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3.19 Discrete Cosine Transforms 99

image files from the hard disk. Applications using color will particularly benefit
from these compression techniques.
A transform coding; scheme based on the discrete cosine transform has been
standardized and is now one of the most widely used compression techniques [5,
34]. This standard, known by the acronym JPEG (joint photographic expert
group), establishes a standard for compression of color and grayscale still images.
The JPEG scheme has been implemented both in software and by using ASIC.
In a typical transform coding scheme, an
input image is divided into nonoverlapping sub-
frames or blocks as shown in Figure 3.30. The
subframes are linearly transformed by using
the discrete cosine transform into the trans-
form domain. The transform has the property
that the signal is concentrated in relatively few
transform coefficients compared to the number
of samples in the original image. Typical sub-
- Blocking of an image
8
frame sizes are in the range of 8 x 8 to 16 x 16 Fi ure 3 30
pixels. The rationale for this data reduction is
that any redundancy in the image, from a visual point of view, can more easily be
removed in the frequency domain due to a masking phenomenon in the human
vision system. A coarse quantization of the frequency components in a region with
high power is perceived as a small distortion of the corresponding image while the
human eye is more sensitive to errors in regions with low power. The energy com-
paction property of the discrete cosine transform is therefore important in this
type of application.

3.19 DISCRETE COSINE TRANSFORMS

The DCT (discrete cosine transform) was first proposed by Ahmed et al. [3] in 1974.
The discrete cosine transform is highly suitable for transform coding of images.
The main reason is that the decorrelation property of the DCT is almost as good as
for the optimal transform, the Karhunen-Loeve transform (1947), but the DCT is
much simpler from a computational point of view, since it is independent of the
signal.
There exist several types of DCTs: even, odd, symmetric, and the modified
symmetric DCT. They possess slightly different properties which are of relevance
for image coding applications. In fact, only the even DCT and modified symmetric
DCT are suitable for image coding [27, 30, 34, 37].

3.19.1 EDCT (Even Discrete Cosine Transform)
The EDCT (even discrete cosine transform) is defined as