11.1 Considerations Based on Information Theory Image data compression is possible due to the following basic characteristics: Code redundancy | all code words pixel values do not occur
Trang 1Image Coding and Data Compression
High spatial resolution and ne gray-scale quantization are often required
in biomedical imaging Digital mammograms are typically represented inarrays of 4 096 4 096 pixels with 12 b=pixel, leading to raw-data les ofthe order of 32MB per image Volumetric data obtained by CT and MRIcould be of size 512 512 64 voxels with 16 b=voxel, occupying 32 MBper examination Patients with undetermined or multiple complications mayundergo several examinations via di erent modalities such as X-ray imaging,ultrasound scanning, CT scanning, and nuclear medicine imaging, resulting
in large collections of image les
Most health-care jurisdictions require medical records, including images, ofadults to be stored for durations of the order of seven years from the date
of acquisition Children's records and images are required to be maintaineduntil at least the time they reach adulthood
With the view to improve the eciency of storage and access, several ing centers and hospitals have moved away from lm-based storage towardelectronic storage Furthermore, most medical imaging systems have moved
imag-to direct digital image acquisition with adequate resolution, putting aside thedebate on the quality of an original lm-based image versus that of its scanned(digitized) representation Since 1980, an entire series of conferences has beendedicated to PACS: see the PACS volumes of the SPIE Medical Imaging con-ference series 956] Networks and systems for PACS are integrated into theinfrastructure of most modern hospitals The major advantages and disad-vantages of digital and lm-based archival systems are listed below
Films deteriorate with age and handling Digital images are una ected
be an expensive option that adds storage and handling complexities
With the proliferation of computers, digital images may be viewed andmanipulated at several convenient locations, including a surgical suite,
a patient's bedside, and one's home or oce Viewing lm-based
Trang 2im-ages with detailed attention requires specialized viewing consoles undercontrolled lighting conditions.
Digital PACS require signicant initial capital outlay, as well as routinemaintenance and upgrading of the computer, storage, and communi-cation systems However, these costs may be o set by the savings inthe continuing costs of lm, as well as the associated chemical process-ing systems and disposal The environmental concerns related to lmprocessing are also removed by digital PACS
Digital images may be compressed via image coding and data sion techniques so as to occupy less storage space
compres-The nal point above forms the topic of the present chapter
Although the discussion above has been in the context of image storage orarchival, similar concerns regarding the size of image les and the desirability
of compression arise in the communication of image data In this chapter,
we shall study the basic concepts of information theory that apply to imagecoding, compression, and communication We shall investigate several tech-niques for encoding image data, including decorrelation procedures to modifythe statistical characteristics of the data so as to permit ecient representa-tion, coding, and compression
The representation of the signicant aspects of an image in terms of a smallnumber of numerical features for the purpose of pattern classication mayalso be viewed as image coding or data compression however, we shall treatthis topic separately (see Chapter 12)
11.1 Considerations Based on Information Theory
Image data compression is possible due to the following basic characteristics:
Code redundancy | all code words (pixel values) do not occur withequal probability
Spatial redundancy | the values of neighboring pixels tend to lie within
a small dynamic range, and exhibit a high level of correlation
Psychovisual redundancy | human analysts can recognize the essentialnature and components of an image from severely reduced versions such
as caricatures, edges, and regions, and need not (or do not) pay attention
to precise numerical values
Information-theoretic considerations are based upon the notion of tion as related to the statistical uncertainty of the occurrence of an event
Trang 3informa-(such as a signal, an image, or a pixel value), rather than the structural, bolic, pictorial, semantic, or diagnostic content of the entity The measure ofentropy is based upon the probabilities of occurrence of the various symbolsinvolved in the representation of a message or image: seeSection 2.8 Despitethe mathematical and theoretical powers of measures such as entropy, thestandpoint of viewing an image as being composed of discrete and indepen-dent symbols (numerical values) removes the analyst from the real-world andphysical properties of the image The use of the underlying assumptions alsolead to severe limitations in entropy-based source coding, with lossless com-pression factors often limited to the order of 2:1 Additional techniques basedupon decorrelation of the image data via the identication and modeling ofthe underlying image-generation phenomena, or the use of pattern recognitiontechniques, could assist in improving the performance of image compressionprocedures.
