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In this paper, we aim to segment an image degraded by blur and Poisson noise. We adopt a smoothing-and-thresholding (SaT) segmentation framework that finds a piecewise-smooth solution, followed by k -means clustering to segment the image. Specifically for the image smoothing step, we replace the least-squares fidelity for Gaussian noise in the Mumford-Shah model with a maximum posterior (MAP) term to deal with Poisson noise and we incorporate the weighted difference of anisotropic and isotropic total variation (AITV) as a regularization to promote the sparsity of image gradients. For such a nonconvex model, we develop a specific splitting scheme and utilize a proximal operator to apply the alternating direction method of multipliers (ADMM). Convergence analysis is provided to validate the efficacy of the ADMM scheme. Numerical experiments on various segmentation scenarios (grayscale/color and multiphase) showcase that our proposed method outperforms a number of segmentation methods, including the original SaT. 

In a class of piecewise-constant image segmentation models, we propose to incorporate a weighted difference of anisotropic and isotropic total variation (AITV) to regularize the partition boundaries in an image. In particular, we replace the total variation regularization in the Chan--Vese segmentation model and a fuzzy region competition model by the proposed AITV. To deal with the nonconvex nature of AITV, we apply the difference-of-convex algorithm (DCA), in which the subproblems can be minimized by the primal-dual hybrid gradient method with linesearch. The convergence of the DCA scheme is analyzed. In addition, a generalization to color image segmentation is discussed. In the numerical experiments, we compare the proposed models with the classic convex approaches and the two-stage segmentation methods (smoothing and then thresholding) on various images, showing that our models are effective in image segmentation and robust with respect to impulsive noises. 

Convolutional neural networks (CNN) have been hugely successful recently with superior accuracy and performance in various imaging applications, such as classification, object detection, and segmentation. However, a highly accurate CNN model requires millions of parameters to be trained and utilized. Even to increase its performance slightly would require significantly more parameters due to adding more layers and/or increasing the number of filters per layer. Apparently, many of these weight parameters turn out to be redundant and extraneous, so the original, dense model can be replaced by its compressed version attained by imposing inter- and intra-group sparsity onto the layer weights during training. In this paper, we propose a nonconvex family of sparse group lasso that blends nonconvex regularization (e.g., transformed ℓ1, ℓ1 − ℓ2, and ℓ0) that induces sparsity onto the individual weights and ℓ2,1 regularization onto the output channels of a layer. We apply variable splitting onto the proposed regularization to develop an algorithm that consists of two steps per iteration: gradient descent and thresholding. Numerical experiments are demonstrated on various CNN architectures showcasing the effectiveness of the nonconvex family of sparse group lasso in network sparsification and test accuracy on par with the current state of the art. 

In the last decade, convolutional neural networks (CNNs) have evolved to become the dominant models for various computer vision tasks, but they cannot be deployed in low-memory devices due to its high memory requirement and computational cost. One popular, straightforward approach to compressing CNNs is network slimming, which imposes an ℓ1 penalty on the channel-associated scaling factors in the batch normalization layers during training. In this way, channels with low scaling factors are identified to be insignificant and are pruned in the models. In this paper, we propose replacing the ℓ1 penalty with the ℓp and transformed ℓ1 (T ℓ1 ) penalties since these nonconvex penalties outperformed ℓ1 in yielding sparser satisfactory solutions in various compressed sensing problems. In our numerical experiments, we demonstrate network slimming with ℓp and T ℓ1 penalties on VGGNet and Densenet trained on CIFAR 10/100. The results demonstrate that the nonconvex penalties compress CNNs better than ℓ1 . In addition, T ℓ1 preserves the model accuracy after channel pruning, and ℓ1/2,3/4 yield compressed models with similar accuracies as ℓ1 after retraining.