A New Method for Signal and Image Analysis: The Square Wave Method
AA New Method for Signal and Image Analysis:The Square Wave Method
Osvaldo Skliar ∗ Ricardo E. Monge † Sherry Gapper ‡ August 13, 2018
Abstract
A brief review is provided of the use of the Square Wave Method(SWM) in the field of signal and image analysis and it is specified howresults thus obtained are expressed using the Square Wave Transform(SWT), in the frequency domain. To illustrate the new approach intro-duced in this field, the results of two cases are analyzed: a) a sequenceof samples (that is, measured values) of an electromyographic recording;and b) the classic image of Lenna.
Mathematics Subject Classification: 94A12, 65F99
Keywords: signal and image analysis, Square Wave Method (SWM), SquareWave Transform (SWT).
It was previously shown how a new method, the Square Wave Method (SWM),for the analysis of signals depending on one variable [1] can be presented in thefrequency domain by using a mathematical tool called Square Wave Transform(SWT) [2] [3]. The SWM was then generalized quite naturally and directly forimage analysis [4].The objectives of this paper are the following:1. To provide a brief review of the use of the SWM for the analysis of signalsand specify the relations existing between a) the sampling frequency f s ,with which the successive values of recordings of biomedical signals (suchas those of an electrocardiogram, electromyogram or electroencephalo-gram) are measured, and b) the frequencies f , f , . . . , f n , correspondingrespectively to the different trains of square waves S , S , . . . , S n obtainedusing the SWM; ∗ [email protected]; Escuela de Inform´atica, Universidad Nacional, Costa Rica. † [email protected]; Escuela de Ciencias de la Computaci´on e Inform´atica, Universidadde Costa Rica, Costa Rica. ‡ [email protected]; Universidad Nacional, Costa Rica. a r X i v : . [ c s . NA ] J a n . To indicate how it also is possible to present in the frequency domain usingthe SWT, the results of the analysis of images obtained with the SWM.The application of the SWM in the field of signal and image analysis isexemplified with the results of an analysis of a) a sequence of samples (that is,measured values from an electromyographic recording); and b) the classic imageof Lenna, using the SWT [5]. Consider a function of time ( t ), in the interval ∆ t , satisfying the conditions ofDirichlet [6]: f ( t ) = (6 − t )(2 cos(2 π t ) + 5 cos(2 π t )) 0 ≤ t ≤ t = 4 s) has been divided into 18 equal sub-intervals.In this case, it will be seen that function (1) can be approximated in ∆ t , usingthe sum of the parts corresponding to ∆ t of 18 trains of square waves. Thesetrains of square waves will be called S , S , S , . . . , S ; the “ S ” being based onthe word “square” in the expression “train of square waves”.If ∆ t has been divided into 100 equal sub-intervals, the approximation to thefunction (1) in interval ∆ t will be carried out by adding the parts correspondingto ∆ t of 100 trains of square waves: S , S , S , . . . , S . In general, if ∆ t isdivided into any natural number n of equal sub-intervals, the approximationin ∆ t to function (1) will be obtained by adding the parts corresponding to∆ t of n trains of square waves: S , S , S , . . . , S n . The Square Wave Method(SWM) described in this section makes it possible to determine those trainsof square waves unambiguously. Therefore, each S i (where i = 1 , , . . . , n ) ofthose trains of square waves will be characterized by a specific frequency f i (i.e., consideration is given to the number of waves in the train of square waves,which is contained in the unit of time 1 s) and a particular coefficient C i , whoseabsolute value is the amplitude of the corresponding train.The function f ( t ) specified in (1) is shown in figure 1.2igure 1: f ( t ) = (6 − t )(2 cos(2 π t ) + 5 cos(2 π t )) 0 ≤ t ≤ n = 18, a description will be provided below ofhow the frequencies f i (where i = 1 , , . . . ,
18) and the values of the coefficients C i (where i = 1 , , . . . ,
18) corresponding to the different trains of square waves S i (where i = 1 , , . . . ,
18) are determined; see figure 2.3igure 2: How to apply the SWM to the analysis of the function represented infigure 1. (See indications in text.)The first row of figure 2 (with coefficients C ) represents half a square wave,the first semi-wave of the train of square waves S . The frequency of S (i.e., f ) is clearly equal to the number of square waves per unit of time (1 s). Toobtain f , the part of S which occupies ∆ t (the half-wave) is divided by ∆ t . f = ∆ t = − To compute f , note that ∆ t is occupied by the sum of that half-wave of thetrain of square waves S and the fraction of the second semi-wave of the firstsquare wave of S . That fraction is represented by the symbol C in the secondrow of figure 2. Thus the following value is obtained for f : f = + (cid:0) · (cid:1) ∆ t = (cid:0) (cid:1) ∆ t = 12 · ∆ t = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − To compute f , note that ∆ t is occupied by the sum of that half-wave of thetrain of square waves S and the fraction of the second semi-wave of the firstsquare wave of S . This fraction (in the third row of figure 2) is represented by4he sequence of