sym-11.1.1 Noiseless coding theorem for binary transmissionGiven a code with an alphabet of two symbols and a sourceAwith an alphabet
of two symbols, the average length of the code words per source symbol may
be made arbitrarily close to the lower bound (entropy) H(A) by encodingsequences of source symbols instead of encoding individual symbols 9, 126].The average lengthL(n) of encoded n-symbol sequences is bounded by
11.1.2 Lossy versus lossless compression
A coding or compression method is considered to be lossless if the originalimage data can be recovered, with no error, from the coded and compresseddata Such a technique may also be referred to as a reversible, bit-preserving,
or error-free compression technique
Trang 4A compression technique becomes lossy or irreversible if the original datacannot be recovered, with complete pixel-by-pixel numerical accuracy, fromthe compressed data In the case of images, the human visual system can tol-erate signicant numerical di erences or error, in the sense that the degradedimage recovered from the compressed data is perceived to be essentially thesame as the original image This arises from the fact that a human ob-server will, typically, not examine the numerical values of individual pixels,but instead assess the semantic or pictorial information conveyed by the data.Furthermore, a human analyst may tolerate more error, noise, or distortion inthe uniform areas of an image than around its edges that attract visual atten-tion Data compression techniques may be designed to exploit these aspects
to gain signicant advantages in terms of highly compressed representation,with high levels of loss of numerical accuracy while remaining perceptuallylossless On the same token, in medical imaging, if the numerical errors in theretrieved and reconstructed images do not cause any change in the diagnosticresults obtained by using the degraded images, one could achieve high levels
of numerically lossy compression while remaining diagnostically lossless
In the quest to push the limits of numerically lossy compression techniqueswhile remaining practically lossless under some criterion, the question arises as
to the worth of such practice Medical practice in the present highly litigioussociety could face large nancial penalties and loss due to errors Radiologi-cal diagnosis is often based upon the detection of minor deviations from thenormal (or average) patterns expected in medical images If a lossy data com-pression technique were to cause such a faint deviation to be less perceptible
in the compressed (and reconstructed) image than in the original image, andthe diagnosis based upon the reconstructed image were to be in error, thenancial compensation to be paid would cost several times the amount saved
in data storage the loss in professional standing and public condence could
be even more damaging In addition, dening the delity of representation interms of the closeness to the original image or distortion measures is a di-cult and evasive activity Given the high levels of the professional care andconcern, as well as the scal and emotional investment, that are part of medi-cal image acquisition procedures, it would be undesirable to use a subsequentprocedure that could cause any degradation of the image In this spirit, onlylossless coding and compression techniques will be described in the presentchapter Regardless, it should be noted that any lossy compression techniquemay be made lossless by providing the numerical error between the originalimage and the degraded image reconstructed from the compressed data Al-though this step will lead to additional storage or transmission requirements,the approach can facilitate the rapid retrieval or transmission of an initial,low-quality image, followed by completely lossless recovery: such a procedure
is known as progressive transmission, especially when performed over multiplestages of image quality or delity
Trang 511.1.3 Distortion measures and delity criteria
Although we have stated our interest in lossless coding of biomedical images,other processes, such as the transmission of large quantities of data over noisychannels, may lead to some errors in the received images Hence, it would berelevant to consider the characterization of the distortion so introduced, andanalyze the delity of the received image with respect to the original 9].The binary symmetric channel is characterized by a single parameter: thebit-error probabilityp(see Figure 11.1) The channel capacity is given by
FIGURE 11.1
Transmission error probabilities in a binary symmetric channel 9]
