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A sample chapter from:

The Scientist and Engineer's Guide to Digital Signal Processing

by Steven W. Smith

(Low resolution graphics are used in this sample to reduce the download time)

Chapter 27 - Data Compression

Data transmission and storage cost money. The more information being dealt with, the more it costs. In spite of this, most digital data are not stored in the most compact form. Rather, they are stored in whatever way makes them easiest to use, such as: ASCII text from word processors, binary code that can be executed on a computer, individual samples from a data acquisition system, etc. Typically, these easy-to-use encoding methods require data files about twice as large as actually needed to represent the information. Data compression is the general term for the various algorithms and programs developed to address this problem. A compression program is used to convert data from an easy-to-use format to one optimized for compactness. Likewise, an uncompression program returns the information to its original form. We examine five techniques for data compression in this chapter. The first three are simple encoding techniques, called: run-length, Huffman, and delta encoding. The last two are elaborate procedures that have established themselves as industry standards: LZW and JPEG.

Data Compression Strategies

Table 27-1 shows two different ways that data compression algorithms can be categorized. In (a), the methods have been classified as either lossless or lossy. A lossless technique means that the restored data file is identical to the original. This is absolutely necessary for many types of data, for example: executable code, word processing files, tabulated numbers, etc. You cannot afford to misplace even a single bit of this type of information. In comparison, data files that represent images and other acquired signals do not have to be keep in perfect condition for storage or transmission. All real world measurements inherently contain a certain amount of noise. If the changes made to these signals resemble a small amount of additional noise, no harm is done. Compression techniques that allow this type of degradation are called lossy. This distinction is important because lossy techniques are much more effective at compression than lossless methods. The higher the compression ratio, the more noise added to the data.

[TAB 27-1]

TABLE 27-1
Compression classifications. Data compression methods can be divided in two ways. In (a), the techniques are classified as lossless or lossy. Lossless methods restore the compressed data to exactly the same form as the original, while lossy methods only generate an approximation. In (b), the methods are classified according to a fixed or variable size of group taken from the original file and written to the compressed file.

Images transmitted over the world wide web are an excellent example of why data compression is important, and the effectiveness of lossless versus lossy compression. Suppose we need to download a detailed color image over a computer's 33.6 kbps modem. If the image is not compressed (a TIFF file, for example), it will contain about 600 kbytes of data. If it has been compressed using a lossless technique (such as used in the GIF format), it will be about one-half this size, or 300 kbytes. If lossy compression has been used (a JPEG file), it will be about 50 kbytes. The point is, the download times for these three equivalent files are 142 seconds, 71 seconds, and 12 seconds, respectively. That's a big difference! It's no wonder you seldom see TIFF images on the web. JPEG is usually used for acquired images, such as photographs; while GIF is used for line art, such as logos.

Our second way of classifying data compression methods is shown in Table 27-1b. Most data compression programs operate by taking a group of data from the original file, compressing it in some way, and then writing the compressed group to the output file. For instance, one of the techniques in this table is CS&Q, short for coarser sampling and/or quantization. Suppose we are compressing a digitized waveform, such as an audio signal that has been digitized to 12 bits. We might read two adjacent samples from the original file (24 bits), discard one of the sample completely, discard the least significant 4 bits from the other sample, and then write the remaining 8 bits to the output file. With 24 bits in and 8 bits out, we have implemented a 3:1 compression ratio using a lossy algorithm. While this is rather crude in itself, it is very effective when used with a technique called transform compression. As we will discuss later, this is the basis of JPEG.

Table 27-1b shows CS&Q to be a fixed-input fixed-output scheme. That is, a fixed number of bits are read from the input file and a smaller fixed number of bits are written to the output file. Other compression methods allow a variable number of bits to be read or written. As you go through the description of each of these compression methods, refer back to this table to understand how it fits into this classification scheme. Why are JPEG and MPEG not listed in this table? These are composite algorithms that combine many of the other techniques. They are too sophisticated to be classified into these simple categories.

Run-Length Encoding

Data files frequently contain the same character repeated many times in a row. For example, text files use multiple spaces to separate sentences, indent paragraphs, format tables & charts, etc. Digitized signals can also have runs of the same value, indicating that the signal is not changing. For instance, an image of the nighttime sky would contain long runs of the character or characters representing the black background. Likewise, digitized music might have a long run of zeros between songs. Run-length encoding is a simple method of compressing these types of files.

