Dear Phil,

I have carefully read and thought through the comments by the reviewers and here are my responses, followed by more specific ones embedded in the text of the reviewers comments.

First, rhere are some comments regarding the existence of similar books and the targeted readers of the proposed book. For clarification, the targeted readers of the proposal the book are expected to have some prior knowledge in fundamental calculus, linear algebra and probability theory. It is reasonable to expect that a science/engineering undergraduate student to have taken the relevant courses by the time they are in their junior (3rd) year. The book can also be used as a reference for graduate students and practicing scientists/engineers interested in transform methods for data processing in a wide variety of applications.

Also, there are some comments regarding the value of the proposed book, in light of many existing books covering similar topics, typified by reviewer B’s comment: “There are many other books out there which are more complete in terms of transform coverage and variety”. I do not quite agree with this view.

It is true that all topics proposed to cover in this book have been discussed else where in various books already in existence, such as systems and signal books and digital signal processing books. Some of the topics may be aimed at graduate students and researchers with greater details and more rigors than intended in the proposed book, while some other topics may be discussed in textbooks with a less rigorous manner at undergraduate level. However, the unique point of the proposed book is to discuss all these transform methods from a unified view in a single volume, with emphasis on a unified treatment, programming implementation, and various applications in practice. The goal is not to present any new materials not covered in any previous books.

This point is realized by reviewers A and D. Reviewer A said: “I have not found a book which is formed like the proposed book, and I think that it can be very interesting for the intended target group…The book would be useful both for undergrad and graduate students, as well as for engineers in industry”, and reviewer D said: “No other book really covers the breadth of transforms and applications proposed here (and this would be even further the case with the inclusions recommended in the previous section). Given the style and philosophy that the book proposes, it would be useful for all levels of readers in technical fields, ranging from university upper division and graduate students through experienced practicing engineers.  For those beginning their training in technical fields, the book would serve as a good educational introduction to transforms, their foundations, details, and applications.  It does not appear, though, that the book would be necessarily useful for graduate students specifically researching transforms per se, since its level would not be technical enough (unless more advanced treatments, exercises, and applications were provided).  For those that are already experienced practitioners in technical fields, the book would serve as a useful reference, especially since it would expose the user to transforms used in completely different fields, and thus likely lead to

new cross-field application of these analysis tools.  I myself would recommend the book to my current colleagues that practice in the microwave engineering area.”

Second they are some criticism in terms of the sample writing. But first of all, it may need to be clarified that the final version of the sample writing was carefully checked (including spell check). However, for some reason, it seems a wrong file of the sample writing containing numerous typos was submitted. It is therefore also possible that the submitted version of the sample writing did not reflect the later revisions of the text. This is certainly very unfortunate due to the carelessness on my part. The latest version of the sample writing could still be submitted if addition reviewers are expected.

Certainly there is a lot of room for improvement of the sample writing, such as more text to explain, and more figures to illustrate the ideas, etc. But when I was preparing for the sample writing (by expanding some existing lecture notes used in a course), I thought the writing would only need to “give a reasonable idea of the style that you’re after”, and it would not be seen as the final version. And I also felt that a 24-page text might already have been too long as sample writing. I could have developed this sample chapter more fully and prepared it more carefully as a polished final version.

Having said this, I totally agree with the various of the comments made by the reviewers in terms of how the chapter can be improved, by, for example, adding some historical background of the methods, relating more to other transform methods, and having more text and figures to better explain the idea of PCA method, etc. I am very happy to see all the positive suggestions for possible improvement in writing the chapter.

Also, I greatly appreciate many other specific suggestions made by the reviewers in terms of what additional topics should be added, for example, some other transform methods not already mentioned in the table of content (see my responses to the specific comments below). I will adopt these suggestions while writing the book, as I definitely believe they will make the book more readable and valuable.

