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

This paper introduces a novel framework for the study of the total-variation model for image denoising. In the model, the denoised image is the proximity operator of the total-variation evaluated at a given noisy image. The totalvariation can be viewed as the composition of a convex function (the 1 norm for the anisotropic total-variation or the 2 norm for the isotropic totalvariation) with a linear transformation (the first-order difference operator). These two facts lead us to investigate the proximity operator of the composition of a convex function with a linear transformation. Under the assumption that the proximity operator of a given convex function (e.g., the 1 norm or the 2 norm) can be readily obtained, we propose a fixed-point algorithm for computing the proximity operator of the composition of the convex function with a linear transformation. We then specialize this fixed-point methodology to the total-variation denoising models. The resulting algorithms are compared with the Goldstein-Osher split-Bregman denoising algorithm. An important advantage of the fixed-point framework leads us to a convenient analysis for convergence of the proposed algorithms as well as a platform for us to develop efficient numerical algorithms via various fixed-point iterations. Our numerical experience indicates that the methods proposed here perform favorably.

Thrombin activates protease-activated receptor-1 (PAR-1) by cleavage of the amino terminus to unmask a tethered ligand. Although peptide analogs can activate PAR-1, we show that the functional responses mediated via PAR-1 differ between the agonists. Thrombin caused endothelial monolayer permeability and mobilized intracellular calcium with EC 50 values of 0.1 and 1.7 nM, respectively. The opposite order of activation was observed for agonist peptide (SFLLRN-CONH 2 or TFLL-RNKPDK) activation. The addition of inactivated thrombin did not affect agonist peptide signaling, suggesting that the differences in activation mechanisms are intramolecular in origin. Although activation of PAR-1 or PAR-2 by agonist peptides induced calcium mobilization, only PAR-1 activation affected barrier function. Induced barrier permeability is likely to be G␣ 12/13 -mediated as chelation of G␣ q -mediated intracellular calcium with BAPTA-AM, pertussis toxin inhibition of G␣ i/o , or GM6001 inhibition of matrix metalloproteinase had no effect, whereas Y-27632 inhibition of the G␣ 12/13 -mediated Rho kinase abrogated the response. Similarly, calcium mobilization is G␣ q -mediated and independent of G␣ i/o and G␣ 12/13 because pertussis toxin and Y-27632 had no effect, whereas U-73122 inhibition of phospholipase C-␤ blocked the response. It is therefore likely that changes in permeability reflect G␣ 12/13 activation, and changes in calcium reflect G␣ q activation, implying that the pharmacological differences between agonists are likely caused by the ability of the receptor to activate G␣ 12/13 or G␣ q . This functional selectivity was characterized quantitatively by a mathematical model describing each step leading to Rho activation and/or calcium mobilization. This model provides an estimate that peptide activation alters receptor/G protein binding to favor G␣ q activation over G␣ 12/13 by ϳ800-fold.

The connection between fluid flow and optical flow is explored in typical flow visualizations to provide a rational foundation for application of the optical flow method to image-based fluid velocity measurements. The projected-motion equations are derived, and the physics-based optical flow equation is given. In general, the optical flow is proportional to the path-averaged velocity of fluid or particles weighted with a relevant field quantity. The variational formulation and the corresponding EulerLagrange equation are given for optical flow computation. An error analysis for optical flow computation is provided, which is quantitatively examined by simulations on synthetic grid images. Direct comparisons between the optical flow method and the correlation-based method are made in simulations on synthetic particle images and experiments in a strongly excited turbulent jet.

We propose orthogonal Fourier-Mellin moments, which are more suitable than Zernike moments, for scaleand rotation-invariant pattern recognition. The new orthogonal radial polynomials have more zeros than do the Zernike radial polynomials in the region of small radial distance. The orthogonal Fourier-Mellin moments may be thought of as generalized Zernike moments and orthogonalized complex moments. For small images, the description by the orthogonal Fourier-Mellin moments is better than that by the Zernike moments in terms of image-reconstruction errors and signal-to-noise ratio. Experimental results are shown.