Figure 27-1 illustrates run-length encoding for a data sequence having frequent runs of zeros. Each time a zero is encountered in the input data, two values are written to the output file. The first of these values is a zero, a flag to indicate that run-length compression is beginning. The second value is the number of zeros in the run. If the average run-length is longer than two, compression will take place. On the other hand, many single zeros in the data can make the encoded file larger than the original. Many different run-length schemes have been developed. For example, the input data can be treated as individual bytes, or groups of bytes that represent something more elaborate, such as floating point numbers. Run-length encoding can be used on only one of the characters (as with the zero above), several of the characters, or all of the characters.

[FIG 27-1]

Example of run-length encoding. Each run of zeros is replaced by two characters in the compressed file: a zero to indicate that compression is occurring, followed by the number of zeros in the run.

A good example of a generalized run-length scheme is PackBits, created for Macintosh users. Each byte (eight bits) from the input file is replaced by nine bits in the compressed file. The added ninth bit is interpreted as the sign of the number. That is, each character read from the input file is between 0 to 255, while each character written to the encoded file is between -255 and 255. To understand how this is used, consider the input file: 1,2,3,4,2,2,2,2,4, and the compressed file generated by the PackBits algorithm: 1,2,3,4,2,-3,4. The compression program simply transfers each number from the input file to the compressed file, with the exception of the run: 2,2,2,2. This is represented in the compressed file by the two numbers: 2,-3. The first number ("2") indicates what character the run consists of. The second number ("-3") indicates the number of characters in the run, found by taking the absolute value and adding one. For instance, 4,-2 means 4,4,4; 21,-4 means 21,21,21,21,21, etc.

An inconvenience with PackBits is that the nine bits must be reformatted into the standard eight bit bytes used in computer storage and transmission. A useful modification to this scheme can be made when the input is restricted to be ASCII text. As shown in Table 27-2, each ASCII character is usually stored as a full byte (eight bits), but really only uses seven of the bits to identify the character. In other words, the values 127 through 255 are not defined with any standardized meaning, and do not need to be stored or transmitted. This allows the eighth bit to indicate if run-length encoding is in progress.

0 - null
1 - start heading
2 - start of text
3 - end of text
4 - end of xmit
5 - enquiry
6 - acknowledge
7 - bell, beep
8 - backspace
9 - horz. tab
10 - line feed
11 - vert. tab, home
12 - formfeed, cls
13 - carriage return
14 - shift out
15 - shift in
16 - data line esc
17 - device control 1
18 - device control 2
19 - device control 3
20 - device control 4
21 - negative ack.
22 - sync. idle
23 - end xmit block
24 - cancel
25 - end of medium
26 - substitute
27 - escape
28 - file separator
29 - group separator
30 - record separator
31 - unit separator

32 - space
33 - !
34 - "
35 - #
36 - $
37 - %
38 - &
39 - '
40 - (
41 - )
42 - *
43 - +
44 - ,
45 - -
46 - .
47 - /
48 - 0
49 - 1
50 - 2
51 - 3
52 - 4
53 - 5
54 - 6
55 - 7
56 - 8
57 - 9
58 - :
59 - ;
60 - <
61 - =
62 - >
63 - ?

64 - @
65 - A
66 - B
67 - C
68 - D
69 - E
70 - F
71 - G
72 - H
73 - I
74 - J
75 - K
76 - L
77 - M
78 - N
79 - O
80 - P
81 - Q
82 - R
83 - S
84 - T
85 - U
86 - V
87 - W
88 - X
89 - Y
90 - Z
91 - [
92 - \
93 - ]
94 - ^
95 - _

96 - `
97 - a
98 - b
99 - c
100 - d
101 - e
102 - f
103 - g
104 - h
105 - i
106 - j
107 - k
108 - l
109 - m
110 - n
111 - o
112 - p
113 - q
114 - r
115 - s
116 - t
117 - u
118 - v
119 - w
120 - x
121 - y
122 - z
123 - {
124 - |
125 - }
126 - ~
127 - del

TABLE 27-2
ASCII codes. This is a long established standard for allowing letters and numbers to be represented in digital form. Each printable character is assigned a number between 32 and 127, while the numbers between 0 and 31 are used for various control actions. Even though only 128 codes are defined, ASCII characters are usually stored as a full byte (8 bits). The undefined values (128 to 255) are often used for Greek letters, math symbols, and various geometric patterns; however, this is not standardized. Many of the control characters (0 to 31) are based on older communications networks, and are not applicable to computer technology.

