目的 针对低质量浅浮雕表面的噪声现象，提出一种二次联合局部自适应稀疏表示和非局部低秩矩阵近似的浅浮雕优化算法。方法 本文方法分两个阶段。第1阶段，将浅浮雕灰度图划分成大小相同的数据块，提取边界块并进行去噪，分别对数据块进行稀疏表示和低秩近似处理。一方面，通过字典学习获得过完备字典和稀疏编码；另一方面，利用K均值聚类算法（K-means）将事先构建的外部字典库划分成k类，从k个簇中心匹配每个数据块的相似块并组成相似矩阵，依次进行低秩近似和特征增强处理。最后通过最小二乘法求解，重建并聚合新建数据块以得到新的高度场。第2阶段与第1阶段的结构相似，主要区别在于改用重建高度场的非局部自身相似性来实现块匹配。结果 在不同图像压缩率下（70%，50%，30%），对比本文方法与BM3D（block-matching and 3D filtering）、WNNM（weighted nuclear norm minimization）、STROLLR（sparsifying transform learning and low-rank）、TWSC（trilateral weighted sparse coding）4个平滑降噪方法的浅浮雕重建结果，发现BM3D和STROLLR方法的特征保持虽好，但平滑效果较差，WNNM方法出现模型破损现象，TWSC方法的平滑效果比BM3D和STROLLR方法更好，但特征也同时被光顺化。阴影恢复形状法（shape from shading，SFS）是一种基于图像的3D建模法，但是其重建结果比较粗糙。相比之下，本文方法生成的浅浮雕模型更加清晰直观，在浅浮雕的特征增强和平滑去噪方面都展现出更好的性能。结论 本文综合数据块的局部稀疏性和数据块之间的非局部相似性对粗糙的浅浮雕模型进行二次高度场重建。本文方法有效改善了现有浅浮雕模型的质量，提高了模型的整体视觉效果，为浅浮雕的优化提供了新方法。

Objective Bas-relief is a semi-stereoscopic sculpture between 2D and 3D space that is typically attached to a plane. It is frequently used to decorate buildings, coins, badges, ceramics, and utensils. It occupies less space and enhances stereo sense mostly through lines. Research on bas-relief is focused on two modeling techniques: 3D models and 2D images. It is performed in three aspects: enhancing the continuity of the height field, preserving the original structure and details, and avoiding the attenuation of the stereoscopic effect. The bas-relief modeling technique based on a 3D model starts from the aspects of spatial, gradient, or normal domains, and compresses the depth values of the 3D model in a given visual direction to generate the bas-relief model. The bas-relief modeling technique based on a 2D image extracts the gray information from the image and converts it into depth information, completing the reconstruction from the image to the bas-relief model. However, minimal attention has been given to the repair and optimization of existing bas-reliefs. Many low-quality bas-relief models exist in practice. Bas-relief models frequently appear as grayscale images during storage or transmission, such as lossy compressed JPG (joint photographic experts group) images, to reduce file size or protect the work. The quality of bas-reliefs directly transformed from such lossy compressed images is rough. The model surfaces present evident block distribution and boundary noise, which seriously reduce the overall visual effect. Moreover, the greater the degree of image compression is, the more obvious the model noise will be. The lossy compression of grayscale image reduces the quality and overall visual effect of corresponding bas-relief model. Considering the large number of bas-relief grayscale images on the network, how to recover high-quality bas-relief models is a problem that is worthy of further study. This study proposes a novel algorithm for bas-relief optimization based on local adaptive sparse representation and nonlocal group low-rank approximation. Method The algorithm has two stages. In the first stage, the initial grayscale image is divided into patches of the same size, and the edge patches are extracted for denoising. Then, patches are processed via sparse representation and low-rank approximation. On the one hand, the edge-denoised patches are trained using the K-SVD (k-singular value decomposition) dictionary learning algorithm to obtain an overcomplete dictionary that is used to sparsely decompose noisy patches to obtain the corresponding sparse coding. On the other hand, the K-means clustering algorithm is used to classify the external dictionary library constructed in advance into k classes. A similar matrix for low-rank approximation is formed by similar patches selected from the k cluster centers. Further feature enhancement processing is performed on the low-rank approximation results. Combined with the results of edge denoising, sparse code reconstruction, and feature enhancement, new patches are generated using a least squares solution to reconstruct a new height field. The second stage is similar to the structure of the first stage, and the primary difference is the block matching operation. In the first stage, although the noisy patches are smoothed and denoised by external clean data, the consistency between patches and similar patches in the external dictionary library is constrained, resulting in insufficient smoothness between the patches of the height field. Therefore, block matching is realized by the nonlocal similarity of the reconstructed height field in the second stage to ensure consistency between patches, thereby improving the smoothing effect on bas-reliefs. Result By comparing the bas-relief reconstruction results of the proposed method with those of block-matching and 3D filtering (BM3D), weighted nuclear norm minimization (WNNM), sparsifying transform learning and low-rank (STROLLR), and trilateral weighted sparse coding (TWSC) at different image compression rates (70%, 50%, and 30%), we determine that BM3D and STROLLR exhibit better feature retention, but their smoothing effect is the worst. WNNM presents model breakage. The smoothing effect of TWSC is better than those of BM3D and STROLLR, but features are also smoothed simultaneously, resulting in inconspicuous details. The proposed method is further compared with shape from shading (SFS), which is an image-based 3D model reconstruction method for transforming 2D images into 3D models. Under different illumination conditions, SFS calculates depth information through the shadow and brightness of an image and then combines it with the reflected illumination model to realize the reconstruction of the 3D model. The overall structure shape of the 3D model generated using SFS is reasonable, but the following shortcomings exist. First, the reconstruction results of SFS are insufficiently fine to restore lines and features. Second, the model produced via SFS is influenced by light source, position, and direction. Third, numerous bump grooves are found in the reconstruction model of SFS. With the influences of light and shadow, local areas are either bright or dark. The model looks complicated because of the lines generated by bump grooves. By contrast, the proposed method is clearer, more intuitive, and exhibits better performance in terms of the smoothing, feature enhancement, and edge denoising of bas-reliefs. The experiment compares the operating efficiencies of different methods. The time taken by the four smoothing methods is in ascending order as follows: BM3D, WNNM, TWSC, and STROLLR. The proposed method takes longer than the four aforementioned methods. For a grayscale image of 1 024×1 024 pixels, the average computing time of the proposed method is 1 000 s, which is approximately 1.5 times of STROLLR. The time spent by SFS is only 40%60% that of the proposed method. Although the computing time of the proposed method is longer, its optimization ability for 3D models is greater, and the visual effect is more significant. Conclusion Sparse representation contains the differences between patches, effectively retaining the details of bas-relief. Low-rank approximation contains the correlation between patches and exhibits a good smoothing effect. The proposed method achieves the complementarity of sparse representation and low-rank approximation, and comprehensively considers the local sparsity of patches and nonlocal similarity between patches. The experiment shows that the proposed method effectively improves the overall visual effect of bas-relief models and provides a new method for bas-relief optimization.