BOOK REVIEWS 527

Is such a concern useful? The answer mustbe a strong yes, because optical processing isconfined primarily to linear oper,ations andanything else requires the use of a computer.A host of activitiesmincluding image inter-pretation, information extraction, imageclassification, image segmentation, and im-age compression--involve operations notamenable to linear optical filtering. Thus wemust involve a computer, requiring that animage be segmented onto a lattice and itsluminance be mapped as discrete numbersorganized into an array. Such arrays are thestarting point of this book.

The book’s strength lies in the largecollection of mathematical tools it providesfor image processing operations. distinguishbetween tools (which the authors furnish inabundance) and algorithms (which they donot). Image processing tools are arithmeticor logical operations that are available to theuser for designing an algorithm to achieve aspecific end. What are such algorithms? Oneis reconstructing an image from projectionssuch as are used in computer-aided tomog-raphy (CAT); sharpening an imaging cor-rupted by motion blurring is another. Ofparticular importance in these days ofrobot-ics are algorithms for object recognition bycomputer, image reconstruction from only afew views, and image segmentation accord-ing to distance.

As stated above, problem-solving algo-rithms are virtually ignored in this book. Butthe tools required to construct the algorithmsare not. Thus a fundamental tool requiredto reconstruct an image from projections isthe Radon transform and a good discussionof the Radon transform is provided. To rec-ognize shape in an image, edge detection isa critical operation. Edge detection proce-dures are furnished and clearly discussed.

In addition to a grabbag of image proc-essing operations, the book contains chaptersdealing with morphological image process-ing, Fourier and other transform methods,projection methods, probabilistic methods,and estimation and detection methods. Oneannoying feature is a frequent switching be-tween analog procedures and their digitalequivalents without establishing a relationbetween them. Amazingly, there is no dis-cussion of fast Fourier techniques, one ofthe central operations in computer imageprocessing.

The book gives a clear discussion ofmorphological methods based on Min-kowski algebra. Its discussion of projectiontechniques is basic but ignores the powerfultools furnished by the theory of convex pro-jections widely used in image processing.There are excellent discussions of chaincodes, chain code representation of images,and transform representation of shapes.

The book has the irritating feature ofintroducing mathematical material withoutconnecting this strongly to image processing.For example, a whole chapter entitled"Probability for Image Processing" virtuallyignores image processing applications exceptfor a few weak examples, almost after-thoughts. The same can be said of the nextchapter, "Inferential Statistical Techniquesin Image Processing," which introduces ad-vanced tools such as the Kalman filter butfails to effectively illustrate its use in imageprocessing.

The deeper issues of image processing,such as the question ofwhat is the minimuminformation that uniquely specifies an image,or of the relation between magnitude andphase, are beyond the level of this book.

In summary, this is basically a soundand good book for what it does, but not forwhat is implied by the title. The title shouldhave been Computer Techniques in ImageProcessing or Methods for Digital ImageProcessing. With this reservation in mind,can recommend this book as a very good,perhaps excellent, choice for a first course incomputer techniques in image processing.

The production is nicely done exceptfor a cover design that resembles those ofhandbooks of tables.

HENRY STARKRensselaer Polytechnic Institute

Preference Modelling. ByMarcRoubens andPhilippe Vincke. Springer-Verlag, Berlin,1985. viii + 94 pp. $12.30, paper. ISBN 3-540-15685-2. Lecture Notes in Economicsand Mathematical Systems, Vol. 250.

Suppose a decision-maker, when pre-sented with two numbers of a set A, musteither prefer x to y (x P y), be indifferent(x y), or else say they are incompatable(x R y). Then P, I, and R form the preferencestructure on A for that decision-maker. In

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528 BOOK REVIEWS

general, a preference structure on a set A isa set of three binary relations {P, I, R} suchthat: P is asymmetric; is reflexive andsymmetric; R is irreflexive and symmetric.Preference modelling is the study of pref-erence structures: specifically, of deducingthem from data by mathematical modellingtechniques, and of their representationalproperties.

The volume under review is a terse,brief survey of some aspects of preferencemodelling. The series in which it appears iscalled Lecture Notes; informality is to beexpected, and polish is sacrificed in order toensure rapid publication. But the book readslike notes taken by a student at a lecturecoursemslightly rewritten but not editedmand it would be a far more useful documentif more care had gone into editing it. Forexample, "preference modelling" (the subjectof the book) is not defined. The authors sayin their Introduction that they will studypreference structures according to threespecific guidelines: graph representation, nu-merical representation, and opinion tableauconfiguration; we would accordingly expectthese three ideas to be very important. Butopinion tableaux are first introduced in anexample and defined only in parentheses.

In many places the authors present in-formation in long lists. For example, the firstchapter (five pages) contains four half-pagelists (and some shorter ones). This style ofexposition may be appropriate as an accom-paniment to blackboard work, but is lessuseful when it stands alone.

The book is written in poor English.While we are willing to forgive clumsy orungrammatical prose, the problem is thatsome passages are quite incomprehensible.For example, here is 2.3 (page 7) in itsentirety:

2.3. IMPORTANT AGREEMENT.Given the preference structure {P, I, R}

one can associate an equivalence relation E(reflexive, symmetric and transitive) asfollows:

la, b-A:aEb iff aPcc,bPc,cPac,cPb,alc,blc,aRc,bRc, lce.A.

We can gain much in the way of econ-omy by dealing with representatives of theclasses of E rather than the elements of A

themselves. If it is not the case, we shouldwork in the quotient of A by E, A/E, can-ceiling out the equivalence relation.

Does the section heading name the equiva-lence relation E, or does it refer to the secondparagraph or to something else? And whatdoes the last sentence mean?

These remarks should not be taken as acondemnation of the volume. However, ifmore care had been taken and time spent onpresentation, this book would be useful asan introduction to preference structure. Asit stands, it would be very difficult for some-one new to this area to read it without ateacher.

The book opens with a sketch of binaryrelations. Chapter 2 defines preference struc-tures, and Chapter 3 discusses common ex-amples. Two new preference structuresmpartial interval order structure and partialsemiorder structuremare introduced inChapter 4.

A valued preference structure can beinterpreted as a map that gives the degree (orintensity) ofpreference: ifx is preferred to y,then u(x, y) may range between zero and 1;z(y, x)=-(x, y). Thus # replaces the rela-tion P in the definition of a preference struc-ture; usually x I y is also replaced by therelation (x, y) 0. A complete-valued pref-erence structure is one for which R is empty.Chapter 5 provides a useful discussion ofcomplete-valued preference structures. If ucan take only two positive values ("strong"or "weak" preference) then the valued pref-erence structure is called two-valued; thesestructures are defined in Chapter 6, the briefconcluding chapter.

In summary, this short book is an inter-esting addition to the (somewhat meager)literature on preference structures. It is noteasy reading, more because of the presenta-tion than the degree of difficulty of thematerial.

W. D. WALLISSouthern Illinois University

Fuzzy Mathematical Techniques with Appli-cations. By Abraham KandeL Addison-Wesley, Reading, MA, 1986. xiv + 274 pp.No price given. ISBN 0-201-11752-5.

This is a graduate-level textbook forstudents and researchers whose interests are

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