Huffman Encoding

This method is named after D.A. Huffman, who developed the procedure in the 1950s. Figure 27-2 shows a histogram of the byte values from a large ASCII file. More than 96% of this file consists of only 31 characters: the lower case letters, the space, the comma, the period, and the carriage return. This observation can be used to make an appropriate compression scheme for this file. To start, we will assign each of these 31 common characters a five bit binary code: 00000 = "a", 00001 = "b", 00010 = "c", etc. This allows 96% of the file to be reduced in size by 5/8. The last of the five bit codes, 11111, will be a flag indicating that the character being transmitted is not one of the 31 common characters. The next eight bits in the file indicate what the character is, according to the standard ASCII assignment. This results in 4% of the characters in the input file requiring 5+8=13 bits. The idea is to assign frequently used characters fewer bits, and seldom used characters more bits. In this example, the average number of bits required per original character is: 0.96x5 + 0.04x 13 = 5.32. In other words, an overall compression ratio of: 8 bits/5.32 bits, or about 1.5:1.

[FIG 27-2]

Histogram of text. This is a histogram
of the ASCII values from a chapter in
this book. The most common characters
are the lower case letters, the space and
the carriage return.

Huffman encoding takes this idea to the extreme. Characters that occur most often, such the space and period, may be assigned as few as one or two bits. Infrequently used characters, such as: !, @, #, $ and %, may require a dozen or more bits. In mathematical terms, the optimal situation is reached when the number of bits used for each character is proportional to the logarithm of the character's probability of occurrence.

A clever feature of Huffman encoding is how the variable length codes can be packed together. Imagine receiving a serial data stream of ones and zeros. If each character is represented by eight bits, you can directly separate one character from the next by breaking off 8 bit chunks. Now consider a Huffman encoded data stream, where each character can have a variable number of bits. How do you separate one character from the next? The answer lies in the proper selection of the Huffman codes that enable the correct separation. An example will illustrate how this works.

Figure 27-3 shows a simplified Huffman encoding scheme. The characters A through G occur in the original data stream with the probabilities shown. Since the character A is the most common, we will represent it with a single bit, the code: 1. The next most common character, B, receives two bits, the code: 01. This continues to the least frequent character, G, being assigned six bits, 000011. As shown in this illustration, the variable length codes are resorted into eight bit groups, the standard for computer use. When uncompression occurs, all the eight bit groups are placed end-to-end to form a long serial string of ones and zeros. Look closely at the encoding table of Fig. 27-3, and notice how each code consists of two parts: a number of zeros before a one, and an optional binary code after the one. This allows the binary data stream to be separated into codes without the need for delimiters or other marker between the codes. The uncompression program looks at the stream of ones and zeros until a valid code is formed, and then starting over looking for the next character. The way that the codes are formed insures that no ambiguity exists in the separation.

A more sophisticated version of the Huffman approach is called arithmetic encoding. In this scheme, sequences of characters are represented by individual codes, according to their probability of occurrence. This has the advantage of better data compression, say 5-10%. Run-length encoding followed by either Huffman or arithmetic encoding is also a common strategy. As you might expect, these types of algorithms are very complicated, and usually left to data compression specialists.

[FIG 27-3]

Huffman encoding. The encoding table assigns each of the seven letters used in this example a variable length binary code, based on its probability of occurrence. The original data stream composed of these 7 characters is translated by this table into the Huffman encoded data. Since each of the Huffman codes is a different length, the binary data need to be regrouped into standard 8 bit bytes for storage and transmission.

To implement Huffman or arithmetic encoding, the compression and uncompression algorithms must agree on the binary codes used to represent each character (or groups of characters). This can be handled in one of two ways. The simplest is to use a predefined encoding table that is always the same, regardless of the information being compressed. More complex schemes use encoding optimized for the particular data being used. This requires that the encoding table be included in the compressed file for use by the uncompression program. Both methods are common.

Delta Encoding

In science, engineering, and mathematics, the Greek letter delta ( ) is used to denote the change in a variable. The term delta encoding, refers to several techniques that store data as the difference between successive samples (or characters), rather than directly storing the samples themselves. Figure 27-4 shows an example of how this is done. The first value in the delta encoded file is the same as the first value in the original data. All the following values in the encoded file are equal to the difference (delta) between the corresponding value in the input file, and the previous value in the input file.

[FIG 27-4]

Example of delta encoding. The first value in the encoded file is the same as the first value in the original file. Thereafter, each sample in the encoded file is the difference between the current and last sample in the original file.

