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Orientation estimation is an important task in three-dimensional cryo-EM image reconstruction. By applying the common line method, the orientation estimation task can be formulated as a least squares (LS) problem or a least un-squared deviation (LUD) problem with orthogonality constraint. However, the non-convexity of the orthogonality constraint introduces numerical difficulties to the orientation estimation. The conventional approach is to reformulate the orthogonality constrained minimization problem into a semi-definite programming problem using convex relaxation strategies. In this paper, we consider a direct way to solve the constrained minimization problem without relaxation. We focus on the weighted LS problem because the LUD problem can be reformulated into a sequence of weighted LS problems using the iteratively re-weighted LS approach. As a classical approach, the projected gradient descent (PGD) method has been successfully applied to solve the convex constrained minimization problem. We apply the PGD method to the minimization problem with orthogonality constraint and show that the constraint set is a generalized prox-regular set, and it satisfies the norm compatibility condition. We also demonstrate that the objective function of the minimization problem satisfies the restricted strong convexity and the restricted strong smoothness over a constraint set. Therefore, the sequence generated by the PGD method converges when the initial conditions are satisfied. Experimental results show that the PGD method significantly outperforms the semi-definite relaxation methods from a computation standpoint, and the mean square error is almost the same or smaller.

Parameter selection is crucial to regularization-based image restoration methods. Generally speaking, a spatially fixed parameter for the regularization term does not perform well for both edge and smooth areas. A larger parameter for the regularization term reduces noise better in smooth areas but blurs edge regions, while a small parameter sharpens edge but causes residual noise. In this paper, an automated spatially adaptive regularization model, which combines the harmonic and total variation (TV) terms, is proposed for the image reconstruction from noisy and blurred observation. The proposed model detects the edges and then spatially adjusts the parameters of Tikhonov and TV regularization terms for each pixel according to the edge information. Accordingly, the edge information matrix will also be dynamically updated during the iterations. Computationally, the newly-established model is convex, which can be solved by the semi-proximal alternating direction method of multipliers with a linear convergence rate. Numerical simulation results demonstrate that the proposed model effectively preserves the image edges and eliminates the noise and blur at the same time. In comparison to state-of-the-art algorithms, it outperforms other methods in terms of peak signal to noise ratio, structural similarity index and visual quality.