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This paper considers the convex minimization problem with linear constraints and a separable objective function which is the sum of many individual functions without coupled variables. An algorithm is developed by splitting the augmented Lagrangian function in a parallel way. The new algorithm differs substantially from existing splitting methods in alternating style which require solving the decomposed subproblems sequentially, while it remains the main superiority of existing splitting methods in that the resulting subproblems could be simple enough to have closed-form solutions for such an application whose functions in the objective are simple. We show applicability and encouraging efficiency of the new algorithm by some applications in image processing.Received by the editor February 16, 2012 and, in revised form, December 1, 2012. 2010 Mathematics Subject Classification. Primary 90C06, 90C25, 94A08.

This paper analyzes the cartoon and texture structures to inspect and visualize defective objects in a patterned fabric image. It presents a method of an image decomposition (ID) and solves it by a convex optimization algorithm. Our experimental results on benchmark fabric images are superior to those by other methods.Note to Practitioners-This paper is motivated by an ID method to examine how to novelly represent defective objects and repeated patterns in fabric images. We decompose a fabric image into two components: cartoon structure as defective objects and texture structure as repeated patterns. The ID is optimized by the largest correlation between a given defect-free fabric image and the texture structure of a testing image. Its merit is requiring only one defect-free image to optimize the inspection. The resulting cartoon structure is identified for inspection and visualization. An intensive performance evaluation is conducted on dot-, star-, and box-patterned fabric images and the detection accuracies range from 94.9% 99.6%. This research is beneficial to the practitioners for quality control in textile, ceramics, tile, wallpaper, printed circuit board, and aircraft window industries.

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