Delta encoding can be used for data compression when the values in the original data are smooth, that is, there is typically only a small change between adjacent values. This is not the case for ASCII text and executable code; however, it is very common when the file represents a signal. For instance, Fig. 27-5a shows a segment of an audio signal, digitized to 8 bits, with each sample between -127 and 127. Figure 27-5b shows the delta encoded version of this signal. The key feature is that the delta encoded signal has a lower amplitude than the original signal. In other words, delta encoding has increased the probability that each sample's value will be near zero, and decreased the probability that it will be far from zero. This uneven probability is just the thing that Huffman encoding needs to operate. If the original signal is not changing, or is changing in a straight line, delta encoding will result in runs of samples having the same value.

[FIG 27-5]

Example of delta encoding. Figure (a) is an audio signal digitized to 8 bits. Figure (b) shows the delta encoded version of this signal. Delta encoding is useful for data compression if the signal being encoded varies slowly from sample-to-sample.

This is what run-length encoding requires. Correspondingly, delta encoding followed by Huffman and/or run-length encoding is a common strategy for compressing signals. The idea used in delta encoding can be expanded into a more complicated technique called Linear Predictive Coding, or LPC. To understand LPC, imagine that the first 99 samples from the input signal have been encoded, and we are about to work on sample number 100. We then ask ourselves: based on the first 99 samples, what is the most likely value for sample 100? In delta encoding, the answer is that the most likely value for sample 100 is the same as the previous value, sample 99. This expected value is used as a reference to encode sample 100. That is, the difference between the sample and the expectation is placed in the encoded file. LPC expands on this by making a better guess at what the most probable value is. This is done by looking at the last several samples, rather than just the last sample. The algorithms used by LPC are similar to recursive filters, making use of the z-transform and other intensively mathematical techniques.

LZW Compression

LZW compression is named after its developers, A. Lempel and J. Ziv, with later modifications by Terry A. Welch. It is the foremost technique for general purpose data compression due to its simplicity and versatility. Typically, you can expect LZW to compress text, executable code, and similar data files to about one-half their original size. LZW also performs well when presented with extremely redundant data files, such as tabulated numbers, computer source code, and acquired signals. Compression ratios of 5:1 are common for these cases. LZW is the basis of several personal computer utilities that claim to "double the capacity of your hard drive." LZW compression is always used in GIF image files, and offered as an option in TIFF and PostScript. LZW compression is protected under U.S. patent number 4,558,302, granted December 10, 1985 to Sperry Corporation (now the Unisys Corporation). For information on commercial licensing, contact: Welch Licensing Department, Law Department, M/SC2SW1, Unisys Corporation, Blue Bell, Pennsylvania, 19424-0001.

[FIG 27-6]

Example of code table compression. This is the basis of the popular LZW compression method. Encoding occurs by identifying sequences of bytes in the original file that exist in the code table. The 12 bit code representing the sequence is placed in the compressed file instead of the sequence. The first 256 entries in the table correspond to the single byte values, 0 to 255, while the remaining entries correspond to sequences of bytes. The LZW algorithm is an efficient way of generating the code table based on the particular data being compressed. (The code table in this figure is a simplified example, not one actually generated by the LZW algorithm).

LZW compression uses a code table, as illustrated in Fig. 27-6. A common choice is to provide 4096 entries in the table. In this case, the LZW encoded data consists entirely of 12 bit codes, each referring to one of the entries in the code table. Uncompression is achieved by taking each code from the compressed file, and translating it through the code table to find what character or characters it represents. Codes 0-255 in the code table are always assigned to represent single bytes from the input file. For example, if only these first 256 codes were used, each byte in the original file would be converted into 12 bits in the LZW encoded file, resulting in a 50% larger file size. During uncompression, each 12 bit code would be translated via the code table back into the single bytes. Of course, this wouldn't be a useful situation.

The LZW method achieves compression by using codes 256 through 4095 to represent sequences of bytes. For example, code 523 may represent the sequence of three bytes: 231 124 234. Each time the compression algorithm encounters this sequence in the input file, code 523 is placed in the encoded file. During uncompression, code 523 is translated via the code table to recreate the true 3 byte sequence. The longer the sequence assigned to a single code, and the more often the sequence is repeated, the higher the compression achieved.

Although this is a simple approach, there are two major obstacles that need to be overcome: (1) how to determine what sequences should be in the code table, and (2) how to provide the uncompression program the same code table used by the compression program. The LZW algorithm exquisitely solves both these problems.

