The task of partitioning convex shape objects from images is a hot research topic, since this kind of object can be widely found in natural images. The difficulties in achieving this task lie in the fact that these objects are usually partly interrupted by undesired background scenes. To estimate the whole boundaries of these objects, different neural networks are designed to ensure the convexity of corresponding image segmentation results. To make use of well-trained neural networks to promote the performances of convex shape image segmentation tasks, in this paper a new image segmentation model is proposed in the variational framework. In this model, a fuzzy membership function, instead of a classical binary label function, is employed to indicate image regions. To ensure fuzzy membership functions can approximate to binary label functions well, an edge-preserving smoothness regularizer is constructed from an off-the-shelf plug-and-play network denoiser, since an image denoising process can also be seen as an edge-preserving smoothing process. From the numerical results, our proposed method could generate better segmentation results on real images, and our image segmentation results were less affected by the initialization of our method than the results obtained from classical methods.

An image recolorization task aims to generate synthetic color images on computers specifically for fabric color design proposals. To enhance processing efficiency and better preserve the original geometric and textile structures, this work proposes a novel fuzzy pretrained neural-network-based image recolorization architecture for fabric and clothing images, consisting of three phases: image decomposition, image segmentation, and image reconstruction. Intrinsic image components, which are initially estimated using a pretrained neural network VOS-Net, formulate an optimization process to achieve high-quality reflectance and shading images. The plug-and-play denoiser network DRU-Net is integrated to smooth these components. The image segmentation phase, which is grounded in the piecewise constant assumption and fuzzy membership functions, partitions image regions to be recolored through only one computation. Recolored images expressed through analytical solutions are produced using the given color palette. Numerical results demonstrate that, compared with classical model-driven methods, our model-data-driven method shows superior efficiency, achieving a remarkable 34.73% reduction in runtime. In terms of quantitative analysis, the proposed method excels at preserving original texture structures and yarn edges with better performance of Learned Perceptual Image Patch Similarity (LPIPS) and structural similarity (SSIM) scores. Furthermore, the color palette distance as an evaluation metric of recoloring indicates that the proposed method effectively reduces the color distortion and color leakage, and the color presentation are closer to the given color palette.

Color transfer on fabric images has become a prevalent computer-aided tool in the fashion design and textile industry, enabling the automatic generation of simulated recolored images available for computer previews. Many existing variational methods are sourced with color palettes due to their ease to use; however, these approaches are often limited by practical inefficiencies, primarily arising from their reliance on cost-intensive iterative calculations. Moreover, the presence of color bleeding and distortion frequently degenerates the visual quality of their recolored fabric images. In this paper, we propose a novel color transfer variational framework on fabric images that consists of three modules: (1) image decomposition, (2) image segmentation, and (3) image recoloring. The first module constructs a variational Wasserstein optimization to generate independent reflectance and shading images, utilizing outputs from the intrinsic image decomposition network OS-Net for efficient modeling and initialization. Building on the piecewise constant approximation of the reflectance image, the second module formulates a variational model to partition the reflectance image into multiple regions of interest, represented by fuzzy membership functions. For image recoloring, color transfer results are produced by reintegrating the shading image with the recolored reflectance image, modified according to the given color palette. Experiment results demonstrate that, compared to classical fabric color transfer variational methods and recent learning-based approaches, our framework significantly enhances operational efficiency and achieves superior color transfer performance. It effectively eliminates unexpected color bleeding and distortion while preserving the original geometric structures and intricate fabric details in recolored images.

Foreground and background image segmentation plays an important role in the field of image processing. Convex shape priors are introduced in some image segmentation methods to estimate those implicit boundaries of image objects when these convex objects are partly obscured by unrelated surroundings that appeared in images. Classical convex shape prior based image segmentation methods use binary label functions to distinguish different image regions, which are sensitive to model parameter choices and often show poor performance in segmentation results. To solve these problems, in this paper fuzzy membership functions, which can be seen as relaxed forms of typical binary label functions, are chosen to represent different image parts, and a new variational model with respect to fuzzy membership functions is proposed in which a convex shape prior on fuzzy membership functions is designed to guarantee estimated image foregrounds having convex shapes in geometry, while to avoid the indicator function from being over-smoothed, a minimax concave regularization term is introduced to measure the smoothness of fuzzy membership functions. In order to construct an efficient algorithm to obtain desired solutions, the original model we proposed is transformed into an equivalent constrained minimization problem using the variable splitting technology, and then numerical solutions for the equivalent constrained model can be solved by an iteration algorithm which integrates the alternating direction method of multipliers, the fast Fourier transform, and the soft thresholding method. Numerical results demonstrate our proposed image segmentation method can keep estimated foreground regions convex in geometry. Compared with classical image segmentation methods, our proposed method shows more robust in model parameter choices and can obtain better convex shape in image segmentation results.

Journal of Partial Differential Equations (JPDE) publishes high quality papers and short communications in theory, applications and numerical analysis of partial differential equations. Papers addressing new theoretical techniques, novel ideas, and new analysis tools are suitable topics for the journal. JPDE is a peer-reviewed international journal which is founded by Peking University, Institute of Applied Physics and Computational Mathematics, and Zhengzhou University, with an editorial office based in Zhengzhou, China.