2258 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 8, AUGUST 2000 Wavelets and Differential-Dilation Equations Todor Cooklev, Member, IEEE, Gheorghe I. Berbecel, and A. N. Venetsanopoulos Abstract—In this paper, it is shown how differential-dilation equations can be constructed using iterations, similar to the itera- tions with which wavelets and dilation equations are constructed. A continuous-time wavelet is constructed starting from a differ- ential-dilation equation. It has compact support and excellent time domain and frequency domain localization properties. The wavelet is infinitely differentiable and therefore cannot be obtained using digital filter banks. In addition, the wavelet has excellent approx- imation properties. New sampling and differentiation techniques are also introduced. Results on image interpolation using the solu- tion of the differential-dilation equation are presented. Examples are given, demonstrating the suitability of the new wavelet function for signal analysis. Index Terms—Compact support, continuous time, differential- dilation equation, image interpolation, signal processing, wavelets. I. INTRODUCTION W AVELETS are an important tool in signal processing. They are more versatile and powerful than Fourier-based techniques because they have more degrees of freedom and in particular the basis function is not fixed. The continuous-time wavelet transform (CWT) is defined as [1] (1) Two standard choices of the basis function are the Mexican hat wavelet (2) and the Morlet wavelet (3) In addition to the basis function, the continuous-time wavelet transform depends on two parameters: dilation and shift . The wavelet transform involves basis functions which do not have a constant length: very short basis functions are used to achieve good time resolution, while longer basis functions can be used to obtain fine frequency analysis. When and are continuous the set of basis functions does not constitute an orthonormal basis, i.e., the representation is redundant. The discrete wavelet trans- Manuscript received December 8, 1998; revised March 29, 2000. The asso- ciate editor coordinating the review of this paper and approving it for publication was Dr. Shubha Kadambe. T. Cooklev is with Aware, Inc., Lafayette, CA 94549 USA (e-mail: cooklev@juno.com). G. Berbecel is with Genesis Microchip Inc., Thornhill, Ont., Canada. A. N. Venetsanopoulos is with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, Ont., M5S 3G4 Canada. Publisher Item Identifier S 1053-587X(00)06102-X. form can be obtained by quantizing and . The basis functions become , where and . The case where and is the most common and the corresponding grid is called dyadic. The discrete-time wavelet transform (DTWT) corresponds to a filter bank iterated a finite number of times along the lowpass channel. A very important property is the ability of the filter bank to represent polynomials, which is equivalent to the number of vanishing moments of the wavelet. Tremendous amount of research activity has been devoted to wavelets, designed using filter banks, while the continuous-time wavelet transform has been much less studied. In this paper we shall be concerned only with the CWT. A very important question in wavelet analysis is choosing the basis function, and this is the focus of our concern in this paper. We are looking for a continuous-time wavelet that is infinitely dif- ferentiable, has compact support and provides good frequency localization. A wavelet with these three properties has not been used in signal processing. The Mexican hat and Morlet wavelets are infinitely differentiable, but do not have compact support. Wavelets generated from filter banks can have compact support, but cannot be infinitely differentiable. The paper is organized as follows. In Section II, the itera- tive scheme, through which a differential-dilation equation is obtained is presented. In Section III, we discuss the properties of the solution of the differential-dilation equation and its ap- plication as a window function in STFT analysis. Results on image restoration and interpolation are also presented. The first derivative of the solution of the differential-dilation equation is a wavelet and forms a frame. This is proven in Section IV. Sec- tion V gives a generalization. II. BASIC CONSTRUCTION Iteration of a digital filter followed by downsampling (Fig. 1) leads to a limit function provided the filter satisfies certain con- straints [1]. We would like to explore the counterpart of Fig. 1 in the continuous-time domain. Downsampling and upsampling are discrete-time multirate operations. We are going to find it useful to define a continuous-time dilator [Fig. 2(a)]. Note that the block in Fig. 2(a) is purely a mathematical tool that is only conceptually similar to the discrete-time decimator. Note also that, by definition, the continuous-time dilator performs am- plification in addition to dilation. Suppose now that the blocks of continuous-time filtering and dilation are cascaded and iter- ated [Fig. 2(b)]. The properties of this iteration are known in the discrete-time domain. It appears that this paper is the first attempt to study similar iterations in the continuous-time do- main. Note that such continuous-time iterations without dilating blocks, however, have trivial properties and have been used nu- merous times, and in particular in the construction of contin- 1053–587X/00$10.00 © 2000 IEEE