Tight Frame Based Method for High-Resolution Image Reconstruction

High-resolution image reconstruction is to reconstruct a high-resolution (HR) image from multiple, undersampled, shifted, degraded and noisy frames obtained by using multiple images shifted from each other by sub-window shifts. Problems of high-resolution restoration for images arise in a variety of scientific, medical, and engineering applications. The problem of HR image reconstruction is becoming a very hot field. Two special issues on this topic are launched recently in IEEE Signal Processing Magazine (Volume 20, Issue 3, May 2003) and International Journal of Imaging Systems and Technology (vol. 14, no. 2, 2004). The earliest formulation of the problem of HR image reconstruction was motivated by the need of improved resolution images from Landsat image data. In [28], Huang and Tsay used the frequency domain approach to demonstrate reconstruction of one improved resolution image from several down-sampled noisefree versions of it. Later on, Kim el al. [30] generalized this idea to noisy and blurred images. Both methods in [28, 30] are computational efficiency, but, they are prone to model errors, and that limit their use [1]. Statistical methods have appeared recently for super-resolution image reconstruction problems. In this direction, tools such as a maximum a posteriori (MAP) estimator with the Huber-Markov random filed prior and a Gibbs image prior are proposed in [25, 43]. In particular, the task of simultaneous image registration and super-resolution image reconstruction are studied in [25, 45]. Iterative spatial domain methods are popular class of methods for solving the problems of resolution enhancement [3, 21, 22, 23, 27, 31, 32, 36, 38, 39, 41]. The problems are formulated as Tikhonov regularization. A great deal of work has been devoted to the efficient calculation of the reconstruction and the estimation of the associated hyperparameters by taking advantage of the inherent structures in the HR system matrix. Bose and Boo [3] use a block semi-circulant matrix decomposition in order to calculate the MAP reconstruction. Ng et al. [36] and Ng and Yip [37] proposed a fast discrete cosine transform based approach for HR image reconstruction with Neumann boundary condition. Nguyen et al. [40, 41] also addressed the problem of efficient calculation. The proper choice of the regularization tuning parameter is crucial to achieving robustness in the presence of noise and avoiding trial-and-error in the selection of an optimal tuning parameter. To this end, Bose et al. [4] used a L-curve based approach. Nguyen et al. [41] used a generalized cross-validation method. Molina et al. [33] used an expectation-maximization algorithm. Lu et al. [32] proposed multiparameter regularization methods which introduce different regularization parameters for different frequency bands of the regularization operator. Low-resolution images can be viewed as outputs of the original high-resolution image passing through a low-pass filter followed by a decimation process. This viewpoint suggests that a framework of multiresolution analysis can be naturally adopted to produce a HR image from a set of low-resolution images of the same