SHAPE FROM FOCUS USING LULU OPERATORS AND DISCRETE PULSE TRANSFORM IN THE PRESENCE OF NOISE

A study of three dimension (3D) shape recovery is an interesting and challenging area of research. Recovering the depth information of an object from normal two dimensional (2D) images has been studied for a long time with different techniques. One technique for 3D shape recovery is known as Shape from Focus (SFF). SFF is a method that depends on different focused values in reconstructing the shape, surface, and depth of an object. The different focus values are captured by taking different images for the same object by varying the focus length or varying the distance between object and camera. This single view imaging makes the data gathering simpler in SFF compared to other shape recovery techniques. Calculating the shape of the object using different images with different focused values can be done by applying sharpness detection methods to maximize and detect the focused values. However, noise destroys many information in an image and the result of noise corruption can change the focus values in the images. This thesis presents a new 3D shape recovery technique based on focus values in the presence of noise. The proposed technique is based on LULU operators and Discrete Pulse Transform (DPT). LULU operators are nonlinear rank selector operators that hold consistent separation, total variation and shape preservation properties. The proposed techniques show better and more accurate performance in comparison with the existing SFF techniques in noisy environment

Sparse Gradient Image Reconstruction from Incomplete Fourier Measurements and Prior Edge Information

In many imaging applications, such as functional Magnetic Resonance Imaging (fMRI), full, uniformly- sampled Cartesian Fourier (frequency space) measurements are acquired to reconstruct an image. In order to reduce scan time and increase temporal resolution for fMRI studies, one would like to accurately reconstruct these images from the smallest possible set of Fourier measurements. The emergence of Compressed Sensing (CS) has given rise to techniques that can provide exact and stable recovery of sparse images from a relatively small set of Fourier measurements. In particular, if the images are sparse with respect to their gradient, e.g., piece-wise constant, total-variation minimization techniques can be used to recover those images from a highly incomplete set of Fourier measurements. In this paper, we propose a new algorithm to further reduce the number of Fourier measurements required for exact or stable recovery by utilizing prior edge information from a high resolution reference image. This reference image, or more precisely, the fully sampled Fourier measurements of this reference image, is obtained prior to an fMRI study in order to provide approximate edge information for the region of interest. By combining this edge information with CS techniques for sparse gradient images, numerical experiments show that we can further reduce the number of Fourier measurements required for exact or stable recovery by an additional factor of 1.6 − 3 , compared with CS techniques alone, without edge information

Untrained neural network embedded Fourier phase retrieval from few measurements

Fourier phase retrieval (FPR) is a challenging task widely used in various applications. It involves recovering an unknown signal from its Fourier phaseless measurements. FPR with few measurements is important for reducing time and hardware costs, but it suffers from serious ill-posedness. Recently, untrained neural networks have offered new approaches by introducing learned priors to alleviate the ill-posedness without requiring any external data. However, they may not be ideal for reconstructing fine details in images and can be computationally expensive. This paper proposes an untrained neural network (NN) embedded algorithm based on the alternating direction method of multipliers (ADMM) framework to solve FPR with few measurements. Specifically, we use a generative network to represent the image to be recovered, which confines the image to the space defined by the network structure. To improve the ability to represent high-frequency information, total variation (TV) regularization is imposed to facilitate the recovery of local structures in the image. Furthermore, to reduce the computational cost mainly caused by the parameter updates of the untrained NN, we develop an accelerated algorithm that adaptively trades off between explicit and implicit regularization. Experimental results indicate that the proposed algorithm outperforms existing untrained NN-based algorithms with fewer computational resources and even performs competitively against trained NN-based algorithms

Stable image reconstruction using total variation minimization

This article presents near-optimal guarantees for accurate and robust image recovery from under-sampled noisy measurements using total variation minimization. In particular, we show that from O(slog(N)) nonadaptive linear measurements, an image can be reconstructed to within the best s-term approximation of its gradient up to a logarithmic factor, and this factor can be removed by taking slightly more measurements. Along the way, we prove a strengthened Sobolev inequality for functions lying in the null space of suitably incoherent matrices.Comment: 25 page

High-quality Image Restoration from Partial Mixed Adaptive-Random Measurements

A novel framework to construct an efficient sensing (measurement) matrix, called mixed adaptive-random (MAR) matrix, is introduced for directly acquiring a compressed image representation. The mixed sampling (sensing) procedure hybridizes adaptive edge measurements extracted from a low-resolution image with uniform random measurements predefined for the high-resolution image to be recovered. The mixed sensing matrix seamlessly captures important information of an image, and meanwhile approximately satisfies the restricted isometry property. To recover the high-resolution image from MAR measurements, the total variation algorithm based on the compressive sensing theory is employed for solving the Lagrangian regularization problem. Both peak signal-to-noise ratio and structural similarity results demonstrate the MAR sensing framework shows much better recovery performance than the completely random sensing one. The work is particularly helpful for high-performance and lost-cost data acquisition.Comment: 16 pages, 8 figure