FRACTIONAL SPLINE WAVELET SOFTWARE FOR MATLAB Thierry Blu The Chinese University of Hong Kong June 2014 A. TEST PROGRAMS DEMO1D: Demonstration program that exemplifies the use of the 1D Fractional Spline Wavelet transform and the associated helper functions DEMO2D: Demonstration program that exemplifies the use of the 2D Fractional Spline Wavelet transform and the associated helper functions B. CORE WAVELET TRANSFORM FUNCTIONS FFTFRACTSPLINEFILTERS: Computes the frequency responses of the wavelet analysis and synthesis filters (lowpass and highpass) for various types ('ortho','bspline','dual') of orthonormal or semi-orthonormal fractional spline wavelet transforms, and for various parameters (alpha,tau) as described in Ref. [3]. FFTWAVELETANALYSIS1D : Performs an FFT-based computation of the 1D wavelet transform of a signal to a given depth J. FFTWAVELETSYNTHESIS1D: Performs an FFT-based computation of the 1D inverse wavelet transform. FFTWAVELETANALYSIS2D : Performs an FFT-based computation of the 2D wavelet transform of an image to a given depth J. FFTWAVELETSYNTHESIS2D: Performs an FFT-based computation of the 2D inverse wavelet transform. C. OTHER ROUTINES WEXTRACT1D : Extraction of a subband of a 1D wavelet transform. WAVELETPLOT1D: Show the 1D wavelet transform as several staged plots. WEXTRACT2D : Extraction of a subband of a 2D wavelet transform. WAVELETPLOT2D: Show the 2D wavelet transform using a standard quadrant image. FRACTSPLINEFUNCTION : Computes a scaling function with the input (alpha,tau) parameters FRACTSPLINEWAVELETFUNCTION: Computes a wavelet function with the input (alpha,tau) parameters FRACTSPLINEAUTOCORR: Frequency domain computation of fractional spline autocorrelation function. References: [1] Unser, M. & Blu, T.,"Fractional Splines and Wavelets", SIAM Review, Vol. 42 (1), pp. 43-67, March 2000. [2] Blu, T. & Unser, M.,"The Fractional Spline Wavelet Transform: Definition and Implementation", Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'00), Istanbul, Turkey, Vol. {I}, pp. 512-515, June 5--9, 2000. [3] Blu, T. & Unser, M.,"A Complete Family of Scaling Functions: The $($-Fractional Splines", Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'03), Hong Kong, China, Vol. {VI}, pp. 421-424, April 6--10, 2003. Demo at http://bigwww.epfl.ch/demo/jfractsplinewavelet/