Eleven
year later (2012) both reviews proved to be totally inept.

Editor |
Reviewer: 1 | Reviewer: 2 |

Dear Mr Larkin, Your manuscript, "Natural demodulation of two-dimensional patterns", has now been seen by two referees, whose reports I enclose. I am sorry to say that both referees raise what appear to be serious criticisms, in view of which we cannot offer to publish your paper in [name of journal removed]. Moreover, we have to doubt that a revised version would be likely to meet with a more favourable response; rather, we feel that the work would ultimately find a more suitable outlet in a specialist journal. I am sorry that we cannot respond more positively, but I hope that you will find the referees' comments helpful. Yours sincerely, Editor |
The paper is definitely not suitable for publication in [name of journal removed]. The work does fall short of the mark in spite of the authors' attempts to repeatedly beat their own drums coupled with the annoyance caused by the often unjustified belittling of past contributions. The manuscript pages are unnumbered and what I found particularly distasteful was the remark that the work was "quite literally a revolution" (top of page 4). In fact the authors quote several classic papers by mathematicians published more than six or seven decades back and I wonder if they read carefully the items cited e.g. ref. 20, a 83-page paper in Russian? Then there are papers like ref 12 which are commented upon but which we are unable to access. The authors have defined the 2-D Hilbert transform in a certain way and then justified their choice of the definition by particular simulations. There is hardly any analysis. If they had chosen to convey these findings in some engineering journal that might cater to utilitarian demands, I could have been more sympathetic. As it stands, it lacks in scholarship, hard analysis and a lack of focus (too many citations, too many unsubstantiated comments). REJECT
the paper. |
I read this paper with great interest. I have worked on 1D and 2D Hilbert transforms and was aware that there were few problems, either conceptual or mathematical, in defining the HT in higher dimensions. The logarithmic HT is a different matter, of course. Several of the papers cited by the authors discuss 2D .HTs which makes the opening remarks in the paper somewhat off-putting and In need of more careful wording. Even in the abstract, I would dispute the statement that "It is generally believed that… does not extend naturally beyond one dimension". On the second page of the manuscript, I strongly disagree that it is "Unknown to most researchers in signal processing ........ " (There are too many references to cite but see for example various chapters in Image Recovery: Theory and Applications, Ed. H. Stark, Academic Press, Florida, 1987, in particular chapter 13). |

The authors argue that it was believed that the HT did not extend naturally beyond 1D and that they have found an elegant isotropic extension. As a simple function of more than one complex variable, there is little problem in defining a multidimensional causality condition, ensuring that the analytic function IS entire and of exponential type and then invoking Reimann's (sic) lemma to allow the Cauchy integral formula to be applied with a contour including the real plane and the upper (or lower) half space. Taking real and imaginary parts of the resulting integral generates the HT pair. Do the authors agree? |
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The authors seems (sic) to lock on to the apparent directionality associated with a HT as a problem. Presumably this results from the choice of variables being Cartesian rather than some other preferred system and is not a fundamental problem. It is true that if generating a HT by first taking a FT and then multiplying by a generalized i_signum function, an arbitrary directionality is imposed. Could the authors address this point? It is assumed by the authors from the beginning that there is a problem and that there exists a need for an isotropic 2D HT but at no point is this explained or at all obvious to this reviewer. Please expand. |
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The phase function proposed to play the role of the 2D signum function is the phase that occurs in a wavefront at a field zero, i.e. it defines a wavefront dislocation. It still possess an arbitrary directionality depending on the angle at which the branch cut in the phase is directed. Is this correct? As a quadrature function it appears to suit the need but I would like to see the discussion of its Fourier transform. Why do the authors not make their paper more complete and include the 2D Fourier domain analysis of what they are doing in this manuscript rather than leave it for a forthcoming paper (?). I think it would assist with the clarity and impact of the present paper if they did so. ( I would note an old paper by Kuo and Freeney from the '60s on HTs in which the HT property that only the modulating frequency component and not the signal envelope function is transformed if the carrier frequency is sufficiently high. A "natural" quadrature function as the authors discuss it may not be what is required in most cases when the HT of a function is to be calculated, consider the 2D top-hat function for example.) |
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Is there an implication in the paper that the spiral phase kernel function now has to be applied locally depending on the local orientation of the fringes in an interference pattern? This needs to be clarified in the text and generalized to the wider context in which fringes per se may not be as apparent. Your approach requires that an orientation estimator be employed and gives the impression that a spatially variant solution is being proposed. Is this correct? It is also well known that most 2D band limited functions are littered with point zeros in which case phase unwrapping which seems to be necessary for finding the orientation estimator, becomes impossible. Do the authors agree? In summary, I believe that the authors have found an interesting approach to generating a quadrature function which, in the examples given, appears to behave better for some classes of images when implemented numerically. A concern that I have is that the paper gives the impression that a fundamental mathematical solution is being proposed, while it seems that it might be more of an approximation with some numerical advantages for certain classes of image. Do the authors agree? |

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