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ALMA MATER STUDIORUM – UNIVERSITÀ DI BOLOGNA ARCES – ADVANCED RESEARCH CENTER ON ELECTRONIC SYSTEMS Design and Computation of Warped Time-Frequency Transforms Salvatore Caporale SUPERVISOR Professor Guido Masetti COORDINATOR Professor Claudio Fiegna EDITH – EUROPEAN DOCTORATE ON INFORMATION TECHNOLOGY JANUARY 2006 – DECEMBER 2008 XXI CYCLE – ING-INF/01

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“If you try and take a cat apart to see how it works, the ﬁrst thing you have on your hands is a non-working cat.” Douglas Adams

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Preface his work mainly concerns warping techniques for the ma- Tnipulation of signals. Our approach on this topic will be guided by theoretical issues rather than experimental ones. So, we will not dedicate much space to explain what warping is in a practical sense. In order to compensate the excess of theory which will be experienced by the reader in the this work, here we want to introduce some basic concepts behind frequency warping in an easy way. Generically, a signal is described as a measurable quantity which is able to vary through time and over space. Although warping could be applied on any kind of signals, as an example we consider those signals which are intrinsically perceived by human visual observation, i.e. images. As a signal has to be measured, the visual information related to a subject which produces an image can be stored in many ways, determining a diﬀerent kind of measure. In modern electronic sensor devices are employed, in traditional cameras light was stored by a chem- ical reaction and in humans the storage process is devolved upon biological sensors. Referring to humans, themeasurement is not completely carried out by the eyes, since the light information is reported to the brain which makes some further elaborations before memorizing it in synapses. Since signals concern the transport of information, or rather the communication through time and space, before the inven- iii

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iv Preface Figure 1: Escher’s lithograph “Print Gallery” (1956). M. C. Escher “Prentententoonstelling”© 2003CordonArt-Baarn-Holland. All rights reserved. tion of cameras humans have developed alternative methods to store and communicate images beyond the time and place where they were living. Of course we are talking about ﬁgura- tive art. e measurement performed by a man and reported on a painting or any other kind of ﬁgurative representation shows the importance of the way the perceived information is weighted according to speciﬁc patterns which are enclosed in the measurement instrument. Figurative art taught that, since there is not a single way to represent reality, then there is not a single way to observe reality. Works of art are always aﬀected by a kind of signal processing, including simple ﬁltering oper- ations or complicated non-linear eﬀects. Furthermore, we can notice that during the last centuries, ﬁgurative art deliberately abandoned the aim of giving a faithful representation of reality and expressed the willing of going beyond what can be directly experienced by human senses.

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v Figure 2: Warping Grid used by Escher to draw the “Print Gallery”. In this background, we consider the work ofM. C. Escher. In his prints he took advantage of some concepts akin tomathemat- ics, like self-reference, inﬁnite and recursive processes. In par- ticular, recursion is themain concept in his print titled the “Print Gallery”, which is reported in ﬁgure 1. An accurate description of the mathematical structure of this work can be found in [1]. e print have been drawn starting from an unwarped image, representing a man observing a print which illustrates himself watching the same print recursively (this recursion is called Droste eﬀect). On this image, a warping have bee applied according to grid which is shown in ﬁgure 2. e new warped image is created by making the tales of a square grid built on the original image correspond to the tales on the new grid. e performed operation is more than a deformation, since the grid is designed such that it contains a progressive scaling operation which makes the recursive spaces reconnect together. But apart from the scaling, we want to focus on the global eﬀect that the

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vi Preface author’s point of view has given to the content. anks to the grid, the tales of the image have been re-weighted according to a new sampling, so that some details which were not visible and recognizable in the original version have been increased in importance. A very interesting consideration to be done is that trough the warping operation there is no increase in the global information contained in the picture. Instead, the way the space of observation (the square frame) is split among the various part of the image has been modiﬁed. It is quite intuitive that the problem of recovering the orig- inal image, which has been treated in [1], is actually the same problem as drawing the warping image. In fact, one can assume that the image in ﬁgure 1 is the original one, and then draw a new image through a grid which nullify the eﬀect of the grid in ﬁgure 2. rough this example, we have already illustrated some of the basic properties and concepts behind thewarping technique. Possible aims of such an operation can be easily imagine by comparison with the shown example. For instance, one could need to exalt some parts of a signal despite to others in order to perform an accurate feature extraction. is approach can be categorized as a direct application of a warping technique, since the starting point is the unmodiﬁed signal. Otherwise, it could be necessary to remove the eﬀects of an acquisition process which weights non-uniformly the diﬀerent parts of the incoming signal. is approach would be labeled as an inverse use of a warping technique, since the starting point is an already warped signal. As we suggested before, there is an intrinsic duality between the direct and the inverse approach. e possibility of recovering the original signal by the war- ped one, that is the capability of deﬁne an inverse unwarping which exactly inverts the direct one, is a very important issue when dealing with warping technique from a mathematical point of view. Invertibility is the major problem which will be considered in this work. Furthermore we will cope with the way the warping operation should to be designed, which means, by comparisonwith the Escher’s print example, what kind of curves should compose the grid in ﬁgure 2. We ﬁnally report other hints suggested from Escher’s litho- graph. Although these consists in conceptual observations ra- ther than mathematical ones, they reveal to make sense in hindsight. We notice that the center of 1 was le unpainted. We

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vii also learn from [1] that the unwarped picture used by Escher was not complete, since the unpainted spot gives rise to an empty spiral. ese observations can be translated to our perspective in the following metaphorical meaning. When warping a signal from a ﬁnite-dimensional domain to another ﬁnite-dimensional one (i.e. a domain having an upper limited resolution), some information is necessarily discarded. Maybe a perfect recon- struction could be achieved anyway, but it involves something more than merely inverting the steps employed for warping.

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