Review A

I think there is a market for an introductory book on orthogonal transform methods and their applications – in particular if it is really formed in the fashion that is described by the authors.   There are some sentences in the background text that I find particularly

interesting and promising. I have commented these sentences below: 1.        “The main approach … is the unified view and treatment of these transform methods” Good!   2.        “… present the materials in such a way that the common philosophical principles and similar properties of these transform methods will be emphasized” “… the different orthogonal transforms will be compared in terms of their pros and cons, and implementations” I think this is one of the core characteristics of the book. Then, it is desirable that the comparisons are made regularly through the book – not only after Part One, Part Two, etc., and not even only in the end of each main chapter! Otherwise, the reader (in particular the undergraduate reader) may not successfully grasp the contents and the book will not fulfil its purpose. For example, the authors could present some data processing problems in the beginning of the book and then, in each chapter, investigate (when applicable) the efficiency of the present transform method compared to the efficiency of the transform methods in the previous chapters.

Thanks to this suggestion. Indeed, one or two data processing problems can be addressed by all different transform methods through out the book as an example for the comparison of these methods.  3.        “this book would also emphasize the importance of the conceptual understanding and the intuitive interpretation of the results derived mathematically.” Good – very important!   4.        “In addition to the explanation in the text, many of the concepts and examples will be illustrated by Matlab demonstrations for the student to see how the concepts and ideas discussed in the chapter can be actually carried out using a computer.” This is very good!   5.        “the book will also discuss how the different orthogonal transform methods are actually used in the real world” Also very important, in particular as motivating reasons for the student to learn the subject! Too many books lack in this respect. In this book, it is

important to exemplify and demonstrate the wide area of applications.   6.        “this book will emphasize the importance of hands-on skills and capabilities of applying the theories learned to solving actual problems.” I agree!   7.        “The reader of the book will gain a more thorough and profound understanding of such transform methods as a general methodology along with their applications to a wide variety of different fields.” Yet another statement that strengthens the reason for writing this book. The authors promise a lot… but if they succeed, this can be a very valuable book.   Some additional comments

The promised comparison philosophy, which I am very positive about, can not be seen so clearly in the Table of Contents. Part 1 and Part 2 (Ch. 1-9) are the typical/traditional chapters on Fourier transforms, Laplace transforms, z-transforms, and filtering that is included in most signal processing books. It would be interesting to see the titles of the subchapters – concerning how the authors compare and evaluate the pros and cons of the different transform methods through each chapter of the book. I hope that the comparisons do not come mainly in the end of the book!?

The scope of the proposed book is the standard transform methods (ch. 1-9), which are included in every traditional signal analysis and linear systems book, and some other important transform methods (ch. 10-). Orthogonal transforms are very important, efficient and widely used tools in many signal analysis and filtering areas. It is, however, not easy to fully evaluate the book with help only from the background information from the authors and the excerpt from the book.

I have not found a book which is formed like the proposed book, and I think that it can be very interesting for the intended target group.   The book would be useful both for undergrad and graduate students, as well as for engineers in industry.  

As I mentioned above, it is not clear from the table of contents how the intended philosophy will make this book so different from other similar books on signal analysis and filtering! I find it hard to judge this from only the brief table of contents.   Some comments about the excerpt from the book: When reading the excerpt, I was actually a bit disappointed that it didn’t appeal to me so much. After reading the supplemented description of the book (from the authors), I expected more. The first 13 pages are mostly text and formulas that I don’t think increases the student’s willingness to learn more. Maybe this is just a draft, and the authors will add more clarifying figures and examples later? I also would like more comparisons with other transforms earlier in the text. The interesting part doesn’t show until the second part of the material (pages 14-24).   I am not saying that the text is “bad”, but I would like it to be more interesting to read. The section “Comparison with other orthogonal transforms” was, however, more interesting!

***************************** Review B

This book on orthogonal transforms proposed by the authors covers some well-known transforms in the literature. The book is divided into four parts. The first two parts and more than 80% of part three cover basis orthogonal transforms. These materials are suitable for first and second year undergraduate students. The remaining of part three (mainly ch 4 PCA) and part four are suitable for first year graduate students. One class of transforms not being covered in the proposal is the modified DCTs (MDCTs). They are widely used especially in audio compression and processing applications. Other notable transforms are the Hartley transforms, the lapped orthogonal transform (LOT) and its generalized version (GenLOT) although the authors may argue that these two are special cases of filter banks. In my opinion, this book, if published, would be suitable as a reference and majority readers would be undergraduate students.

It is a good idea to include additional transforms in the discussion.