symbols − C − C . Therefore, the following value is obtainedfor f : f = + (cid:0) · (cid:1) ∆ t = (cid:0) (cid:1) ∆ t = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − With a precision of 7 decimal places, the values are given below not only for f , f , f , but also for those corresponding to f , f , . . . , f , f . f = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − = 0 . − f = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − = 0 . − f = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − = 0 . − f = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − = 0 . − f = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − = 0 . − f = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − = 0 . − f = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − = 0 . − f = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − = 0 . − f = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − = 0 . − f = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − = 0 . − f = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − = 0 . − f = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − = 0 . − f = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − = 0 . − f = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − = 0 . − = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − = 0 . − f = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − = 0 . − f = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − = 1 . − f = 12∆ t (cid:18) − (cid:19) = 18 (cid:18) − (cid:19) s − = 2 . − Observe that any of the 18 values of f i , where i = 1 , , . . . ,
18, can be com-puted with the following equation: f i = 12∆ t (cid:18) − ( i − (cid:19) = 18 (cid:18) − ( i − (cid:19) s − ; i = 1 , , . . . , t , whose value, of course, may be different from4 s, is divided into n equal sub-intervals, the frequencies corresponding to eachof the n trains of square waves are as follows: f i = 12∆ t (cid:18) nn − ( i − (cid:19) s − ; i = 1 , , . . . , n (2)It has been explained how to compute each f i corresponding to each S i ,where i = 1 , , . . . ,
18, for the case of the approximation to f ( t ) specified in (1)when dividing ∆ t into 18 equal sub-intervals ( n = 18). Indications will now begiven on how to compute the C i for each S i .The vertical arrow pointing down at the right of figure 2 indicates how toadd the terms corresponding to each of the 18 sub-intervals of ∆ t . Thus, toobtain the values of the coefficients C , C , . . . , C and C , corresponding to S , S , . . . , S and S , the following system of linear equations must be solved.6 + C + C + C + C + C + C + C + C + C + C + C + C + C + C + C + C + C = V C + C + C + C + C + C + C + C + C + C + C + C + C + C + C + C + C − C = V C + C + C + C + C + C + C + C + C + C + C + C + C + C + C + C − C − C = V C + C + C + C + C + C + C + C + C + C + C + C + C + C + C − C − C − C = V C + C + C + C + C + C + C + C + C + C + C + C + C + C − C − C − C − C = V C + C + C + C + C + C + C + C + C + C + C + C + C − C − C − C − C − C = V C + C + C + C + C + C + C + C + C + C + C + C − C − C − C − C − C − C = V C + C + C + C + C + C + C + C + C + C + C − C − C − C − C − C − C − C = V C + C + C + C + C + C + C + C + C + C − C − C − C − C − C − C − C − C = V C + C + C + C + C + C + C + C + C − C − C − C − C − C − C − C − C − C = V C + C + C + C + C + C + C + C − C − C − C − C − C − C − C − C + C + C = V C + C + C + C + C + C + C − C − C − C − C − C − C − C + C + C + C + C = V C + C + C + C + C + C − C − C − C − C − C − C + C + C + C + C + C + C = V C + C + C + C + C − C − C − C − C − C + C + C + C + C + C − C − C − C = V C + C + C + C − C − C − C − C + C + C + C + C − C − C − C − C + C + C = V C + C + C − C − C − C + C + C + C − C − C − C + C + C + C − C − C − C = V C + C − C − C + C + C − C − C + C + C − C − C + C + C − C − C + C + C = V C − C + C − C + C − C + C − C + C − C + C − C + C − C + C − C + C − C = V (3)7n the preceding system of linear algebraic equations (3), V , V , . . . , V and V are the values for f ( t ) as specified in (1) at the midpoints of the first, second,third, . . . , seventeenth and eighteenth sub-intervals, respectively, of interval ∆ t ,in which f ( t ) is analyzed. It follows that the values V i (where i = 1 , , , . . . , f ( t ) has been specified in (1). These valuesare as follows: V = − . V = − . V = 30 . V = 22 . V = − . V = − . V = − . V = − . V = 49 . V = 35 . V = − . V = − . V = − . V = − . V = 25 . V = 17 . V = − . V = − . V i , where i = 1 , , . . . ,
18, has been computed witha precision of seven decimal digits.The 18 unknowns of the systems of equations specified in (3) are C , C , . . . , C , and C . Thus | C i | refers to the amplitude of the train of square waves S i ,where i = 1 , , . . . ,
