Objective

2

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

2

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

2

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

2

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.