One unique feature that interests me in this proposal is that the authors try

to unify all the transforms by viewing them as special cases of the Fourier transform. There is an interesting paper published in 2002 for viewing sinusoidal transforms as special cases of the DFT [1]. The authors propose to develop a book along the same line as that by N. Ahmed and K.R. Rao (1975). In fact, there is another book recently published last year by V. Britanak, P. Yip and K.R. Rao on Discrete Cosine and Sine Transforms: General Properties, Fast Algorithm and Integer Approximations. K.R. Rao also has another handbook on signal transforms published a few years ago. Therefore, there are many more competitors than what are listed in the proposals. Another feature the authors might want to emphasize is the combination of continuous and discrete domains. This would add to the uniqueness of the book.

In summary, with the current contents, the book would not be of interest to many graduate students or researchers. It might interest some undergraduate students but there are many other books out there which are more complete in terms of transform coverage and variety. The only selling point of this book I can see, if successfully done, is the unification of all transforms and between the continuous and discrete domains.

It is true that all topics proposed to cover in this book have been discussed else where in various books already in existence. Some of the topics may be discussed in certain books for graduate students and researchers with greater details and more rigors than intended in the proposed book, while some other topics may be discussed in textbooks with a less rigorous manner at undergraduate level. However, as also realized by reviewer B, the point of the proposed book is to discuss all these transform methods from a unified view in a single volume at undergraduate level, instead of presenting any new materials not covered in any previous books. Also, it is not necessarily the case that “there are many other books out there which are more complete in terms of transform coverage and variety”.

References: [1] Oraintara, S.; The unified discrete Fourier-Hartley transforms: Theory and Structure, Circuits and Systems, 2002. ISCAS 2002. IEEE International Symposium on , Volume: 3, 26-29 May 2002 [2] V. Britanak, P. Yip and K.R. Rao, ``Discrete Cosine and Sine Transforms: General Properties, Fast Algorithms and Integer Approximations,” Academic Press, 2007.

Thanks for these very helpful references. Certain contents of these references

could be included in the proposed book to make it more comprehensive.

************************** Review C

I have looked closely at the table of contents and the chapter that was sent to me. Frankly I was not very impressed. First because this area is heavily populated with both undergraduate signals and systems books as well as graduate text books in signal processing. Although it is true that neither cover all the orthogonal transforms as the authors propose, I do not see this difference sufficiently relevant in terms of marketing. The reason is that electrical engineering students focus on Fourier and Laplace and computer science students tend to focus on the others (principal components and wavelets). Through the example chapter that the authors’ attached I see that the level requires knowledge of linear algebra. This leaves the junior students behind, i.e. unable to fully utilize the book. So it is unclear which the target audience is.

For the graduate students, I was a bit worried with the lack of rigor that is presented in the Kahunen-Loeve Transform (KLT) chapter!  The KLT was originally formulated for continuous time signals (which are functionals) on an Hilbert space, however the authors start with a vector space formulation, which is OK for windowed discrete signals, but does not fully utilize the concept of KLT. The way it is presented, it is simply principal component analysis (PCA) in the signal space, so the reference to KLT is disturbing. In my opinion adds to the confusion that already exists between the two transforms. The authors are so much more careful with the Fourier of continuous and discrete signals, and they should be as careful with KLT and PCA.

In conclusion, I feel that the information provided does not provide me with a clear view of the audience, and that the material is not as rigorous as it should be!

Again the issue here is how the various topics are conventionally/traditionally presented vs. how they are to be presented in the proposed book. Traditionally, EE students learn certain materials such as Fourier, Laplace and Z transforms, while the CS students learn some others such as PCA. However, one of the motivations of the proposed book is to deviate from this convention so that the EE students will also learn about PCA, and CS students learn Fourier transform, simply because there are the

needs for them to be familiar with both aspects of the transform methods in general. This point seems to be better appreciated by reviewer D below.

Also, various transform methods have been taught at different levels. Some at introductory level for undergraduate students, some with more rigor for graduate students. But, again, the difference of this proposed book is to put all such similar transform methods in the same volume with a unified treatment and at the introductory level. It is certainly not realistic to expect the rigor for graduate students on the one hand while requiring little math (such as linear algebra) to accommodate low-level undergraduate students on the other hand.

