Transform Methods and Image Compression

An introduction to JPEG and wavelet transform techniques using Octave and Matlab.
The Cosine Transform and JPEG

In this section, several examples using the cosine transform are presented. This transform is used by JPEG, applied to 8x8 portions of an image. An NxN cosine transform exists for every N, which exchanges spatial information for frequency information. For the case N=4, a given 4x4 portion of an image can be written as a linear combination of the 16 basis images which appear in Figure 1(a).

The transform provides the coefficients in the linear combination, allowing approximations or adjustments to the original image based on frequency content. One possibility is simply to eliminate certain frequencies, obtaining a kind of partial sum approximation. The implicit assumption in JPEG, for example, is that the higher-frequency information in an image tends to be of less importance to the eye.

The images in Figure 1 can be obtained from the scripts supplied on our web site (see Resources 4) as follows. We'll use “>” to denote the prompt printed by Matlab or Octave, but this will vary by platform.

Define the test image:

> x = round(rand(4)*50) % 4x4 random matrix,
                       % integer entries in

This will display some (random) matrix, perhaps

    | 10 20 10 41 |
    | 40 30 2  12 |
    | 20 35 20 15 |

and we can view this “image” with the instructions:

> imagesc(x);      % Matlab users
> imagesc(x, 8);   % Octave users

Something similar to the smaller image at the lower left in Figure 1(b) will be displayed. (We chose the 4x4 example for clarity; however, the viewer in Octave may fail to display it properly. In this case, either the image can be padded before display or a larger image can be chosen.) Now ask for the matrix of partial sums (the larger image in Figure 1(b)):

> imagesc(psumgrid(x)); % Display the 16 partial
                        % sums

The partial sums are built up from the basis elements in the order shown in the zigzag sequence above. This path through Figure 1(a) is based on increasing frequency of the basis elements. Roughly speaking, the artificial image in Figure 1(b) is the worst kind as far as JPEG compression is concerned. Since it is random, it will likely have significant high-frequency terms. We can see these by performing the discrete cosine transform:

> Tx = dct(x, 4) % 4
                 % of x

For the example above, this gives the matrix

     | 79.25   9.47  4.75 -11.77 |
     |  6.25 -19.69 -4.25 -11.60 |
     |  5.97   8.02 12.73 -15.64 |

of coefficients used to build the partial sums in Figure 1 from the basis elements. The top left entry gets special recognition as the DC coefficient, representing the average gray level; the others are the AC coefficients, AC0,1 through AC3,3.

The terms in the lower right of Tx correspond to the high-frequency portion of the image. Notice that even in this “worst case”, Figure 1 suggests that a fairly good image can be obtained without using all 16 terms.

The process of approximation by partial sums is applied to a “real” image in Figure 2, where 1/4, 1/2 and 3/4 of the 1024 terms for a 32x32 image are displayed. These can be generated with calls of the form:

> x = getpgm('math4.pgm'); % Get a graymap image
> n = length(x);        % n is the number of rows
                        % in the square image
> y = psum(x, n*n / 2); % y is the partial sum
                        % using 1/2 of the terms
> imagesc(y);           % Display the result

Our approximations retain all of the frequency information corresponding to terms from the zigzag sequence below some selected threshold value; the remaining higher-frequency information is discarded. Although this can be considered a special case of a JPEG-like scheme, JPEG allows more sophisticated use of the frequency information.

JPEG exploits the idea of local approximation for its compression: 8x8 portions of the complete image are transformed using the cosine transform, then each block is quantized by a method which tends to suppress higher-frequency elements and reduce the number of bits required for each term. To “recover” the image, a dequantizing step is used, followed by an inverse transform. (We've ignored the portion of JPEG which does lossless compression on the output of the quantizer, but this doesn't affect the image quality.) The matrix operations can be diagrammed as:

  transform    quantize     dequantize     invert
x  ------>  Tx  ----->  QTx  -------> Ty  ---> y

In Octave or Matlab, the individual steps can be written:

> x = getpgm('bird.pgm'); % Get a graymap image
> Tx = dct(x);       % Do the 8
> QTx = quant(Tx);   % Quantize, using standard
                     % 8
> Ty = dequant(QTx); % Dequantize
> y = invdct(Ty);    % Recover the image
> imagesc(y);        % Display the image

To be precise, a rounding procedure should be done on the matrix y. In addition, we have ignored the zero-shift specified in the standard, which affects the quantized DC coefficients.

It should be emphasized that we cannot recover the image completely—there has been loss of information at the quantizing stage. It is illustrative to compare the matrices x and y, and the difference image x-y for this kind of experiment appears in Figure 3(f). There is considerable interest in measuring the “loss of image quality” using some function of these matrices. This is a difficult problem given the complexity of the human visual system.

The images in Figure 3 were generated at several “quality” levels, using software from the Independent JPEG Group (see Resources 5). The sizes are given in bits per pixel (bpp); i.e., the number of bits, on average, required to store each of the numbers in the matrix representation of the image. The sizes for the GIF and PNG versions are included for reference. (“Bird” is part of a proposed collection of standard images at the Waterloo BragZone [see Resources 11] and has been modified for the purposes of this article.)