The least-squares single-letter delity criterion is dened as 9]
n(x y) = 1n
n
X
l =1(xl ;yl)22( l ; 1) (11.3)where x and y are the transmitted and received n-bit vectors (blocks orwords), respectively
The Hamming distance between the vectorsxandyis dened as
DH(x y) = 1n
n
X
l =1 (xl ;yl)2: (11.4)Measures of delity may also be dened based upon entire images by den-ing an error image as
e(m n) =g(m n) f(m n) (11.5)
Trang 6whereg(m n) is the received (degraded) version of the original (transmitted)imagef(m n), and then dening the RMS value of the error as
11.2 Fundamental Concepts of Coding
In general, coding could be dened as the use of symbols to represent mation The following list provides the denitions of a few basic terms andconcepts related to coding 9]:
infor- An alphabet is a predened set of symbols
A word is a nite sequence of symbols from an alphabet
A code is a mapping of words from a source alphabet into the words of
A desirable property of a uniquely decodable code is that it be decodable
on a word-to-word basis This is ensured if no code word may be a prex
to another the code is then instantaneously decodable
A code is said to be optimal if it is instantaneously decodable and hasthe minimum average length for a given source PDF
Examples of symbols are f0 1gin the binary alphabetf0 1 2 3 4 5 6 7g
in the octal systemf0 1 2 3 4 5 6 7 8 9g in the decimal systemf0 1 2 3
4 5 6 7 8 9 A B C D E Fg in the hexadecimal systemfI V X L C D
Mgin the Roman system (with the decimal equivalents of 1 5 10 50 100 500and 1 000, respectively) and fA;Z a;zg in the English alphabet (not
Trang 7considering punctuation marks and special symbols) An example of a word
in the context of image coding is 00001011 in 8 b binary coding, standingfor the gray level 11 in the decimal system Table 11.1 lists the codes forintegers in the range 0 20] in the Roman, decimal, binary, Gray 957], octal,and hexadecimal codes 958] The Gray code has the advantageous featurethat only one digit is changed from one number to the next Observe that, ingeneral, all of the codes described here (including the English language) failthe conditions dened above for an optimal code
11.3 Direct Source Coding
Pixels generated by real-life sources of images bear limitations in dynamicrange and variability within a small spatial neighborhood Therefore, codesused to represent pixel data at the source may be expected to demonstratecertain patterns of limited variation and high correlation Furthermore, real-life sources of images do not generate random, uncorrelated values that areequally likely instead, it is common to encounter PDFs of gray levels thatare nonuniform Some of these characteristics may be exploited to achieveecient representation of images by designing coding systems tuned to specicproperties of the source Because the coding method is applied directly to pixelvalues generated by the source (without processing them by an algorithm togenerate a di erent series of values), such techniques are categorized as directsource coding procedures
a word-by-word basis, which implies that no code word may be a prex toanother Hu man devised a coding scheme to meet these requirements andlead to average code-word lengths lower than that provided by xed-lengthcodes Hu man coding provides an average code-word lengthLthat is limited
by the zeroth-order entropy of the sourceH0(see Equation 2.18) andH0+1:
The procedure to generate the Hu man code is as follows 9, 959]:
Trang 8Leading zeros have been removed in the decimal and hexadecimal (Hex) codes,but retained in the binary, Gray, and octal codes.
Trang 91 Prepare a table listing the symbols (gray levels) in the source (image)sorted in decreasing order of the probabilities of their occurrence.
2 Combine the last two probabilities The list of probabilities now hasone less entry than before
3 Copy the reduced list over to a new column, rearranging (as necessary)such that the probabilities are in decreasing order
4 Repeat the procedure above until the list of probabilities is reduced toonly two entries
5 Assign the code digits 0 and 1 to the two entries in the nal column
of probabilities (Note: There are two possibilities of this assignmentthat will lead to two di erent codes however, their performance will beidentical.)
6 Working backwards through the columns of probabilities, assign tional bits of 0 and 1 to the two entries that resulted in the last com-pounded entry in the column
addi-7 Repeat the procedure until the rst column of probabilities is reachedand all symbols have been assigned a code word
It should be noted that a Hu