Search

The Lawrence Berkeley National Laboratory working group was presented with three problems involving the improvement of images from an AFM scanner. The first concerned the inpainting of an image for which data is collected via a spiral scan waveform. The second concerned the exploitation of data obtained at different times to obtain better spatial resolution. The third concerned the correction of horizontal positions reported by the sensor, particularly when the scanner is operating at high frequency. Herein is a brief report on the progress and ideas generated by the working group on each of these problems.

A sum of different frames under the action of a bounded linear operator is studied with the help of analysis, synthesis and frame operators. In particular, it is shown that the sum of a pair of orthogonal frames is a frame. This provides an easy construction of a frame where the frame bounds can be computed easily. For a pair of orthogonal frames, the necessary and sufficient condition is presented for their alternate duals to be orthogonal.

A system using an oversampled Fourier transform for hiding data is given in [J.R. Miotke, L. Rebollo-Neira, Oversampling of Fourier coefficients for hiding messages, Appl. Comput. Harmon. Anal. 16 (2004) 203–207]. When viewed as a cryptographic algorithm, we demonstrate here that the system is susceptible to a known plaintext attack.

Motivated by the notion of orthogonal frames, we describe sufficient conditions for the construction of orthogonal MRA wavelet frames in L 2 ( R ) from a suitable scaling function. These constructions naturally lead to filter banks in l 2 ( Z ) with similar orthogonality relations and, through these filter banks, the orthogonal wavelet frames give rise to a vector-valued discrete wavelet transform (VDWT). The novelty of these constructions lies in their potential for use with vector-valued data, where the VDWT seeks to exploit correlation between channels. Extensions to higher dimensions are natural and the constructions corresponding to the bidimensional case are presented along with preliminary results of numerical experiments in which the VDWT is applied to color image data.