When the LZW program starts to encode a file, the code table contains only the first 256 entries, with the remainder of the table being blank. This means that the first codes going into the compressed file are simply the single bytes from the input file being converted to 12 bits. As the encoding continues, the LZW algorithm identifies repeated sequences in the data, and adds them to the code table. Compression starts the second time a sequence is encountered. The key point is that a sequence from the input file is not added to the code table until it has already been placed in the compressed file as individual characters (codes 0 to 255). This is important because it allows the uncompression program to reconstruct the code table directly from the compressed data, without having to transmit the code table separately.

Figure 27-7 shows a flowchart for LZW compression. Table 27-3 provides the step-by-step details for an example input file consisting of 45 bytes, the ASCII text string: the/rain/in/Spain/falls/mainly/on/the/plain. When we say that the LZW algorithm reads the character "a" from the input file, we mean it reads the value: 01100001 (97 expressed in 8 bits), where 97 is "a" in ASCII. When we say it writes the character "a" to the encoded file, we mean it writes: 000001100001 (97 expressed in 12 bits).

[FIG 27-7]

LZW compression flowchart. The variable, CHAR, is a single byte. The variable, STRING, is a variable length sequence of bytes. Data are read from the input file (box 1 & 2) as single bytes, and written to the compressed file (box 4) as 12 bit codes. Table 27-3 shows an example of this algorithm.

[TAB 27-3]
TABLE 27-3
LZW example. This shows the compression of the phrase: the/rain/in/Spain/falls/mainly/on/the/plain/.

The compression algorithm uses two variables: CHAR and STRING. The variable, CHAR, holds a single character, i.e., a single byte value between 0 and 255. The variable, STRING, is a variable length string, i.e., a group of one or more characters, with each character being a single byte. In box 1 of Fig. 27-7, the program starts by taking the first byte from the input file, and placing it in the variable, STRING. Table 27-3 shows this action in line 1. This is followed by the algorithm looping for each additional byte in the input file, controlled in the flow diagram by box 8. Each time a byte is read from the input file (box 2), it is stored in the variable, CHAR. The data table is then searched to determine if the concatenation of the two variables, STRING+CHAR, has already been assigned a code (box 3).

If a match in the code table is not found, three actions are taken, as shown in boxes 4, 5 & 6. In box 4, the 12 bit code corresponding to the contents of the variable, STRING, is written to the compressed file. In box 5, a new code is created in the table for the concatenation of STRING+CHAR. In box 6, the variable, STRING, takes the value of the variable, CHAR. An example of these actions is shown in lines 2 through 10 in Table 27-3, for the first 10 bytes of the example file.

When a match in the code table is found (box 3), the concatenation of STRING+CHAR is stored in the variable, STRING, without any other action taking place (box 7). That is, if a matching sequence is found in the table, no action should be taken before determining if there is a longer matching sequence also in the table. An example of this is shown in line 11, where the sequence: STRING+CHAR= in, is identified as already having a code in the table. In line 12, the next character from the input file, /, is added to the sequence, and the code table is searched for: in/. Since this longer sequence is not in the table, the program adds it to the table, outputs the code for the shorter sequence that is in the table (code 262), and starts over searching for sequences beginning with the character, /. This flow of events is continued until there are no more characters in the input file. The program is wrapped up with the code corresponding to the current value of STRING being written to the compressed file (as illustrated in box 9 of Fig. 27-7 and line 45 of Table 27-3).

A flowchart of the LZW uncompression algorithm is shown in Fig. 27-8. Each code is read from the compressed file and compared to the code table to provide the translation. As each code is processed in this manner, the code table is updated so that it continually matches the one used during the compression. However, there is a small complication in the uncompression routine. There are certain combinations of data that result in the uncompression algorithm receiving a code that does not yet exist in its code table. This contingency is handled in boxes 4,5 & 6.

[FIG 27-8]

LZW uncompression flowchart. The variables, OCODE and NCODE (oldcode and newcode), hold the 12 bit codes from the compressed file, CHAR holds a single byte, STRING holds a string of bytes.

Only a few dozen lines of code are required for the most elementary LZW programs. The real difficulty lies in the efficient management of the code table. The brute force approach results in large memory requirements and a slow program execution. Several tricks are used in commercial LZW programs to improve their performance. For instance, the memory problem arises because it is not know beforehand how long each of the character strings for each code will be. Most LZW programs handle this by taking advantage of the redundant nature of the code table. For example, look at line 29 in Table 27-3, where code 278 is defined to be ainl. Rather than storing these four bytes, code 278 could be stored as: code 269 + l, where code 269 was previously defined as ain in line 17. Likewise, code 269 would be stored as: code 261 + n, where code 261 was previously defined as ai in line 7. This pattern always holds: every code can be expressed as a previous code plus one new character.