18. The (constant) value of each positive square semi-waveof the train of square waves S i is | C i | and the (constant) value of each negativesquare semi-wave of that S i is −| C i | .The system of equations (3) has been solved by using LAPACK [7], and thefollowing results were obtained for the unknowns: C = 117 . C = 4 . C = 50 . C = − . C = − . C = 12 . C = − . C = 8 . C = 9 . C = 60 . C = 12 . C = − . C = 61 . C = 28 . C = 49 . C = − . C = 12 . C = − . S , S , S , . . . , S and S have been shown forinterval ∆ t in figures 33.1, 33.2, 33.3, . . . , 33.18, respectively.8 S ( t )(3.2) : S ( t ) Figure 39 S ( t )(3.4) : S ( t ) Figure 310 S ( t )(3.6) : S ( t ) Figure 311 S ( t )(3.8) : S ( t ) Figure 312 S ( t )(3.10) : S ( t ) Figure 313 S ( t )(3.12) : S ( t ) Figure 314 S ( t )(3.14) : S ( t ) Figure 315 S ( x )(3.16) : S ( t ) Figure 316 S ( t )(3.18) : S ( t ) Figure 3: Trains of square waves S , S , . . . , S and S f ( t ) (as specified in (1), in interval ∆ t , byadding the 18 trains of square waves) is displayed in figure 4.Figure 4: The dashed line indicates the approximation to f ( t ), specified in (1),by X i =1 S i ( t ).If one requires a better approximation to f ( t ), by adding the trains of squarewaves, then ∆ t should be divided into a larger number of equal sub-intervals.The higher the number of sub-intervals, the better the approximation.Suppose that interval ∆ t is divided into n sub-intervals of equal duration. Inequation (2) it was specified how to compute each f i (where i = 1 , , , . . . , n )for each train of square waves S , S , S , . . . , S n , which must be added in ∆ t to obtain, in that interval, the corresponding approximation to f ( t ) specifiedin (1).To obtain the coefficients C , C , C , . . . , C n corresponding respectively tothose square waves, a system of linear algebraic equations must be solved. Thissystem can be obtained by using the same type of approach as that used toobtain the system of equations specified in (3).Approximations to f ( t ) specified in equation (1) when dividing ∆ t into 100and into 1000 intervals respectively, are displayed in (5.1) and (5.2) in figure 5.The SWM cannot be considered a branch of Fourier analysis; the trains ofsquare waves S i , where i = 1 , , , . . . , n , do not make up a system of orthogonalfunctions. 18 f ( t ), specified in (1), by X i =1 S i ( t ).(5.2) : The dashed line indicates the approximation to f ( t ), specified in (1) by X i =1 S i ( t ). Figure 519he results obtained upon carrying out the type of analysis described ofa function characterized in an interval ∆ t divided into n sub-intervals with anequal duration can be presented in a sequence of dyads (ordered pairs) such thatthe first element of the first dyad is the frequency f corresponding to S andthe second element of the first dyad is the coefficient C ; the first element of thesecond dyad is the frequency f corresponding to S and the second elementof that dyad is the coefficient C ; and so on successively, such that the firstelement of the n th dyad is the frequency f n corresponding to S n and the secondelement of that n th dyad is the coefficient C n .Consider, for example, the sequence of 18 dyads obtained when carrying outthe type of analysis described of the f ( t ) specified in (1), if the interval ∆ t isdivided into 18 sub-intervals:( f ; C ) = (0 . . f ; C ) = (0 . − . f ; C ) = (0 . . f ; C ) = (0 . . f ; C ) = (0 . . f ; C ) = (0 . − . f ; C ) = (0 . . f ; C ) = (0 . − . f ; C ) = (1 . − . f ; C ) = , (0 . . f ; C ) = (0 . − . f ; C ) = (0 . . f ; C ) = (0 . . f ; C ) = (0 . . f ; C ) = (0 . . f ; C ) = (0 . . f ; C ) = (0 . . f ; C ) = (2 . − . f ( t ) specified in (1) can be expressed inthe frequency domain. To achieve this objective, for each of the frequenciesconsidered f , f , f , . . . , f , the corresponding coefficients C , C , C , . . . , C must be indicated.The expression in the frequency domain of this approximation to f ( t ) willbe called the Square Wave Transform (SWT) of that approximation to f ( t ).Note that previously (in figures 4 and 5, for example), each approximation to f ( t ) was represented in the time domain. This SWT is displayed in figure 6.20igure 6: SWT of the approximation to f ( t ) obtained by dividing ∆ t into 18sub-intervals.Of course, the SWTs corresponding to numbers as large as desired of equalsub-intervals into which ∆ t is divided can be obtained for the f ( t ) specifiedin (1), or for any other function of the time which, in a particular interval ∆ t ,satisfies the conditions of Dirichlet.In (7.1), (7.2), and (7.3) of figure 7, the SWTs obtained for the approxima-tions to the f ( t ) specified in (1) are shown for n = 100, n = 1000, and n = 2000,respectively. 21 f ( t ), specified in equation (1), for n = 100.(7.2) : SWT of the approximation to the f ( t ), specified in equation (1), for n = 1000. Figure 722 f ( t ), specified in equation (1), for n = 2000. Figure 7In (8.1), (8.2), (8.3), and (8.4) of figure 8, partial representations can beseen (up to frequency f i = 2) of the SWTs of the different approximations tothe f ( t ) specified in (1). 23 f ( t ) specified in (1) for n = 1000(8.2) : Partial representation of the SWT of the approximation to the f ( t ) specified in (1) for n = 2000 Figure 824 f ( t ) specified in (1) for n = 4000(8.4) : Partial representation of the SWT of the approximation to the f ( t ) specified in (1) for n = 8000 Figure 825n figure 8, it can be observed that in all 4 cases considered the “promi-nent coefficients” correspond to certain frequencies. (They are prominent in thesense that their moduli are quite larger than the moduli of the coefficients cor-responding to frequencies near to those considered.) They have been indicatedby the letters A, B, C, and D. The dyads corresponding to the coefficients forthese cases are the following: n = 1000 A : (0 . . B : (0 . . C : (0 . . D : (1 . . n = 2000 A : (0 . . B : (0 . . C : (0 . . D : (1 . . n = 4000 A : (0 . . B : (0 . . C : (0 . . D : (1 . . n = 8000 A : (0 . . B : (0 . . C : (0 . . D : (1 . . n from 1000 to 2000, from 2000 to 4000,and from 4000 to 8000. 26 Analysis of Sequences of Samples (MeasuredValues) from an Electromyographic Record-ing
