A Cartoon-Texture Approach for JPEG JPEG 2000 Decompression Based on TGV and Shear let Transform
Abstract: In this paper, we propose a new artifact-free variation model for JPEG/JPEG 2000 decompression based on a cartoon-texture decomposition scheme. The new animal convolution type regularization associated with total generalized variation (TGV) and shear let transform can reconstruct piecewise smooth images with structured textures well, due to the property of shear let of representing the positions and orientations of singularities, which can be interpreted as the oscillation texture parts. In order to enhance the qualities of reconstructed images, we incorporate an L2 cost functional into the model, then the discretization of such functional can be easily solved by the generic proximal Primal-Dual method. Numerical experiments show that our proposed model is competitive with the learning method-Trainable Nonlinear Reaction Dilution (TNRD) [33, 34] in term of texture preservation, and outperforms the TV-based and TGV-based variation methods. Existing system: In this section, we show the performance of our animal convolution type model for JPEG/JPEG 2000 decompression compared with some existing state-of-the-art
methods. The standard JPEG/JPEG 2000 decompression model, TV based model , TGV of second order , a post-processing with BM3D ltering , and a learning method (TNRD). However, there exists some slight artifacts, for instance, see the closeup of \barbara", the line textures are not connected well and also result some chessboard-like textures which are di_erent from the original image. In contrast, the proposed TGV+Sh, TGV+Sh2 and TGV+Sh0:3 models are able to recover this kind of stripes which are closer to the original. However, our three models result some ghost artifacts due to the shearlet transform, see the boundary between texture part and smooth cartoon part. Furthermore, our models maybe lose the contrast, while one can see that our results are slightly gray compared to the original images, that might be the reason that TNRD outperforms the proposed methods in term of the evaluation standards, see Tables 1 and 2. As ones see, our cartoon/texture models can obtain higher evaluation values than TV and TGVbased models. Proposed system: Numerical experiments show that our proposed model is competitive with the learning method-Trainable Nonlinear Reaction Disunion (TNRD) in term of texture preservation, and outperforms the TV-based and TGV-based variation methods. Subsequently, Bredies and Holler gave a theoretical analysis for this continuous JPEG decompression with TV regularizer in an innate dimensional setting in], moreover, they proposed a high-order decompression model associated with TGV regularizer to reduce staircase e_ects in , which also can be extended to JPEG 2000 decompression and zooming applications. Advantages: In order to speed up the computation, we do the projection un 2 U for every 5 iterations, as shown in Figure 7(b), then TGV+SH will take about 40 seconds to obtain the same PSNR as the former case. Indeed, this is not a precise approach to speed up for practical usage, and how to reduce the computation burden, e.g., implementation on GPU, is our further work.
It is well known that the continuous wavelet transform can identify the set of singularities of a function u, recently, some generalizations of the wavelet transform are useful to capture extra details of these singularities, e.g., shear let transform, which is able to exactly identify the location and orientation of edges of planar objects. That is because the latter model reduces the projection interval to be smaller such that the reconstruction is closer to the original standard JPEG result. Focusing on the texture parts, the TV and TGV models are not able to preserve textures which are designed for piecewise constant and piecewise smooth images, respectively, and do not speci_cally take textures into account. Disadvantages: One can see that its frequency support is a pair of trapezoids which are symmetric with respect to the origin, oriented along a line of slope s. This is called bandlimited system and the drawback is that its support becomes increasingly thin as a ! In order to overcome this disadvantage, ones proposed the cone-adapted system, and we refer readers to for more details. Due to its outstanding property, the regularization has been widely applied in many image processing problems, such as linear inverse problems , tensor image problems , image decompression , MRI reconstruction and so on. In this paper we just employ the second order TGV (TGV2) in our applications to model the cartoon component of an image which can be regarded as piecewise smooth. We assume R2 in the following of the paper. for u 2 L1( ;RL), the vectorvalued version of second orderTGV is assigned by the following functional.
The shear let can detect the positions and orientations of singularities, which implies that it can represent the edge-like textures in 2D image. As done in scarcity problem, the L1 norm of shear let transform in the functional can also regularize the texture to be sparse. One can see that our model can recover the texture parts of an image well compared to the learning TNRD method. Modules:
Jpeg decompression: Where A is a Riesz basis transformation operator and Ji represent nonempty closed intervals. In the application of JPEG decompression, A is the block-wise cosine transform (for JPEG 2000, A is the wavelet transform), and intervals Ji are the sets of possible source images associated with quantized coe_cients. Notice that the regularizer J is responsible for the qualitative properties of the solutions, therefore it is signi_cant to be chosen appropriately. In , the authors employed total variation (TV) to approximate the image as piecewise constant to reduce the artifact e_ects. Subsequently, Bredies and Holler gave a theoretical analysis for this continuous JPEG decompression with TV regularizer in an innate dimensional setting in , moreover, they proposed a high-order decompression model associated with TGV regularizer to reduce staircase e_ects in , which also can be extended to JPEG 2000 decompression and zooming applications. Shearlet transform: It is well known that the continuous wavelet transform can identify the set of singularities of a function u, recently, some generalizations of the wavelet transform are useful to capture extra details of these singularities, e.g., shear let transform, which is able to exactly identify the location and orientation of edges of planar objects. However, our three models result some ghost artifacts due to the shear let transform, see the boundary between texture part and smooth cartoon part. Furthermore, our models maybe lose the contrast, while one can see that our results are slightly gray compared to the original images, that might be the reason that TNRD outperforms the proposed methods in term of the evaluation standards, see Tables 1 and 2. As ones see, our cartoon/texture models can obtain higher evaluation values than TV and TGV-based models. Moreover, the model with L2 norm performs better than TGV+Sh in PSNR and SSIM values, but it is still worse than TGV+Sh.