As a special shift-invariant spaces,spline subspaces yield many advantages so that there are many practical applications for signal or image processing.In this paper,we pay attention to the sampling and reconstruction problem in spline subspaces.We improve lower bound of sampling set conditions in spline subspaces.Based on the improved explicit lower bound,a improved explicit convergence ratio of reconstruction algorithm is obtained.The improved convergence ratio occupies faster convergence rate than old one.At the end,some numerical examples are shown to validate our results.
This paper discusses conditions under which the solution of linear system with minimal Schatten-p norm, 0 < p 1, is also the lowest-rank solution of this linear system. To study this problem, an important tool is the restricted isometry constant (RIC). Some papers provided the upper bounds of RIC to guarantee that the nuclear-norm minimization stably recovers a low-rank matrix. For example, Fazel improved the upper bounds to δA4r < 0.558 and δA3r < 0.4721, respectively. Recently, the upper bounds of RIC can be improved to δA2r < 0.307. In fact, by using some methods, the upper bounds of RIC can be improved to δA2r < 0.4931 and δ A r < 0.309. In this paper, we focus on the lower bounds of RIC, we show that there exists linear maps A with δA2r > 1/ 2^(1/2) or δAr > 1/3 for which nuclear norm recovery fail on some matrix with rank at most r. These results indicate that there is only a little limited room for improving the upper bounds for δA2r and δAr . Furthermore, we also discuss the upper bound of restricted isometry constant associated with linear maps A for Schatten p (0 < p < 1) quasi norm minimization problem.
We consider efficient methods for the recovery of block sparse signals from underdetermined system of linear equations. We show that if the measurement matrix satisfies the block RIP with δ2s < 0.4931, then every block s-sparse signal can be recovered through the proposed mixed l2 /l1 -minimization approach in the noiseless case and is stably recovered in the presence of noise and mismodeling error. This improves the result of Eldar and Mishali (in IEEE Trans. Inform. Theory 55: 5302-5316, 2009). We also give another sufficient condition on block RIP for such recovery method: δs < 0.307.