The execution time of the compression algorithm is limited by searching the code table to determine if a match is present. As an analogy, imagine you want to find if a friend's name is listed in the telephone directory. The catch is, the only directory you have is arranged by telephone number, not alphabetical order. This requires you to search page after page trying to find the name you want. This inefficient situation is exactly the same as searching all 4096 codes for a match to a specific character string. The answer: organize the code table so that what you are looking for tells you where to look (like a partially alphabetized telephone directory). In other words, don't assign the 4096 codes to sequential locations in memory. Rather, divide the memory into sections based on what sequences will be stored there. For example, suppose we want to find if the sequence: code 329 + x, is in the code table. The code table should be organized so that the "x" indicates where to starting looking. There are many schemes for this type of code table management, and they can become quite complicated.

This brings up the last comment on LZW and similar compression schemes: it is a very competitive field. While the basics of data compression are relatively simple, the kinds of programs sold as commercial products are extremely sophisticated. Companies make money by selling you programs that perform compression, and jealously protect their trade-secrets through patents and the like. Don't expect to achieve the same level of performance as these programs in a few hours work.

JPEG (Transform Compression)

Many methods of lossy compression have been developed; however, a family of techniques called transform compression has proven the most valuable. The best example of transform compression is embodied in the popular JPEG standard of image encoding. JPEG is named after its origin, the Joint Photographers Experts Group. We will describe the operation of JPEG to illustrate how lossy compression works. We have already discussed a simple method of lossy data compression, coarser sampling and/or quantization (CS&Q in Table 27-1). This involves reducing the number of bits per sample or entirely discard some of the samples. Both these procedures have the desired effect: the data file becomes smaller at the expense of signal quality. As you might expect, these simple methods do not work very well.

Transform compression is based on a simple premise: when the signal is passed through the Fourier (or other) transform, the resulting data values will no longer be equal in their information carrying roles. In particular, the low frequency components of a signal are more important than the high frequency components. Removing 50% of the bits from the high frequency components might remove, say, only 5% of the encoded information.

As shown in Fig. 27-9, JPEG compression starts by breaking the image into 8x8 pixel groups. The full JPEG algorithm can accept a wide range of bits per pixel, including the use of color information. In this example, each pixel is a single byte, a grayscale value between 0 and 255. These 8x8 pixel groups are treated independently during compression. That is, each group is initially represented by 64 bytes. After transforming and removing data, each group is represented by, say, 2 to 20 bytes. During uncompression, the inverse transform is taken of the 2 to 20 bytes to create an approximation of the original 8x8 group. These approximated groups are then fitted together to form the uncompressed image. Why use 8x8 pixel groups instead of, for instance, 16x16? The 8x8 grouping was based on the maximum size that integrated circuit technology could handle at the time the standard was developed. In any event, the 8x8 size works well, and it may or may not be changed in the future.

[FIG 27-9]

JPEG image division. JPEG transform compression starts by breaking the image into 8x8 groups, each containing 64 pixels. Three of these 8x8 groups are enlarged in this figure, showing the values of the individual pixels, a single byte value between 0 and 255.

Many different transforms have been investigated for data compression, some of them invented specifically for this purpose. For instance, the Karhunen-Loeve transform provides the best possible compression ratio, but is difficult to implement. The Fourier transform is easy to use, but does not provide adequate compression. After much competition, the winner is a relative of the Fourier transform, the Discrete Cosine Transform (DCT).

Just as the Fourier transform uses sine and cosine waves to represent a signal, the DCT only uses cosine waves. There are several versions of the DCT, with slight differences in their mathematics. As an example of one version, imagine a 129 point signal, running from sample 0 to sample 128. Now, make this a 256 point signal by duplicating samples 1 through 127 and adding them as samples 255 to 130. That is: 0,1,2,3...127,128,127...2,1. Taking the Fourier transform of this 256 point signal results in a frequency spectrum of 129 points, spread between 0 and 128. Since the time domain signal was forced to be symmetrical, the spectrum's imaginary part will be composed of all zeros. In other words, we started with a 129 point time domain signal, and ended with a frequency spectrum of 129 points, each the amplitude of a cosine wave. Voila, the DCT!