To clarify, the mathematics expected of the reader is basic linear algebra, and probability theory as well as calculus. (The basic results in linear algebra and probability needed in the text of the book will be provided in the Appendix for the reader to review.) Most engineering students should have learned these basic math courses by their junior year, so the book should be accessible for both junior and senior students.

******************************** Review D

The philosophy of the book is sound in several respects:There are not many books that contain a unifying and comprehensive treatment of the transforms used in various technical fields.  As the authors indicate, such transforms are treated on an individual basis across a broad array of disparate books, and there is a strong need to bring this information together in a cohesive whole.  Another positive feature of such unification is the fact that these transforms are all tied together through representation and harmonic analysis theory.  From what was provided, it seems that the authors were to begin the book with the concepts of function spaces, linear independence, basis functions, and the like.  It would have been good if the sample chapter had come from this part of the book, since it is by far the most important.  There are many books out there, especially about wavelets, for example, which do not provide this much needed framework description that goes a long ways towards the understanding of this more sophisticated basis.  It is good that the authors want to provide a pedagogical treatment of the subject, and this would be especially the case for the representation framework just mentioned. Another advantage of this unification philosophy is that the transforms can

be compared and contrasted side by side, thereby enhancing the understanding and application of all of them.

The comments above are greatly appreciated; especially the point that all transform methods should be derived from the fundamental ideas of function space, linear independence, basis functions, etc. In the sample chapter, these ideas are discussed in the first section (which should have been given a subtitle), and then later in the section entitled “Geometric Interpretation of KLT”. This theme should be more clear and obvious once chapter 0 is fully developed and all subsequent transform methods are viewed as different ways of representing the points in the function or vector space.

The other important aspect of the proposed treatment of the subject is its intention on providing real-world application examples for the various transforms, as well as imparting to the reader a working knowledge of the transforms so that when a new problem arises, there will be insight and confidence to proceed in its solution without the use of packaged computer-aided tools.  Again there are many examples in the wavelet arena wherein there is much theory but no interpretation or application, and I strongly agree with the author’s sentiments concerning the blind and uninsightful applications of general purpose commercial tools instead of writing a simple program that can effectively solve the problem.  The automatic reliance on these packaged tools is likewise becoming a problem in engineering, for example, since now engineering working knowledge is being encapsulated in the tool instead of the practitioner using the tool.  Thus when a new problem arises, there is an immediate urge to apply a sledgehammer instead of the ordinary hammer that would likely do.

The basic organization of the book is good, again with the most important subject of representation and harmonic analysis theory being its first chapter.  The scope of the book is appropriate as already mentioned above, while the outline is somewhat complete, with some omissions that need to be considered for inclusion because of their importance in engineering applications.  Specifically, it seems that both the Hilbert and the Radon transforms should be included in Part Three of the book, with the first widely used in electrical engineering and the second primarily used for image analysis.  In addition, for Part Four, a treatment should be considered for the ambiguity function that has similarities to the wavelet transform, and is heavily used for radar and sonar analysis.  It must be emphasized here that the inclusion of these other transforms would make

the book substantially stronger in impact and usefulness to its engineering-related readership.  

These suggested topics are greatly appreciated. I totally agree that adding these transform methods will enhance the book significantly.

In addition, it must be said that the use of the word “orthogonal” in this book can be somewhat of a misnomer since, for example, you can have a wavelet basis and accompanying transform that are not orthogonal, yet faithfully represent the function space, and can very well be useful for applications.  What is more appropriate is that the basis functions span a space of interest and can thus be used for decomposition, synthesis, and analysis purposes (a topic that definitely must be included in Chapter 0). Thus a change in title of the book and some of its parts maybe called for here (e.g., replacing the word “orthogonal” with “mathematical” and using an alternative book title such as “Introduction to the Theory, Algorithms, and Applications of Mathematical Transforms”).  

Again this is definitely a profound point!

