Wavelet Algorithms for High-Resolution Image Reconstruction

Abstract

Abstract. High-resolution image reconstruction refers to the reconstruction of high-resolution images from multiple low-resolution, shifted, degraded samples of a true image. In this paper, we analyze this problem from the wavelet point of view. By expressing the true image as a function in L(R 2 ), we derive iterative algorithms which recover the function completely in the L sense from the given low-resolution functions. These algorithms decompose the function obtained from the previous iteration into different frequency components in the wavelet transform domain and add them into the new iterate to improve the approximation. We apply wavelet (packet) thresholding methods to denoise the function obtained in the previous step before adding it into the new iterate. Our numerical results show that the reconstructed images from our wavelet algorithms are better than that from the Tikhonov least-squares approach. Extension to super-resolution image reconstruction, where some of the low-resolution images are missing, is also considered.Key words. wavelet, high-resolution image reconstruction, Tikhonov least square method AMS subject classifications. 42C40, 65T60, 68U10, 94A08 PII. S10648275003831231. Introduction. Many applications in image processing require deconvolving noisy data, for example the deblurring of astronomical images [11]. The main objective in this paper is to develop algorithms for these applications using a wavelet approach. We will concentrate on one such application, namely, the high-resolution image reconstruction problem. High-resolution images are often desired in many situations, but made impossible because of hardware limitations. Increasing the resolution by image processing techniques is therefore of great importance. The earliest formulation of the problem was proposed by Tsai and Huang [24] in 1984, motivated by the need of improved resolution images from Landsat image data. Kaltenbacher and Hardie [14], and Kim, Bose, and Valenzuela [15] applied the work of [24] to noisy and blurred images, using least-squares minimization. The high-resolution image reconstruction also can be obtained by mapping several low-resolution images onto a single high-resolution image plane, then interpolating it between the nonuniformly spaced samples [3,23]. The high-resolution image reconstruction can also be put into a Bayesian framework by using a Huber-Markov random field; see, for example, Schultz and Stevenson [22].Here we follow the approach in Bose and Boo [1] and consider creating highresolution images of a scene from the low-resolution images of the same scene. When

High‐resolution image reconstruction with displacement errors: A framelet approach

“…Recently, new approaches for HR image reconstruction problems using wavelet techniques have been proposed by Chan et al in [5,6,7]. The problem of HR image reconstruction is understood and analyzed under the framework of multi-resolution analysis of L 2 (R 2 ) by recognizing the blurring kernel H as a low-pass filter associated with a multi-resolution analysis.…”

“…The reasoning within the wavelet framework provides the intuition for new algorithms. The wavelet-based HR image reconstruction algorithms in [5,6,7] are developed through the perfect reconstruction formula of a bi-orthogonal wavelet system which has (2) as its primary low-pass filter. The algorithms approximate iteratively the wavelet coefficients folded by the given low-pass filter.…”

“…The first requirement is the construction of a bi-orthogonal wavelet system with (2) as its primary low-pass filter. Examples for L = 2 and 4 are given in [6] for = 0 and in [7] for = 0. Minimally supported bi-orthogonal wavelet systems with (2) as primary low-pass filter are constructed for arbitrary integer L ≥ 2 and any real number | | < 1/2 in [45].…”

“…For the case without displacement error (i.e., when all = 0), the corresponding blurring kernel H is spatially invariant and (1) is actually a de-convolution problem. The proposed algorithm in [6] outperforms the least squares method in terms of peak signal-to-noise ratio (PSNR).…”

“…The performance of the proposed algorithm in [7] is comparable with that of the least squares method. We note that the algorithm in [7] is a nontrivial extension of the algorithmic framework of [6], which applies only to spatially invariant blurring operators. There are several issues affecting the performance of the wavelet approach for problems with displacement errors.…”