0 \leq u < 8</math>.
* <math>\ F_{u,v}</math> is the approximated DCT coefficient at coordinates <math>\ (u,v).</math>
* <math>\ f_{x,y}</math> is the reconstructed pixel value at coordinates <math>\ (x,y)</math>

Rounding the output to integer values (since the original had integer values) results in an image with values (still shifted down by 128)

{{Multiple image|align=right|direction=vertical|image1=JPEG example image.jpg|alt1=|image2=JPEG example image decompressed.jpg|alt2=|width=180|footer=Slight differences are noticeable between the original (top) and decompressed image (bottom), which is most readily seen in the bottom-left corner.}}

:<math>
\left[
\begin{array}{rrrrrrrr}
-66 & -63 & -71 & -68 & -56 & -65 & -68 & -46 \\
-71 & -73 & -72 & -46 & -20 & -41 & -66 & -57 \\
-70 & -78 & -68 & -17 & 20 & -14 & -61 & -63 \\
-63 & -73 & -62 & -8 & 27 & -14 & -60 & -58 \\
-58 & -65 & -61 & -27 & -6 & -40 & -68 & -50 \\
-57 & -57 & -64 & -58 & -48 & -66 & -72 & -47 \\
-53 & -46 & -61 & -74 & -65 & -63 & -62 & -45 \\
-47 & -34 & -53 & -74 & -60 & -47 & -47 & -41
\end{array}
\right]
</math>

and adding 128 to each entry

:<math>
\left[
\begin{array}{rrrrrrrr}
62 & 65 & 57 & 60 & 72 & 63 & 60 & 82 \\
57 & 55 & 56 & 82 & 108 & 87 & 62 & 71 \\
58 & 50 & 60 & 111 & 148 & 114 & 67 & 65 \\
65 & 55 & 66 & 120 & 155 & 114 & 68 & 70 \\
70 & 63 & 67 & 101 & 122 & 88 & 60 & 78 \\
71 & 71 & 64 & 70 & 80 & 62 & 56 & 81 \\
75 & 82 & 67 & 54 & 63 & 65 & 66 & 83 \\
81 & 94 & 75 & 54 & 68 & 81 & 81 & 87
\end{array}
\right].
</math>

This is the decompressed subimage. In general, the decompression process may produce values outside the original input range of <math>[0, 255]</math>. If this occurs, the decoder needs to clip the output values so as to keep them within that range to prevent overflow when storing the decompressed image with the original bit depth.

The decompressed subimage can be compared to the original subimage (also see images to the right) by taking the difference (original − uncompressed) results in the following error values:

:<math>
\left[
\begin{array}{rrrrrrrr}
-10 & -10 & 4 & 6 & -2 & -2 & 4 & -9 \\
6 & 4 & -1 & 8 & 1 & -2 & 7 & 1 \\
4 & 9 & 8 & 2 & -4 & -10 & -1 & 8 \\
-2 & 3 & 5 & 2 & -1 & -8 & 2 & -1 \\
-3 & -2 & 1 & 3 & 4 & 0 & 8 & -8 \\
8 & -6 & -4 & -0 & -3 & 6 & 2 & -6 \\
10 & -11 & -3 & 5 & -8 & -4 & -1 & -0 \\
6 & -15 & -6 & 14 & -3 & -5 & -3 & 7
\end{array}
\right]
</math>

with an average absolute error of about 5 values per pixels (i.e., <math>\frac{1}{64} \sum_{x=0}^7 \sum_{y=0}^7 |e(x,y)| = 4.8750</math>).

The error is most noticeable in the bottom-left corner where the bottom-left pixel becomes darker than the pixel to its immediate right.