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Mask-based lensless cameras replace the lens of a conventional camera with a custom mask. These cameras can potentially be very thin and even flexible. Recently, it has been demonstrated that such mask-based cameras can recover light intensity and depth information of a scene. Existing depth recovery algorithms either assume that the scene consists of a small number of depth planes or solve a sparse recovery problem over a large 3D volume. Both these approaches fail to recover the scenes with large depth variations. In this paper, we propose a new approach for depth estimation based on an alternating gradient descent algorithm that jointly estimates a continuous depth map and light distribution of the unknown scene from its lensless measurements. We present simulation results on image and depth reconstruction for a variety of 3D test scenes. A comparison between the proposed algorithm and other method shows that our algorithm is more robust for natural scenes with a large range of depths. We built a prototype lensless camera and present experimental results for reconstruction of intensity and depth maps of different real objects.

Mask-based lensless cameras replace the lens of a conventional camera with acustom mask. These cameras can potentially be very thin and even flexible.Recently, it has been demonstrated that such mask-based cameras can recoverlight intensity and depth information of a scene. Existing depth recoveryalgorithms either assume that the scene consists of a small number of depthplanes or solve a sparse recovery problem over a large 3D volume. Both theseapproaches fail to recover the scenes with large depth variations. In thispaper, we propose a new approach for depth estimation based on an alternatinggradient descent algorithm that jointly estimates a continuous depth map andlight distribution of the unknown scene from its lensless measurements. Wepresent simulation results on image and depth reconstruction for a variety of3D test scenes. A comparison between the proposed algorithm and other methodshows that our algorithm is more robust for natural scenes with a large rangeof depths. We built a prototype lensless camera and present experimentalresults for reconstruction of intensity and depth maps of different realobjects.

We consider the problem of sparse coding, where each sample consists of asparse linear combination of a set of dictionary atoms, and the task is tolearn both the dictionary elements and the mixing coefficients. Alternatingminimization is a popular heuristic for sparse coding, where the dictionary andthe coefficients are estimated in alternate steps, keeping the other fixed.Typically, the coefficients are estimated via $\ell_1$ minimization, keepingthe dictionary fixed, and the dictionary is estimated through least squares,keeping the coefficients fixed. In this paper, we establish local linearconvergence for this variant of alternating minimization and establish that thebasin of attraction for the global optimum (corresponding to the truedictionary and the coefficients) is $\order{1/s^2}$, where $s$ is the sparsitylevel in each sample and the dictionary satisfies RIP. Combined with the recentresults of approximate dictionary estimation, this yields provable guaranteesfor exact recovery of both the dictionary elements and the coefficients, whenthe dictionary elements are incoherent.

We consider the problem of sparse coding, where each sample consists of asparse linear combination of a set of dictionary atoms, and the task is tolearn both the dictionary elements and the mixing coefficients. Alternatingminimization is a popular heuristic for sparse coding, where the dictionary andthe coefficients are estimated in alternate steps, keeping the other fixed.Typically, the coefficients are estimated via $\ell_1$ minimization, keepingthe dictionary fixed, and the dictionary is estimated through least squares,keeping the coefficients fixed. In this paper, we establish local linearconvergence for this variant of alternating minimization and establish that thebasin of attraction for the global optimum (corresponding to the truedictionary and the coefficients) is $\order{1/s^2}$, where $s$ is the sparsitylevel in each sample and the dictionary satisfies RIP. Combined with the recentresults of approximate dictionary estimation, this yields provable guaranteesfor exact recovery of both the dictionary elements and the coefficients, whenthe dictionary elements are incoherent.