IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 9, NO. 5, MAY 2000 949 [9] B. Kisaˇ canin and D. Schonfeld, “A fast threshold linear convolution representation of morphological operations,” IEEE Trans. Image Pro- cessing, vol. 3, pp. 455–457, July 1994. [10] H. V. Poor, An Introduction to Signal Detection and Estimation, New York: Springer-Verlag, 1994. [11] H. J. A. M. Heijmans, Morphological Image Operators. San Diego, CA: Academic, 1994. [12] D. Schonfeld, “Fast parallel nonlinear filtering based on the FFT,” in Proc. Int. Conf. Digital Signal Processing, vol. 1, Limassol, Cyprus, 1995, pp. 338–341. Examples of Bivariate Nonseparable Compactly Supported Orthonormal Continuous Wavelets Wenjie He and Ming-Jun Lai Abstract—We give many examples of bivariate nonseparable compactly supported orthonormal wavelets whose scaling functions are supported over [0, 3] × [0, 3]. The Hölder continuity properties of these wavelets are studied. Index Terms—Compact support, continuous, nonseparable, or- thonormal, wavelet. I. INTRODUCTION Univariate wavelets have found successful applications in signal pro- cessing since wavelet expansions are more appropriate than conven- tional Fourier series to represent the abrupt changes in nonstationary signals. To apply wavelet methods to digital image processing, we have to construct vibariate wavelets. The most commonly used method is the tensor product of univariate wavelets. This construction leads to a separable wavelet which has a disadvantage of giving a particular importance to the horizontal and vertical directions. Much effort (cf., e.g., [1]–[3]) has been spent on constructing nonseparable bivariate wavelets. In this paper, we construct vibariate nonseparable compactly supported orthonormal wavelets based on the commonly used uniform dilation matrix Let be a trigonometric polynomial. We will construct which satisfies the following requirements: 1 : = 1; 2 : with and Let be generated by Then 1 im- plies the convergence of this infinite product and hence is a well-de- fined continuous function. 2 implies Thus, by Plancheral’s Theorem. For a fixed ordering which maps bi-inte- gers (0, 0) ≤ ≤ into positive integers with let be a matrix of size with entries Manuscript received July 24, 1997; revised August 4, 1999. The associate ed- itor coordinating the review of this manuscript and approving it for publication was Prof. Kannan Ramchandran. The authors are with the Department of Mathematics, University of Georgia, Athens, GA 30602 USA (e-mail: mjlai@math.uga.edu). Publisher Item Identifier S 1057-7149(00)01153-2. for (0, 0) ≤ ≤ In order to make to be an orthonormal set, we need to have the bivariate generalizationof the Lawton condition 3 (cf. [4]). One is a nondegenerate eigenvalue of We then further study the coefficients of such that 4 : with After these preparations, we shall construct such that 5: To make to be a low-pass filter, we require that have a factor That is, 6 : for all For we are able to give a complete solution set of all satisfying 1 , 2 , and 6 . We identify many families of solutions which further satisfy 3 and 4 . For example, a tensor product of Daubechies’ scaling function is included. It is known that with [5]. We can expect other solutions to have certain Hölder’s exponents. We study the regularity of those filters. Finally, we construct to satisfy 5 for any given satisfying 1 and 2 . In Section III, we present some numerical experiments using our nonseparable wavelets. II. CONSTRUCTION OF SCALING FUNCTIONS AND WAVELETS Rewrite as with and Also write in its polyphase form: It is well-known that a polynomial satisfying 2 is equivalent to (1) From now on, we only consider Thus, we write ν = 0, 1, 2, 3. We now present one of the main results in this paper. Theorem 2.1.: Let (2) with (3) Then, satisfies 2 if satisfy the following: (4) Proof: It is straightforward to verify that and satisfy (1) if and only if (5) 0018-9219/00$10.00 © 2000 IEEE