Class of smooth nonseparable N-dimensional scaling functions

Mohsen Maesumi

doi:10.1117/12.188810

11 October 1994 Class of smooth nonseparable N-dimensional scaling functions

Mohsen Maesumi

Proceedings Volume 2303, Wavelet Applications in Signal and Image Processing II; (1994) https://doi.org/10.1117/12.188810
Event: SPIE's 1994 International Symposium on Optics, Imaging, and Instrumentation, 1994, San Diego, CA, United States

Abstract

Wavelets of compact support are an important tool in many areas of signal analysis. A starting point for the construction of such wavelets is the scaling function, the solution of the dilation equation. We study the dilation equation (phi) (X) equals (Sigma) _KC_K(phi )(2X-K) where K (epsilon) {0...m}^N, (phi) :R^N yields R, C_K (epsilon) R. This paper gives a set of sufficient conditions on C_K under which the solution of the dilation equation has a specific degree of regularity. We will construct (phi) through infinite products of 2^N associated matrices with entries in terms of C_K. The conditions for regularity are based on certain sum rules that triangularize all of the associated matrices and on certain inequalities that control the eigenvalues of the matrices. The net effect of the sum rules is to specify the coefficients C_K in terms of a binomial interpolation of their values at the corners of the N-cube, {0,m}^N. The inequalities are based on sums of the coefficients at the corners of the various faces of the N-cube. The number of derivatives that (phi) possesses and the Holder exponent of the last derivative can be determined for the sums.

Citation Download Citation

Mohsen Maesumi "Class of smooth nonseparable N-dimensional scaling functions", Proc. SPIE 2303, Wavelet Applications in Signal and Image Processing II, (11 October 1994); https://doi.org/10.1117/12.188810

ACCESS THE FULL ARTICLE

INSTITUTIONAL
Select your institution to access the SPIE Digital Library.

SELECT YOUR INSTITUTION

PERSONAL
Sign in with your SPIE account to access your personal subscriptions or to use specific features such as save to my library, sign up for alerts, save searches, etc.

PERSONAL SIGN IN

No SPIE Account? Create one

PURCHASE THIS CONTENT

SUBSCRIBE TO DIGITAL LIBRARY

50 downloads per 1-year subscription

Members: $195

Non-members: $335 ADD TO CART

25 downloads per 1 - year subscription

Members: $145

Non-members: $250 ADD TO CART

PURCHASE SINGLE ARTICLE

Includes PDF, HTML & Video, when available