When the DCT is taken of an 8x8 group, it results in an 8x8 spectrum. In other words, 64 numbers are changed into 64 other numbers. All these values are real; there is no complex mathematics here. Just as in Fourier analysis, each value in the spectrum is the amplitude of a basis function. Figure 27-10 shows 6 of the 64 basis functions used in an 8x8 DCT, according to where the amplitude sits in the spectrum. The 8x8 DCT basis functions are given by:

[EQ 27-1]

DCT basis functions. The variables x & y are the indexes in the spatial domain, and u & v are the indexes in the frequency spectrum. This is for an 8x8 DCT, making all the indexes run from 0 to 7.

The low frequencies reside in the upper-left corner of the spectrum, while the high frequencies are in the lower-right. The DC component is at [0,0], the upper-left most value. The basis function for [0,1] is one-half cycle of a cosine wave in one direction, and a constant value in the other. The basis function for [1,0] is similar, just rotated by 90 degrees.

[FIG 27-10]

FIGURE 27-10
The DCT basis functions. The DCT spectrum consists of an 8x8 array, with each element in the array being an amplitude of one of the 64 basis functions. Six of these basis functions are shown here, referenced to where the corresponding amplitude resides.

The DCT calculates the spectrum by correlating the 8x8 pixel group with each of the basis functions. That is, each spectral value is found by multiplying the appropriate basis function by the 8x8 pixel group, and then summing the products. Two adjustments are then needed to finish the DCT calculation (just as with the Fourier transform). First, divide the 15 spectral values in row 0 and column 0 by two. Second, divide all 64 values in the spectrum by 16. The inverse DCT is calculated by assigning each of the amplitudes in the spectrum to the proper basis function, and summing to recreate the spatial domain. No extra steps are required. These are exactly the same concepts as in Fourier analysis, just with different basis functions.

Figure 27-11 illustrates JPEG encoding for the three 8x8 groups identified in Fig. 27-9. The left column, Figs. a, b & c, show the original pixel values. The center column, Figs. d, e & f, show the DCT spectra of these groups.

The right column, Figs. g, h & i, shows the effect of reducing the number of bits used to represent each component in the frequency spectrum. For instance, (g) is formed by truncating each of the samples in (d) to ten bits, taking the inverse DCT, and then subtracting the reconstructed image from the original. Likewise, (h) and (i) are formed by truncating each sample in the spectrum to eight and five bits, respectively. As expected, the error in the reconstruction increases as fewer bits are used to represent the data. As an example of this bit truncation, the spectra shown in the center column are represented with 8 bits per spectral value, arranged as 0 to 255 for the DC component, and -127 to 127 for the other values.

[FIG 27-11]

FIGURE 27-11
Example of JPEG encoding. The left column shows three 8x8 pixel groups, the same ones shown in Fig. 27-9. The center column shows the DCT spectra of these three groups. The third column shows the error in the uncompressed pixel values resulting from using a finite number of bits to represent the spectrum.

The second method of compressing the frequency domain is to discard some of the 64 spectral values. As shown by the spectra in Fig. 27-11, nearly all of the signal is contained in the low frequency components. This means the highest frequency components can be eliminated, while only degrading the signal a small amount. Figure 27-12 shows an example of the image distortion that occurs when various numbers of the high frequency components are deleted. The 8x8 group used in this example is the eye image of Fig. 27-10. Figure (d) shows the correct reconstruction using all 64 spectral values. The remaining figures show the reconstruction using the indicated number of lowest frequency coefficients. As illustrated in (c), even removing three-fourths of the highest frequency components produces little error in the reconstruction. Even better, the error that does occur looks very much like random noise.

[FIG 27-12]

FIGURE 27-12
Example of JPEG reconstruction. The 8 8 pixel group used in this example is the eye in Fig. 27-9. As shown, less than 1/4 of the 64 values are needed to achieve a good approximation to the correct image.

JPEG is good example of how several data compression schemes can be combined for greater effectiveness. The entire JPEG procedure is outlined in the following steps. First, the image is broken into the 8x8 groups. Second, the DCT is taken of each group. Third, each 8x8 spectrum is compressed by the above methods: reducing the number of bits and eliminating some of the components. This takes place in a single step, controlled by a quantization table. Two examples of quantization tables are shown in Fig. 27-13. Each value in the spectrum is divided by the matching value in the quantization table, and the result rounded to the nearest integer. For instance, the upper-left value of the quantization table is one, resulting in the DC value being left unchanged. In comparison, the lower- right entry in (a) is 16, meaning that the original range of -127 to 127 is reduced to only -7 to 7. In other words, the value has been reduced in precision from eight bits to four bits. In a more extreme case, the lower- right entry in (b) is 256, completely eliminating the spectral value.