The SWM and the corresponding expression of the results obtained with theSWT can be used for the analysis of sequences of “samples”, or measured values,from different types of recordings. In particular, they can be used for the anal-ysis of recordings which are important in medicine, as are those of the electro-cardiogram (ECG), the electroencephalogram (EEG), and the electromyogram(EMG). Results obtained by using the SWM for the analysis of a sequence ofsamples from an electroencephalographic recording were described in [2].Suppose that, to take samples from a recording, a sampling frequency f s of250 Hz is used. In this case let it be admitted that every one-second lapse (1 s)has been divided into 250 sub-intervals of equal duration and that the values ofthe samples “correspond” to the values (mentioned in section 2) of the functionanalyzed, at the midpoints of those different sub-intervals. In general, for anynumeric value of the f s used, the number of sub-intervals into which the unit oftime 1 s has been divided is equal to that numeric value.Thus if with a specific f s , such as one of 250 Hz, a sequence of samples istaken for a certain ∆ t , equal to 5 s, for example, to compute the total number n of samples taken during that ∆ t , the f s must be multiplied by that ∆ t : n = f s · ∆ t (4)For the values of f s and ∆ t specified in the above paragraph, the followingequality is found: 1250 = 250 s − · f ( t ) characterized analyticallyin an interval ∆ = 5 s. If n (the number of sub-intervals into which ∆ t is divided)is equal to 1250, for example, the sequence of values V , V , V , . . . , V will beobtained for the midpoints of the specific sub-intervals as follows: V will bethe value of f ( t ) at the midpoint of the first sub-interval, V will be the valueof f ( t ) at the midpoint of the second sub-interval, and so on successively, with V the value of f ( t ) for the last of the sub-intervals considered.When operating with f s = 250 Hz, and taking samples for ∆ t = 5 s, 1250samples will be obtained. To apply the SWM to this sequence of samples, itis considered that the first “corresponds” to V , the second to V , and so on,successively such that the last sample “corresponds” to V . In other words,the sequence of samples is treated the same as the sequence V , V , V , . . . , V ,when applying the SWM to an analytically characterized function.Equation (2), which makes it possible to compute each f i corresponding toeach S i (where i = 1 , , , . . . , n ), remains valid for sequences of samples of arecording: f i = 12∆ t (cid:18) nn − ( i − (cid:19) s − ; i = 1 , , . . . , n (2)27f (4) is substituted in (2), the following equation 5 is obtained: f i = 12 (cid:18) f s f s ∆ t − ( i − (cid:19) ; i = 1 , , , . . . , n (5)Note the equations obtained if in (5), a) i = 1; and b) i = n .For a), ( i = 1) −→ f = t . In this first case, f does not depend on f s ,but rather only on ∆ t .For b), ( i = n ) −→ f n = f s . In this second case, f n does not depend on ∆ t ,but rather only on f s .The graph corresponding to an electromyographic recording made during∆ t = 5 s with f s = 250 Hz is presented in figure 9 [8].Figure 9: Electromyographic recording made during ∆ t = 5 s with f s = 250 Hz.Extending only to f i = 2 Hz, the graph in figure 10 is a section of the SWTcorresponding to the recording represented in figure 9.28igure 10: Partial plot of the SWT corresponding to the electromyographicrecording in figure 9. The moduli of the coefficients not represented (the C forwhich f i >
2) are much smaller than the moduli of C which were represented.The sequence of dyads for values of f i < .
12 is as follows:( f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , . f , C ) = (0 . , − . f , C ) = (0 . , . n ≤ The SWM is also useful for the analysis of images and the corresponding resultscan be expressed with the SWM.First, a “pseudo-image” (shown in figure 11) has been chosen to illustratehow the SWM may be used for analysis purposes. The method applied for theanalysis of a pseudo-image is the same as that used for the analysis of genuineimages. The term “pseudo-image” indicates that no reference is made to agenuine image of something real, such as an object or a living being, or abstractart with aesthetic value. Given that the method used, the SWM, for the analysisof a genuine image and of a pseudo-image is the same, the latter will serve toprovide a clear and simple example of how that method can be applied.32 y P , P , P , P , P , P , P , P , P , P , P , P , P , P , P , P ,
811 2 3 41234Figure 11: Representation of a pseudo-image. The name of each pixel is givenat the top, and the level of gray corresponding to that pixel is given at thebottom.The name of each pixel in the pseudo-image displayed in figure 11 is givenat the top. The numerical value (ranging from 0 for black to 255 for white)corresponding to the level of gray of the pixel is given at the bottom. (Thelevels of gray were chosen arbitrarily.)Although figure 11 can be considered as a representation of a matrix of pixels,the diverse elements in that matrix ( P i,j , where i = 1 , , ,
