Optimising Spatial and Tonal Data for PDE-based Inpainting

Some recent methods for lossy signal and image compression store only a few selected pixels and fill in the missing structures by inpainting with a partial differential equation (PDE). Suitable operators include the Laplacian, the biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The quality of such approaches depends substantially on the selection of the data that is kept. Optimising this data in the domain and codomain gives rise to challenging mathematical problems that shall be addressed in our work. In the 1D case, we prove results that provide insights into the difficulty of this problem, and we give evidence that a splitting into spatial and tonal (i.e. function value) optimisation does hardly deteriorate the results. In the 2D setting, we present generic algorithms that achieve a high reconstruction quality even if the specified data is very sparse. To optimise the spatial data, we use a probabilistic sparsification, followed by a nonlocal pixel exchange that avoids getting trapped in bad local optima. After this spatial optimisation we perform a tonal optimisation that modifies the function values in order to reduce the global reconstruction error. For homogeneous diffusion inpainting, this comes down to a least squares problem for which we prove that it has a unique solution. We demonstrate that it can be found efficiently with a gradient descent approach that is accelerated with fast explicit diffusion (FED) cycles. Our framework allows to specify the desired density of the inpainting mask a priori. Moreover, is more generic than other data optimisation approaches for the sparse inpainting problem, since it can also be extended to nonlinear inpainting operators such as EED. This is exploited to achieve reconstructions with state-of-the-art quality. We also give an extensive literature survey on PDE-based image compression methods

Image compression with anisotropic diffusion

Compression is an important field of digital image processing where well-engineered methods with high performance exist. Partial differential equations (PDEs), however, have not much been explored in this context so far. In our paper we introduce a novel framework for image compression that makes use of the interpolation qualities of edge-enhancing diffusion. Although this anisotropic diffusion equation with a diffusion tensor was originally proposed for image denoising, we show that it outperforms many other PDEs when sparse scattered data must be interpolated. To exploit this property for image compression, we consider an adaptive triangulation method for removing less significant pixels from the image. The remaining points serve as scattered interpolation data for the diffusion process. They can be coded in a compact way that reflects the B-tree structure of the triangulation. We supplement the coding step with a number of amendments such as error threshold adaptation, diffusion-based point selection, and specific quantisation strategies. Our experiments illustrate the usefulness of each of these modifications. They demonstrate that for high compression rates, our PDE-based approach does not only give far better results than the widely-used JPEG standard, but can even come close to the quality of the highly optimised JPEG2000 codec

Characterization and adaptive texture synthesis-based compression scheme

International audienceThis paper presents an adaptive texture synthesis-based compression scheme, where textured regions are detected and removed at encoder side, allowing the decoder to use texture synthesis to fill them. The detection relies on locally adaptive resolution segmentation. According to results shown by synthesis algorithms, they need to be parameterized according to the patterns to be synthesized. In this framework, the synthesizer gets its parameters from DCT feature-based texture descriptors. An adaptive pixel-based algorithm is used, relying on the comparison between current pixel neighborhood and those in an atypically shaped sample. Different neighborhood sizes are considered to better match texture patterns. The framework has been validated within an H.264/AVC video codec. Experimental results show significant bit-rate saving at similar visual quality

A total variation regularization based super-resolution reconstruction algorithm for digital video

Super-resolution (SR) reconstruction technique is capable of producing a high-resolution image from a sequence of low-resolution images. In this paper, we study an efficient SR algorithm for digital video. To effectively deal with the intractable problems in SR video reconstruction, such as inevitable motion estimation errors, noise, blurring, missing regions, and compression artifacts, the total variation (TV) regularization is employed in the reconstruction model. We use the fixed-point iteration method and preconditioning techniques to efficiently solve the associated nonlinear Euler-Lagrange equations of the corresponding variational problem in SR. The proposed algorithm has been tested in several cases of motion and degradation. It is also compared with the Laplacian regularization-based SR algorithm and other TV-based SR algorithms. Experimental results are presented to illustrate the effectiveness of the proposed algorithm.£.published_or_final_versio

Selected Topics in Bayesian Image/Video Processing

In this dissertation, three problems in image deblurring, inpainting and virtual content insertion are solved in a Bayesian framework.;Camera shake, motion or defocus during exposure leads to image blur. Single image deblurring has achieved remarkable results by solving a MAP problem, but there is no perfect solution due to inaccurate image prior and estimator. In the first part, a new non-blind deconvolution algorithm is proposed. The image prior is represented by a Gaussian Scale Mixture(GSM) model, which is estimated from non-blurry images as training data. Our experimental results on a total twelve natural images have shown that more details are restored than previous deblurring algorithms.;In augmented reality, it is a challenging problem to insert virtual content in video streams by blending it with spatial and temporal information. A generic virtual content insertion (VCI) system is introduced in the second part. To the best of my knowledge, it is the first successful system to insert content on the building facades from street view video streams. Without knowing camera positions, the geometry model of a building facade is established by using a detection and tracking combined strategy. Moreover, motion stabilization, dynamic registration and color harmonization contribute to the excellent augmented performance in this automatic VCI system.;Coding efficiency is an important objective in video coding. In recent years, video coding standards have been developing by adding new tools. However, it costs numerous modifications in the complex coding systems. Therefore, it is desirable to consider alternative standard-compliant approaches without modifying the codec structures. In the third part, an exemplar-based data pruning video compression scheme for intra frame is introduced. Data pruning is used as a pre-processing tool to remove part of video data before they are encoded. At the decoder, missing data is reconstructed by a sparse linear combination of similar patches. The novelty is to create a patch library to exploit similarity of patches. The scheme achieves an average 4% bit rate reduction on some high definition videos

