Periodic spline-based frames for image restoration

Amir  Averbuch; Pekka Neittaanmäki; Valery  Zheludev

doi:10.3934/ipi.2015.9.661

Article Contents

2015, Volume 9, Issue 3: 661-707. Doi: 10.3934/ipi.2015.9.661

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Periodic spline-based frames for image restoration

1.
School of Computer Science, Tel Aviv University, Tel Aviv 69978
2.
Dept. of Mathematical Information Technology, Faculty of Information Technology, Agora, P.O. Box 35, FI-40014, University of Jyväskylä

Received: November 2011

Revised: July 2014

Published: August 2015

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Abstract

We present a design scheme that generates tight and semi-tight frames in discrete-time periodic signals space originated from four-channel perfect reconstruction periodic filter banks. Filter banks are derived from interpolating and quasi-interpolating polynomial and discrete splines. Each filter bank comprises one linear phase low-pass filter (in most cases interpolating) and one high-pass filter, whose magnitude's response mirrors that of a low-pass filter. These filter banks comprise two band-pass filters. We introduce local discrete vanishing moments (LDVM). When the frame is tight, analysis framelets coincide with their synthesis counterparts. However, for semi-tight frames, we swap LDVM between synthesis and analysis framelets. The design scheme is generic and it enables us to design framelets with any number of LDVM. The computational complexity of the the framelet transforms, which consists of calculating the forward and the inverse FFTs, does not depend on the number of LDVM and does depend on the size of the the impulse response filters. The designed frames are used for image restoration tasks, which were degraded by blurring, random noise and missing pixels. The images were restored by the application of the Split Bregman Iterations method. The frames performances are evaluated. A potential application of this methodology is the design of a snapshot hyperspectral imager that is based on a regular digital camera. All these imaging applications are described.

Keywords:

Spline frames,
tight and semi-tight frames,
interpolating and quasi-interpolating polynomial,
vanishing moments,
split Bregman iterations,
image restoration.

Mathematics Subject Classification: Primary: 65D07, 06D22, 42C40, 94A08.

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