4; and j = 1 , , , x -axis is that of the abscissas, and the y -axis is that of the ordinates),the point ( i, j ) (where i = 1 , , ,
4; and j = 1 , , ,
4) whose abscissa is i andwhose ordinate is j , is the center point of the pixel P i,j .In figure 12, it has been shown how each of the axes ( x and y ) in figure 11has been processed with the same approach as that of figure 2 with the x -axis(the only axis of coordinates used in this case).33 y P , P , P , P , P , P , P , P , P , P , P , P , P , P , P , P , C C C C C C C − C C C − C − C C − C C − C C C C C C C C − C C C − C − C C − C C − C ∆ x ∆ y Figure 12: How to apply the SWM to the analysis of the pseudo-image repre-sented in figure 11. (See indications in text.) The superscript 1 located on theleft of the coefficient indicates that the coefficient corresponds to the x -axis andthe superscript 2 on the left of the coefficient indicates that it corresponds tothe y -axis.Note, for example, how the equation corresponding to pixel P , is formed:that pixel belongs to both column 4 and row 3 in figure 12:1. Take the sequence of these two sets: a) the set of elements underneathcolumn 4 in figure 12; and b) the set of elements at the left of row 3 inthe same figure: (cid:0) C , − C , − C , − C (cid:1) (cid:0) C , C , − C , C (cid:1) Note that the above expression is an ordered pair of sets.2. The Cartesian product is found for the two sets corresponding to thatordered pair of sets: (cid:0) C , − C , − C , − C (cid:1) × (cid:0) C , C , − C , C (cid:1) = P ∗ , = { ( C , C ) , ( C , C ) , ( C , − C ) , ( C , C ) , ( − C , C ) , ( − C , C ) , ( − C , − C ) , ( − C , C ) , − C , C ) , ( − C , C ) , ( − C , − C ) , ( − C , C ) , ( − C , C ) , ( − C , C ) , ( − C , − C ) , ( − C , C ) } The preceding Cartesian product corresponding to pixel P , has beendesignated as P ∗ , .3. The set P ∗ , is acted on by an operator O such that:(a) it converts each element P ∗ , (a certain ordered pair) into a singleelement which will be a coefficient with two subscripts: the first isthe same as that of the first element of the ordered pair, and thesecond subscript is the same as that of the second element of theordered pair; and(b) if the signs preceding the two elements of the ordered pair have thesame sign (+ and +; or − and − ), the coefficient is positive; if they aredifferent (+ and − ; or − and +), the coefficient obtained is negative.Therefore, for the case considered, the following result is obtained:O( P ∗ , ) =( C , , C , , − C , , C , , − C , , − C , , C , , − C , , − C , , − C , , C , , − C , , − C , , − C , , C , , − C , )The right-hand member of the above equation is a set of coefficients, eachof which has two subscripts and is preceded by the + sign or by the − sign. That set will be called C ∗ , . Of course, C ∗ , must not be confusedwith the coefficient C , .4. The algebraic sum is figured for all the elements in C ∗ , , and the result isequated to the numeric value of the level of gray corresponding to pixel P , (i.e., 255).Therefore, the linear algebraic equation corresponding to pixel P , is ob-tained: C , + C , − C , + C , − C , − C , + C , − C , − C , − C , + C , − C , − C , , − C , + C , − C , = 255If the same procedure is applied to the set of elements under column 3 in figure12 and to the set of elements appearing at the left of row 1 in that same figure,the following linear algebraic equation may be specified for pixel P , : C , + C , + C , + C , + C , + C , + C , + C , − C , − C , − C , − C , + C , + C , + C , + C , = 2535f the same procedure is carried out for each of the remaining pixels of thepseudo-image in figure 11, the following system of linear algebraic equations isobtained: C , + C , + C , + C , + C , + C , + C , + C , + C , + C , + C , + C , + C , + C , + C , + C , = 100 C , + C , + C , − C , + C , + C , + C , − C , + C , + C , + C , − C , + C , + C , + C , − C , = 98 C , + C , − C , + C , + C , + C , − C , + C , + C , + C , − C , + C , + C , + C , − C , + C , = 195 C , − C , − C , − C , + C , − C , − C , − C , + C , − C , − C , − C , + C , − C , − C , − C , = 55 C , + C , + C , + C , + C , + C , + C , + C , + C , + C , + C , + C , − C , − C , − C , − C , = 38 C , + C , + C , − C , + C , + C , + C , − C , + C , + C , + C , − C , − C , − C , − C , + C , = 3 C , + C , − C , + C , + C , + C , − C , + C , + C , + C , − C , + C , − C , − C , + C , − C , = 6 C , − C , − C , − C , + C , − C , − C , − C , + C , − C , − C , − C , − C , + C , + C , + C , = 4 C , + C , + C , + C , + C , + C , + C , + C , − C , − C , − C , − C , + C , + C , + C , + C , = 25 C , + C , + C , − C , + C , + C , + C , − C , − C , − C , − C , + C , + C , + C , + C , − C , = 77 C , + C , − C , + C , + C , + C , − C , + C , − C , − C , + C , − C , + C , + C , − C , + C , = 249 C , − C , − C , − C , + C , − C , − C , − C , − C , + C , + C , + C , + C , − C , − C , − C , = 69 C , + C , + C , + C , − C , − C , − C , − C , − C , − C , − C , − C , − C , − C , − C , − C , = 55 C , + C , + C , − C , − C , − C , − C , + C , − C , − C , − C , + C , − C , − C , − C , + C , = 12 C , + C , − C , + C , − C , − C , + C , − C , − C , − C , + C , − C , − C , − C , + C , − C , = 255 C , − C , − C , − C , − C , + C , + C , + C , − C , + C , + C , + C , − C , + C , + C , + C , = 81 (6)The 16 unknowns of the preceding system of linear algebraic equations are36he 16 coefficients C i,j , where i = 1 , , ,
