Estimation of the parameters in regularized simultaneous super-resolution

Abstract

We describe a method for automatic determination of the regularization parameters for the class of simultaneous super-resolution (SR) algorithms. This method, proposed in (Zibetti et al., 2008c), is based on the joint maximum a posteriori (JMAP) estimation technique, which is a fast alternative to estimate the parameters. However, the classical JMAP technique can be unstable and may generate multiple local minima. In order to stabilize the JMAP estimation, while achieving a cost function with a unique global solution, we derive an improved solution by modeling the JMAP hyperparameters with a gamma prior distribution. In this work, experimental results are provided to illustrate the effectiveness of the proposed method for automatic determination of the regularization parameters for the simultaneous SR. Moreover, we contrast the proposed method to a reference method with known fixed parameters as well as to other parameter selection methods based on the L-curve. These results validate the proposed method as a very attractive alternative for estimating the regularization parameters.

Introduction

Many applications are benefited by acquisition systems that provides images with the best possible resolution, while introducing minimum distortions due to imperfections of the image sensor and the optical system. However, the cost of image acquisition systems, like digital cameras, camcorders and scanners, increases with the resolution of the sensor and with the quality of the optical system. An interesting alternative to improve the resolution and the quality of captured images, without increasing the cost of the system, is to employ digital processing techniques to achieve super-resolution (SR).

Research on super-resolution algorithms began in the 1990s with pioneer works, as Tsai and Huang (1984), which employed Fourier domain methods. Since then, different approaches have been developed, including projections onto convex sets (POCS) (Stark and Oskoui, 1989, Tekalp et al., 1992), non-uniform interpolation (Aizawa et al., 1992) and iterative back-projection (Irani and Peleg, 1991, Irani and Peleg, 1993). Regularized SR approaches based on maximum a posteriori (MAP) and regularized least squares appeared in (Schultz and Stevenson, 1996, Hong et al., 1997). Traditionally, regularized approaches minimize a cost function composed by the residual associated with the estimated high-resolution (HR) frame plus another term, called the prior term, used to regularize the problem (Park et al., 2003). In these approaches, the regularization parameter scales the influence of the prior term in the resulting solution. In most SR methods, the parameter is assumed to be known by some way. In the majority of practical cases, the parameter as well as the HR images must be estimated from the data.

In simultaneous SR methods, proposed in (Borman and Stevenson, 1999, Zibetti and Mayer, 2005, Zibetti and Mayer, 2006, Zibetti and Mayer, 2007), all frames of an image sequence are estimated in a single process. Two different kinds of priors are employed, one to achieve spatial smoothness and other to achieve higher similarity of the HR frames in the motion trajectory. In these problems at least two parameters are necessary. Interesting multiple parameter selection methods for general inverse problems have been studied in (Belge et al., 2002, Hansen, 1998). These methods, however, have not been applied for simultaneous super-resolution. We have addressed this problem in (Zibetti et al., 2008c). In this work, which is an extended version of Zibetti et al. (2008c), the following novelties are provided: (a) more details in the derivation of the method proposed in (Zibetti et al., 2008c); (b) two possible implementations of the method, explained in details and with a pseudocode; and (c) more comprehensive experiments to evaluate the performance of the proposed method, contrasting to other competitive multiple parameter methods such as Belge et al. (2002).

In this paper, we address the problem with two parameters in the simultaneous SR. Section 2 provides a detailed description of system models used in this work. The simultaneous SR algorithm with fixed regularization parameters is reviewed in Section 2.2. In Section 3, the new method with automatic determination of the parameters is proposed using the joint maximum a posteriori (JMAP) estimation technique (Mohammad-Djafari, 1996). The classical JMAP approach, which assumes uniform density for the hyperparameters is, in general, unstable (Mohammad-Djafari, 1996). To circumvent this, we assume a gamma probability density for the hyperparameters which results in a stable algorithm with a unique global solution. In Section 3.3, two detailed implementations of the method are proposed. One is based on non-linear conjugated gradient (Vogel, 2002), the other proposed implementation is based on alternated minimization of updating the parameters and re-solving a linear system with conjugated gradient (Vogel, 2002, Golub and Loan, 1996). Section 4 presents experiments, comparisons and discussions to illustrate the performance. Some new experiments, which provide some insights about the proposed algorithm are shown in this section. Section 5 concludes this paper.