Finally, it would be recommended that the book include a brief history of the origin and theoretical development of each transform.  It is this reviewer’s strong feeling that such inclusions always makes the material more interesting and less impersonal and dry. As indicated above, the proposed book is relatively unique in its philosophy and features.  For example, because of its unified nature, a consistent nomenclature can be used throughout, which of course cannot be found across several books covering different transforms.  This one feature will contribute much to the understanding of the subject, especially by those learning it for the first time.  The inclusion of wavelets with other transforms is also quite unique, since most combination books only have the Fourier transform included with the wavelet treatment.  Another important feature is the inclusion of the algorithmic and software implementation of the transforms that can be applied immediately to problems.  

Yes, a very good suggestion that the history of the development of each transform method be included.

Finally, I agree with the authors that principal component analysis, also known as the Karhunen-Loève transform (KLT), is not well known or treated in standard engineering textbooks, and really for no good reason, since it provides what one could call the “ideal” orthogonal transform that is

instructive for learning about transforms in general.  No other book really covers the breadth of transforms and applications proposed here (and this would be even further the case with the inclusions recommended in the previous section). Given the style and philosophy that the book proposes, it would be useful for all levels of readers in technical fields, ranging from university upper division and graduate students through experienced practicing engineers.  For those beginning their training in technical fields, the book would serve as a good educational introduction to transforms, their foundations, details, and applications.  It does not appear, though, that the book would be necessarily useful for graduate students specifically researching transforms per se, since its level would not be technical enough (unless more advanced treatments, exercises, and applications were provided).  For those that are already experienced practitioners in technical fields, the book would serve as a useful reference, especially since it would expose the user to transforms used in completely different fields, and thus likely lead to new cross-field application of these analysis tools.  I myself would recommend the book to my current colleagues that practice in the microwave engineering area.

It is one of the motivations for the proposed the book that many knowledgeable engineering students do not have much idea about PCA, which is widely used in many different fields, while, on the other hand, many CS students lack the basic understanding of the general transform methods, such as Fourier transform. The hope is to bring such topics conventionally covered in different books for different disciplines together as a reference book (not necessarily as a text book for one specific course) to raise the awareness of the transforms as a general data analysis method.

Other Comments — Draft Chapter The draft chapter reviewed was entitled “Principal Component Analysis and Karhunen-Loeve Transform.” The material was relatively complete, with just about the right detail for the theoretical aspects.  However, the applications portion of the chapter appeared to be essentially only descriptive, with few technical details provided to really appreciate how the KLT was specifically used.  It is recommended that all application descriptions include sufficient technical detail to make them more effective as an educational and reference tool.  If possible, the application portion of the book should provide enough detail so that the reader can become a practitioner of the specific application being presented (as opposed to just gaining an appreciation for the transform’s application range).  For example, it would be especially useful if any algorithms associated with the application be provided in some

usable detail.  Although the organization and style of the chapter was good (the LaTeX formatting was much appreciated by this reviewer), its English needs to be corrected in several locations, and there were several minor notation and technical mistakes, as well as some illustrations that were quite inadequate in their annotation.  As a consequence, a separate electronically marked up version of the sample chapter will be provided to CUP for passing on to the authors to aid in their revision of the chapter.

The comments and suggestions made by this reviewer are greatly appreciated. Indeed, the text of the sample chapter should have been checked more carefully to avoid all the typos.

**********************

Review E The authors propose to write a textbook covering the topic of orthogonal transforms.  The book will treat various transforms, such as Fourier transforms, Karhunen-Loeve transform, wavelet transforms, etc., in a unified manner.  The book is divided into four parts and will include Matlab and C code in the appendices and accompanying CD. I agree with the authors that such a book is needed.  Having all of these transforms under one roof (and described with consistent notation) will facilitate easy comparison between them and yield insight into their relationships.  Such insight is difficult to glean from existing textbooks (at least the ones with which I am aware), which treat the transforms in a more scattered manner. Having said that, I was somewhat disappointed by the book proposal. First, the proposal contains dozens and dozens of typographical errors.   This is unacceptable.  Since such errors could be easily found by a careful reading of the manuscript, or by running a spellchecker, it leads me to believe that the authors hastily put the proposal together.   Such sloppiness does not bode well for the future of the planned project.  If the authors cannot collaborate on such a short proposal, how will they write an entire book? Some examples of the more obvious typos are: - p. 1 of proposal:  "being processing" should be "being processed" - p. 2 of proposal:  "practical problem" should be "practical problems" - p. 3 of proposal:  "to tour" should be "tour" - p. 3 of proposal:  "Bankds" should be "Banks" Similar errors can be found in the example chapter, e.g., "zeor", "menas", "definte", "thse", "secnd", "eiganvectors", "expectated", "smae", "columes", "obtaine", etc.