4; and j = 1 , , , C , = 112 . C , = − . C , = − . C , = − . C , = 15 . C , = 32 . C , = 28 . C , = − . C , = 27 . C , = 51 . C , = 22 . C , = − . C , = − . C , = 13 . C , = 42 . C , = − . P i,j (where i = 1 , , , . . . , n ; and j = 1 , , , . . . , n ) of an image composed of n rows and n columns of pixels (i.e., with n pixels), the following procedure may be used:1. Using the same criterion as that of the case of the pseudo-image discussedabove, it is possible to determine the set of elements underneath column i ,and the set of elements at the left of row j of the image analyzed. Recallthat the abscissa i of the center of pixel P i,j indicates the column wherethat pixel is located; and j , the ordinate of the center of P i,j indicates therow in which that pixel is found.2. The Cartesian product is found for these two sets, and the result obtainedis referred to as P ∗ i,j .3. An action is carried out on the set P ∗ i,j by an operator O such that:(a) it converts each element of P ∗ i,j (a particular ordered pair) into aunique element which will be a coefficient with two subscripts: thefirst is the same as the subscript of the first element of that orderedpair and the second subscript is the same as the subscript of thesecond element of that ordered pair; and(b) if the signs preceding the two elements of the ordered pair are thesame (both positive or both negative), then the sign of the coefficientobtained is positive. If, on the other hand, the signs are different(positive and negative, or negative and positive), then the sign of thecoefficients is negative. 37. C ∗ i,j is used to refer to the set obtained by having the operator O act onthe set P ∗ i,j : C ∗ i,j = O( P ∗ i,j )5. The algebraic sum of all the elements is of C ∗ i,j is calculated and the resultis equated to the numeric value of the level of gray corresponding to pixel P i,j .When doing the same with each pixel P i,j (where i = 1 , , , . . . , n ; and j = 1 , , , . . . , n ), the result is a system of n linear algebraic equations with n unknowns. Those unknowns, which can be found, are the n coefficients C i,j (where i = 1 , , , . . . , n ; and j = 1 , , , . . . , n ).The classic image of Lenna to be analyzed using the SWM is displayed infigure 13. xy Figure 13: Image of LennaThe image in figure 13 is composed of 262 ,
144 pixels. It can be considereda matrix of pixels with 512 rows, each made up of 512 pixels, and 512 columns,each of which is also made up of 512 pixels. Each of the 262 ,
144 pixels belongsto only one of the 512 rows and only one of the 512 columns. In other words,the image of Lenna analyzed can be viewed as a matrix arranged in a 512 × , ×
32 matrix, consisting of:1. 16 rows of sub-images, each composed of 16 sub-images; or2. 16 columns of sub-images, each composed of 16 sub-images.In other words, the image of Lenna analyzed can also be considered as amatrix arranged in a 16 ×
16 square grid of sub-images.The computational tools available to the authors do have the capacity toapply the SWM to the analysis of each of these 256 sub-images, and this processwas carried out successfully.The columns of sub-images have been numbered from left to right, from thefirst column (1) to the last column (16). The rows of sub-images have alsobeen numbered from bottom up, row 1 to row 16. The sub-image belonging tocolumn k ( k = 1 , , , . . . ,
16) and row l ( l = 1 , , , . . . ,
16) is called I k,l .In figure 14, it can be seen how the image of Lenna has been divided into256 sub-images, so that each sub-image belongs only to one column and onerow of the matrix. xy Figure 14: The image of Lenna is divided here into 256 sub-images.39he notations introduced above for the pixels and for the sub-images canbe used to refer to any pixel of the image of Lenna, by using the letter “P” forpixel and a tetrad of subscripts of that letter. Therefore, P , , , refers to thepixel located in the fourth column and the seventeenth row of the sub-imagelocated, in turn, in the eighth column and the fourteenth row of the matrix ofsub-images of Lenna.The analysis of sub-image I , of the image of Lenna (i.e., the sub-imagelocated in column 9, row 8 of the matrix of the sub-images of the image analyzed)is discussed below. The same approach was used for the analysis of all 255remaining sub-images of the image of Lenna. Sub-image I , of the image ofLenna is displayed in figure 15. xy Figure 15: Sub-image I , of the image of LennaWhen using the SWM to analyze I , , a system of 1024 linear algebraicequations was solved. Recall that each sub-image was composed of a matrix of32 ×
32 pixels. For each of those 1024 pixels, a linear algebraic equation wasobtained using the procedure described.The left-hand member of the linear algebraic equation obtained for each ofthe 1024 pixels of I , is an algebraic sum of the 1024 coefficients C i,j (where i = 1 , , , . . . ,
32; and j = 1 , , , . . . , − − C , = 14 .