[FIG 27-13]

FIGURE 27-13
JPEG quantization tables. These are two example quantization tables that might be used during compression. Each value in the DCT spectrum is divided by the corresponding value in the quantization table, and the result rounded to the nearest integer.

In the fourth step of JPEG encoding, the modified spectrum is converted from an 8x8 array into a linear sequence. The serpentine pattern shown in Figure 27-14 is used for this step, placing all of the high frequency components together at the end of the linear sequence. This groups the zeros from the eliminated components into long runs. The fifth step compresses these runs of zeros by run-length encoding. In the sixth step, the sequence is encoded by either Huffman or arithmetic encoding to form the final compressed file.

[FIG 27-14]

FIGURE 27-14
JPEG serial conversion. A serpentine pattern used to convert the 8x8 DCT spectrum into a linear sequence of 64 values. This places all of the high frequency components together, where the large number of zeros can be efficiently compressed with run-length encoding.

The amount of compression, and the resulting loss of image quality, can be selected when the JPEG compression program is run. Figure 27-15 shows the type of image distortion resulting from high compression ratios. With the 45:1 compression ratio shown, each of the 8 8 groups is represented by only about 12 bits. Close inspection of this image shows that six of the lowest frequency basis functions are represented to some degree.

Why is the DCT better than the Fourier transform for image compression? The main reason is that the DCT has one-half cycle basis functions, i.e., S[0,1] and S[1,0]. As shown in Fig. 27-10, these gently slope from one side of the array to the other. In comparison, the lowest frequencies in the Fourier transform form one complete cycle. Images nearly always contain regions where the brightness is gradually changing over a region. Using a basis function that matches this basic pattern allows for better compression.

[FIG 27-15A] [FIG 27-15B] [FIG 27-15C]

FIGURE 27-15
Example of JPEG distortion. The image on the left is the original, while the center and right are restored images using compression ratios of 10:1 and 45:1, respectively. The high compression ratio used in the right image in each 8x8 pixel group being represented by less than 12 bits.


MPEG is a compression standard for digital video sequences, such as used in computer video and digital television networks. In addition, MPEG also provides for the compression of the sound track associated with the video. The name comes from its originating organization, the Moving Pictures Experts Group. If you think JPEG is complicated, MPEG is a nightmare! MPEG is something you buy, not try to write yourself. The future of this technology is to encode the compression and uncompression algorithms directly into integrated circuits. The potential of MPEG is vast. Think of thousands of video channels being carried on a single optical fiber running into your home. This is a key technology of the 21st century.

In addition to reducing the data rate, MPEG has several important features. The movie can be played forward or in reverse, and at either normal or fast speed. The encoded information is random access, that is, any individual frame in the sequence can be easily displayed as a still picture. This goes along with making the movie editable, meaning that short segments from the movie can be encoded only with reference to themselves, not the entire sequence. MPEG is designed to be robust to errors. The last thing you want is for a single bit error to cause a disruption of the movie.

The approach used by MPEG can be divided into two types of compression: within-the-frame and between-frame. Within-the-frame compression means that individual frames making up the video sequence are encoded as if they were ordinary still images. This compression is preformed using the JPEG standard, with just a few variations. In MPEG terminology, a frame that has been encoded in this way is called an intra-coded or I-picture.

Most of the pixels in a video sequence change very little from one frame to the next. Unless the camera is moving, most of the image is composed of a background that remains constant over dozens of frames. MPEG takes advantage of this with a sophisticated form of delta encoding to compress the redundant information between frames. After compressing one of the frames as an I-picture, MPEG encodes successive frames as predictive-coded or P-pictures. That is, only the pixels that have changed since the I- picture are included in the P-picture.

While these two compression schemes form the backbone of MPEG, the actual implementation is immensely more sophisticated than described here. For example, a P-picture can be referenced to an I-picture that has been shifted, accounting for motion of objects in the image sequence. There are also bidirectional predictive-coded or B-pictures. These are referenced to both a previous and a future I-picture. This handles regions in the image that gradually change over many of frames. The individual frames can also be stored out-of-order in the compressed data to facilitate the proper sequencing of the , P, and B-pictures. The addition of color and sound makes this all the more complicated.

The main distortion associated with MPEG occurs when large sections of the image change quickly. In effect, a burst of information is needed to keep up with the rapidly changing scenes. If the data rate is fixed, the viewer notices "blocky" patterns when changing from one scene to the next. This can be minimized in networks that transmit multiple video channels simultaneously, such as cable television. The sudden burst of information needed to support a rapidly changing scene in one video channel, is averaged with the modest requirements of the relatively static scenes in the other channels.

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