Indeed the text of the proposal should have been checked more carefully to avoid the typos indicated. However, while checking the text of the example chapter, I could not find any of the typos quoted above (such as "zeor", "menas", "definte", "thse", "secnd", "eiganvectors", "expectated", "smae", "columes", "obtaine"). And I did run a spell check before submission, which should have caught all such spelling errors. I really have no explanation why these typos showed up in the text seen by this reviewer, except I might have sent you the wrong file (of an early version), which would show a terrible sloppiness on my part! But in any case these typos no longer exist in the text.

Secondly, in addition to the typos, the sample chapter is not compelling. The figures are unclear, unlabeled, and without meaningful captions.  In particular, I do not understand the figures on p. 16 and p. 18 at all.  The chapter presents a lot of math, without much geometric or intuitive explanation to connect the concepts.  In some places, the explanations are not particularly insightful; on p. 5, for example, the examples are presented, and the conclusion is then repeated.  

The figures should have been better labeled with captions explaining what they each represent. I should have been more careful while preparing the sample chapter, although I never thought it would be the final version of the chapter. Figures on p. 16 and p. 18 are only there to illustrate the idea, actual curves will be plotted in the actual text of the chapter.

Regarding the examples on p. 5, a set of figures illustrating the different distributions of the points will be added to make the idea more intuitive and clear.

While the figure on p. 14 is a good start, it could be improved.  Similarly, although the discussion on pp. 14-19 is helpful, more elaboration is desired.  For example, the fact that the DCT or FT can be performed on a single image, but the KLT/PCA requires an ensemble of images, should be expounded and explored.

Sure, I agree, this point could have been emphasized more.

I also found the notation to be confusing in places.  On p. 3, putting $\sigma_{ij}$ inside brackets surrounded by double dots is something that I have never seen.  In this case, the meaning can probably be inferred by the average reader, but it still seems somewhat sloppy to me.  

This is just to show that $\sigma_{ij}$ is a general element of the ith row and jth column of the array $\Sigma$. Effort will be made to find a better way to represent that idea.

On p.4, the superscript $^(k)$ leads to unnecessarily complicated notation; it would be far better to use different letters for the different signals, which would allow the superscript to be dropped entirely.  

We will try to come up with a better notation to convey the idea.

On p. 10, the line $S_M(A) \rarrow max$ is poor notation whose meaning is not clear.  

It means that $S_M(A)$ is to be maximized. The idea could be described by words for better clarity.

Just below on the same page, what is the asterisk above the equal sign?  I could not decipher the description.

This is explained right underneath the formula.

As far as the topics go, I would suggest that the scope of the book be broadened to include sections on non-orthogonal transforms, such as Independent Component Analysis or Daubechies wavelets.  Without these topics, the book's goal of conveying insight between different transforms will be limited.  Also, terms such as overcomplete should be covered.

This is the issue of trade-off between depth of the discussion and the readability for undergraduate students. Adding the suggested topics such as ICA would indeed make the discussion more thorough and comprehensive, but it would also make it more challenging to keep undergraduate students interested and motivated. After all, as suggested by its title, this book is an introduction of the transform methods. Ideas such ICA may be beyond the intended scope.

 While it is worthwhile to cover the four types of Fourier transforms, it is not clear to me from the proposal how the authors will tackle the difficult problem of ensuring consistent notation among them, which is one of their stated goals.  My preference would be for a book that focuses on discrete transforms, with continuous transforms covered in additional sections to show the connection.

The notations for four types of Fourier transforms should not be a problem. Other text books also deal with the issue, such as Signals and Systems by A. Oppenheim and A. Willsky. The notation system we plan to use should be even clearer (see http://fourier.eng.hmc.edu/e161/lectures/fourier/node1.html).

In general, the sample chapter was prepared with the idea in mind that this was not the final version. The sample writing was intended to give a reasonable idea of the style of the proposed book. Many of the formulas and figures are by no means in final form. They will definitely be worked on during the formal writing of the chapter.