500 gives to the value of gray of each pixelof I , , according to its numeric value (which then remains unchanged for allof the pixels of I , ) and to the preceding sign, which can change according tothe algebraic equation corresponding to each pixel of I , . Blue is used for thepositive contributions of C , , and red for the negative contributions for eachpixel of I , . In figure 16, consideration was not given to the absolute value ofthese contributions to the numeric value of the level of gray of each pixel; it isonly shown whether they are positive or negative. xy Figure 16: Type of contribution (positive or negative) of C , to each pixel I , . The positive contributions are represented in blue; the negative, in red.Like figure 16, figure 17 indicates whether the coefficient C , provides apositive or negative contribution to each pixel I , .41 y Figure 17: Type of contribution (positive or negative) of C , to each pixel of I , . The positive contributions are represented in blue; and the negative, inred.The contributions of each coefficient C i,j (where i = 1 , , , . . . ,
32; and j =1 , , , . . . ,
32) to each pixel of I , have certain bidimensional patterns. If themodulus of each contribution is taken into account in addition to its positive ornegative sign, these bidimensional patterns are the elements “corresponding” tothe trains of square waves S , S , . . . , S n .An image of the type analyzed can be considered as a function of two vari-ables: x and y . In this case, to obtain the different approximations to the imageanalyzed, one must add, for each pixel in every sub-image, the patterns men-tioned, taking into account the modulus of the coefficient corresponding to eachpattern (just as in the cases of the functions of one variable, the trains of squarewaves are added to obtain the approximations to the functions analyzed).In the case of a one-variable function, each coefficient C i (where i = 1 , , , . . . , n )is associated to a specific frequency f i . Given the way in which the coefficients C i,j (where i = 1 , , , . . . , n ; and j = 1 , , , . . . , n ) have been characterized, it isclear that each of the latter coefficients is similarly associated to the two spatialfrequencies f i,x and f j,y , corresponding to the x -axis and y -axis respectively.The first of these frequencies f i,x characterizes the periodicity made evident bythe pattern corresponding to each coefficient C i,j , according to the x -axis, andthe second of these frequencies f j,y characterizes the periodicity shown by the42ame pattern, according to the y -axis.The computations of f i,x and f j,y are carried out by using the followingequations (7) and (8), obtained from equation (2), substituting ∆ t by ∆ x and∆ y , respectively (given that f i,x and f j,y are spatial frequencies) and equating n = 32: f i,x = 12∆ x (cid:18) − ( i − (cid:19) ; i = 1 , , , . . . ,
32 (7) f j,y = 12∆ y (cid:18) − ( j − (cid:19) ; j = 1 , , , . . . ,
32 (8)In this case, rather than taking the length of the side of a pixel as the unitof length, ∆ x (which has the same length as ∆ y ) can be used for that purpose.Both the length of ∆ x and that of ∆ y are equal to the product of 32 timesthe length of one side of a pixel. If that is done, the following equations areobtained for f i,x and f j,y : f i,x = 12 (cid:18) − ( i − (cid:19) ; i = 1 , , , . . . ,
32 (9) f j,y = 12 (cid:18) − ( j − (cid:19) ; j = 1 , , , . . . ,
32 (10)In the first place, equations (9) and (10) are used to consider the case pre-sented in figure 16, which represents the contribution of C , (either positiveor negative) to each pixel of I , . Given that in this case i = 17 and j = 25, thefollowing equations are obtained: f i,x = 12 (cid:18) − (17 − (cid:19) = 12 (cid:18) − (cid:19) = 12 (cid:18) (cid:19) = 1 f j,y = 12 (cid:18) − (25 − (cid:19) = 12 (cid:18) − (cid:19) = 12 (cid:18) (cid:19) = 42 = 2The meaning of the first of the preceding equations is as follows: In interval∆ x there is one wave on the x -axis, of the pattern represented in figure 16. (Seefigure to confirm.)The meaning of the second of the preceding equations is as follows: In inter-val ∆ y there are two waves on the y -axis, of the pattern represented in figure 16.(See figure to confirm.)Secondly, equations (9) and (10) will be used to consider the case in figure 17,which represents the contribution of C , (indicating either positive or negative)to each pixel I , . Given that in this case i = 29 and j = 1, the followingequations are obtained: f i,x = 12 (cid:18) − (29 − (cid:19) = 12 (cid:18) − (cid:19) = 12 (cid:18) (cid:19) = 82 = 443 j,y = 12 (cid:18) − (1 − (cid:19) = 12 (cid:18) − (cid:19) = 12 (cid:18) (cid:19) = 12The meaning of the first of the preceding equations is as follows: In interval∆ x , there are 4 waves on the x -axis, of the pattern represented in figure 17.(See figure to confirm.)The meaning of the second of the preceding equations is as follows: In theinterval ∆ y , there is a half-wave on the y -axis, of the pattern represented infigure 17. (See figure to confirm.)In the case of the images, the SWM generates two spatial frequencies, f i,x and f j,y , associated with C i,j , where i = 1 , , , . . . , n , and j = 1 , , , . . . , n .Hence, when applying the method it is also possible in the frequency domain torepresent the results obtained. This expression in the frequency domain of theresults obtained when analyzing images with SWM will be known as SWT, justas when functions of one variable were analyzed. The SWT will be used hereto express some of the results produced.If one adds, for each pixel of every sub-image of Lenna, the positive (ornegative) contributions corresponding to the different coefficients, one obtainsonce again the image analyzed of Lenna. The analysis process carried out withthe SWM, as indicated above, has made it possible to determine, unambiguously,for each pixel of the sub-image of Lenna, which patterns must be added inorder to obtain the image of Lenna again. (Those patterns, as shown above,“correspond” to the trains of square waves obtained when analyzing functionswith one variable with the SWM.)Suppose that not all but only part of the patterns considered are added.For instance, let us admit that for each pixel of every sub-image of Lenna, thecontributions corresponding to the coefficients C i,j such that i ≤ j ≤ C i,j suchthat i ≤
