DigitalVideoAndHDTVAlgorithmsAndInterfaces.pdf

Figure 38.4 An 8×8 array of
luma samples from an
image is shown. This 8×8
array is known in JPEG
terminology as a block.
Figure 38.5 DCT tends to
concentrate the power of the
image block into low-frequency
DCT coefficients (those coefficients
in the upper left-hand
corner of the matrix). No information
is lost at this stage. The
DCT is its own inverse, so
performing the DCT on these
transform coefficients would
reconstruct the original samples
(subject only to roundoff error).
If implemented directly according to these equations,
an 8×8 DCT requires 64 multiply operations (and 49
additions) for each of the 64 result coefficients, for
a total of 4096 multiplies, an average of 8 multiplication
operations per pixel. However, the DCT is separable:
an 8×8 DCT can be computed as eight 8×1
horizontal transforms followed by eight 1×8 vertical
transforms. This optimization, combined with other
optimizations comparable to those of the fast Fourier
transform (FFT), greatly reduces computational
complexity: A fully optimized 8×8 DCT requires as few
as 11 multiplies for each 8 samples (or in an IDCT,
transform coefficients).
JPEG encoding example
I will illustrate JPEG encoding by walking through
a numerical example. Figure 38.4 represents an 8×8
array of luma samples from an image, prior to level
shifting:
f =
139 144 149 153 155 155 155 155
144 151 153 156 159 156 156 156
150 155 160 163 158 156 156 156
159 161 162 160 160 159 159 159
159 160 161 162 162 155 155 155
161 161 161 161 160 157 157 157
162 162 161 163 162 157 157 157
162 162 161 161 163 158 158 158
The result of computing the DCT, rounded to integers,
is shown in Figure 38.5:
F
=
1260 -1 -12 -5 2 -2 -3 1
-23 -17 -6 -3 -3 0 0 1
-11 -9 -2 2 0 -1 -1 0
-7 -2 0 1 1 0 0 0
-1 -1 1 2 0 -1 1 1
2 0 2 0 -1 1 1 -1
-1 0 0 -1 0 2 1 -1
-3 2 -4 -2 2 1 -1 0
452 DIGITAL VIDEO AND HDTV ALGORITHMS AND INTERFACES