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Super-resolution (SR) reconstruction has been extensively studied and used in many fields, such as medical diagnostics, military surveillance, frame freeze in video, and remote sensing. In many imaging systems, since the low resolution (LR) images observed by detector arrays will be degraded by aliasing effects, some visual charge coupled device (CCD) cameras suffer from under-sampling and LR images also can be blurred by relative motion between sensors and objects, it is essential to develop an effective image restoration algorithm. The goal of SR is to reconstruct a un-aliased high resolution (HR) image from multiple aliased LR images. This is possible if there is sub-pixel motion between the acquired LR frames. Tsai and Huang^[1] first addressed SR reconstruction algorithm in the frequency domain. Their algorithm based on the shift property of the Fourier transform makes explicit use of the aliasing relationship under the assumption that the SR image is band-limited. Because their method operates on the noise-free data, Kim and Su^[2] proposed an extension of Tsai-Huang’s algorithm for a blurred and noisy image, resulting in a weighted least squares algorithm. Although the frequency domain methods are theoretically simple and computationally cheap, they are restricted to global translational motion between LR frames and linear space invariant image blur, limiting their use. So the frequency domain approach has not been the active research direction. Resolution techniques presented in the literature operate in the spatial domain. Typically, the projection onto convex set algorithm accounting for the blur caused by the LR sensor geometry was first suggested by Stark and Oskoui^[3] for SR reconstruction. Patti and Altunbasak^[4] extended the Stark-Oskoui’s algorithm which accounted for space varying blur and used block matching or phase correlation to estimate sub-pixel motion. The POCS-based algorithms have the advantage of simplicity and utilizing the powerful spatial domain observation model. They also allow a convenient inclusion of a priori information. The disadvantages of these methods are nonuniqueness of solution, slow convergence, and a high computational cost^[5].
There also appeared stochastic methods for SR image reconstruction^[6-8] . A maximum a posteriori (MAP) method used to reconstruct HR images from a sequence of LR images was from Schultz and Stevenson^[6] , which used specific Huber-Markov-Gibbs model for a prior SR image model, resulting in a constrained optimization problem with a unique minimum. This was extended by Hardie et al^[7] , which sought to minimize a MAP cost function with respect to the HR image and the registration parameters simultaneously using a cyclic coordinate-descent optimization procedure. Tom and Katsaggelos^[8]  proposed the maximum likelihood (ML) estimation applied to the SR reconstruction. The ML estimation is a special case of MAP estimation with no prior term. The ML technique utilizes the expectation-maximization (EM) algorithm to estimate the sub-pixel shifts, the noise variances of each image and the HR image simultaneously.
Recently, regularized SR reconstruction algorithms are most comprehensive. Kang and Katsaggelos^[9] proposed the use of a regularization function instead of a constant regularization parameter, allowing for the simultaneous estimation of the regularization parameters and restoration of the degraded images without any prior knowledge about the original images. It can greatly improve the quality of image restoration when the Gaussian noise is the only noise source added to the LR images. But besides additive Gaussian noise, the LR images include blur noise and registration noise. Therefore, taking into account the differences of noise levels and components of per frame, He and Kondi^[10] proposed a channel weight coefficient regularized SR reconstruction algorithm. The channel weight coefficient is adjusted according to different noise level of each frame, working as the cross-channel fidelity. So this method is very efficient for the LR images including different noises. However, it needs to compute the value of the channel weight coefficient to per frame at each iterative step. So it costs a high computation at each iterative step, resulting in a slow convergence of the algorithm.

We proposed an adaptive SR reconstruction algorithm, which introduces adaptive regularization function in the cost function and uses the adaptive optimization step size with the progress of the iterative process. On one hand, the man-made factor is reduced in the reconstruction process of the image, the high frequency components are restored well and the quality of the image is improved. On the other hand, the rate of convergence is greatly improved, which is a big advancement to reach the goal of real-time reconstruction in theory. Therefore, in order to achieve real-time reconstruction and greatly improve the quality of the image, step size will be further optimized, and at the same time regularization function will be also improved.

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