25 and j ≤
25 are the only ones added, the image thus obtained willbe referred to as “approximation 25 32” to the image of Lenna. Of course, if foreach pixel of every sub-image of Lenna, the contributions corresponding to allthe coefficients C i,j are added, the “approximation 32 32”, which is the same asthe image itself, will be produced.Several approximations of the image of Lenna have been displayed in (18.1)to (18.5). 44 y (18.1) : Approximation 8 32 to image of Lenna xy (18.2) : Approximation 16 32 to image of Lenna Figure 1845 y (18.3) : Approximation 24 32 to image of Lenna xy (18.4) : Approximation 28 32 to image of Lenna Figure 1846 y (18.5) : Approximation 32 32 to image of Lenna Figure 18In the examples presented below the results obtained by applying the SWMare given in the frequency domain.Take, for instance, the approximation 8 32 to sub-image I , of the image ofLenna. The corresponding SWT can be expressed with a sequence of 64 triads.The third element of each triad is the coefficient considered. The first element ofthat triad is the frequency f i,x (where i = 1 , , , . . . ,
8) on the x -axis, specifiedby the first subscript of that coefficient. The second element of that triad is thefrequency f j,y (where j = 1 , , , . . . ,
8) on the y -axis, specified by the secondsubscript of that coefficient. The complete list of these 64 triads is as follows:( f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . . f ,x , f ,y ; C , ) = (0 . , . − . f ,x , f ,y ; C , ) = (0 . , . − . I , of the image ofLenna. The positive coefficients are shown in blue, and the negative in red.(19.2) : SWT of the approximation 16 32 to sub-image I , of the image ofLenna. The positive coefficients are shown in blue, and the negative in red. Figure 19: Graphic display of SWTs of different approximations to sub-imageI9, 8 of Lenna 50 I , of the image ofLenna. The positive coefficients are shown in blue, and the negative in red.(19.4) : SWT of the approximation 32 32 to sub-image I , of the image ofLenna. The positive coefficients are shown in blue, and the negative in red. Figure 19: Graphic display of SWTs of different approximations to sub-imageI9, 8 of Lenna 51
Discussion and Prospects
Not only has it been shown how the SWM can be applied to the analysis of 1)functions of one variable which in the intervals where they are analyzed satisfythe conditions of Dirichlet; and 2) sequences of samples from different types ofbiomedical recordings (ECGs, EEGs, and EMGs); but it has also been shownhow that method can be applied to the analysis of images.Fourier’s outstanding and influential contributions, along with other ap-proaches based on the development of his ideas, have played a very importantrole in a number of scientific and technological fields. In particular, mathemat-ical tools such as the Fourier series, the Fourier transform, the discrete Fouriertransform (DFT), the fast Fourier transform (FFT) – an algorithm to com-pute the DFT – and wavelets are used in the field of signal and image analysis.Perhaps in this field, the SWM can become a useful tool to complement thetechniques provided by the Fourier approach.Among the advantages which the SWM offers are its simplicity and efficiencywith which it is possible to make the necessary computations for its application,using software and hardware readily available; and 2) the systematic way inwhich it is applied: In all cases its use implies the solution of a system of linearalgebraic equations which can be specified unambiguously.An algorithm making it possible to apply the SWM to functions of n vari-ables, where n = 1 , , , . . . , will be addressed in another article.The authors of this paper are particularly interested in the application ofthe SWM to the analysis of signals and images essential to the field of medicineand expect to devote several articles to go into greater depth on the topic. References [1] Skliar, O.; Medina, V.; Monge, R. E. (2008). “A New Method for the Anal-ysis of Signals: The Square Wave Method”,
Revista de Matem´atica. Teor´ıay Aplicaciones arXiv :1309.3719[cs.NA].[3] Skliar, O.; Monge, R. E.; Oviedo, G.; Gapper, S. (2014). “A New Method forthe Analysis of Signals: The Square Wave Transform”.
Presented at: XIXInternational Symposium on Mathematical Methods Applied to the Sciences(XIX SIMMAC), San Jos´e, 25-28 February 2014 .[4] Skliar, O.; Oviedo G.; Monge, R. E.; Medina, V.; Gapper, S. (2013).“A New Method for the Analysis of Images: The Square Wave Method”,
Revista de Matem´atica. Teor´ıa y Aplicaciones http://sipi.usc.edu/database/database.php?volume=misc&image=12 .[6] Riley, K. F.; Hobson, M. P.; Bence, S. J. (2006).
Mathematical Methods forPhysics and Engineering . Cambridge University Press, Cambridge, p. 415.[7] Anderson, E.; Bai, Z.; Bischof, C.; Blackford, S.; Demmel, J.; Dongarra,J.; Du Croz, J.; Greenbaum, A.; Hammarling, S.; McKenney, A.; Sorensen,D. (1999).
LAPACK Users’ Guide , 3rd ed. Philadelphia, PA: Society forIndustrial and Applied Mathematics. ISBN 0-89871-447-8.[8] Goldberger, A. L.; Amaral, L. A. N; Glass, L.; Hausdorff, J. M.; Ivanov,P. Ch.; Mark, R. G.; Mietus, J. E.; Moody, G. B.; Peng, C. K.; Stanley,H. E. (2000). “PhysioBank, PhysioToolkit, and PhysioNet: Componentsof a New Research Resource for Complex Physiologic Signals”,
Circulation