Kompresija slike korištenjem B-tree algoritma kodiranja poboljšan modeliranjem podataka Burrows-Wheeler transformacijom

The paper shows that the partial differential based compression framework, Edge Enhancing Diffusion Compression (EEDC) on high compression ratios can come close to or even be better than present compression standard - JPEG2000 thus presenting a novel method for image compression. In this paper EEDC will be enhanced by changing its data coding, i.e. Huffman coding will be changed with an entropy coder accompanied with Burrows-Wheeler transformation and context mixing. Images, graphs and tables show image compression results. The purpose of this article is to examine the effectiveness of the PDEs in image compression and to evaluate it by comparing to cosine and wavelet transform based compression methods.Ovaj rad pokazuje kako je kompresija temeljena na parcijalnim diferencijalnim jednadžbama, tj. EEDC na velikom stupnju kompresije dovoljno dobra ili čak bolja od trenutačnog kompresijskog standarda - JPEG2000 time predstavljajući novu metodu kompresije slika. U ovom radu EEDC je poboljšan tako da je promijenjeno kodiranje podataka svjetline slike, tj. Huffmanovo kodiranje je zamijenjeno entropijskim koderom i Burrows-Wheeler transformacijom s miješanjem konteksta. Slike, grafovi i tablice prikazuju rezultate kompresije slika. Svrha ovog rada je ispitati efikasnost parcijalnih diferencijalnih jednadžbi u kompresiji slike te ju usporediti metodama kompresije koje su bazirane na kosinusnoj i wavelet transformaciji

Encoder-Driven Inpainting Strategy in Multiview Video Compression

In free viewpoint video systems, where a user has the freedom to select a virtual view from which an observation image of the 3D scene is rendered, the scene is commonly represented by texture and depth images from multiple nearby viewpoints. In such representation, there exists data redundancy across multiple dimensions: a single visible 3D voxel may be represented by pixels in multiple viewpoint images (inter-view redundancy), a pixel patch may recur in a distant spatial region of the same image due to self-similarity (inter-patch redundancy), and pixels in a local spatial region tend to be similar (inter-pixel redundancy). It isimportant to exploit these redundancies for effective multiview video compression. Existing schemes attempt to eliminate them via the traditional video coding paradigm of hybrid signal prediction/residual coding; typically, the encoder codes explicit information to guide the decoder to the location of the most similar block along with the signal differential. In this paper, we argue that, given the inherent redundancy in the representation, the decoder can often independently recover missing data via inpainting without explicit directions from encoder, resulting in lower coding overhead. Specifically, after pixels in a reference view are projected to a target view via depth image-based rendering (DIBR) at the decoder, the remaining holes in the target view are filled via an inpainting process in a block-by-block manner. First, blocks are ordered in terms of difficulty-to-inpaint by the decoder. Then, explicit instructions are only sent for the reconstruction of the most difficult blocks. In particular, the missing pixels are explicitly coded via a graph Fourier transform (GFT) or a sparsification procedure using DCT, which leads to low coding cost. For the blocks that are easy to inpaint, the decoder independently completes missing pixels via template-based inpainting. We implemented our encoder-driven inpainting strategy as an extension of High Efficiency Video Coding (HEVC). Experimental results show that our coding strategy can outperform comparable implementation of HEVC by up to 0.8dB in reconstructed image qualit

Connections Between Numerical Algorithms for PDEs and Neural Networks

We investigate numerous structural connections between numerical algorithms for partial differential equations (PDEs) and neural architectures. Our goal is to transfer the rich set of mathematical foundations from the world of PDEs to neural networks. Besides structural insights, we provide concrete examples and experimental evaluations of the resulting architectures. Using the example of generalised nonlinear diffusion in 1D, we consider explicit schemes, acceleration strategies thereof, implicit schemes, and multigrid approaches. We connect these concepts to residual networks, recurrent neural networks, and U-net architectures. Our findings inspire a symmetric residual network design with provable stability guarantees and justify the effectiveness of skip connections in neural networks from a numerical perspective. Moreover, we present U-net architectures that implement multigrid techniques for learning efficient solutions of partial differential equation models, and motivate uncommon design choices such as trainable nonmonotone activation functions. Experimental evaluations show that the proposed architectures save half of the trainable parameters and can thus outperform standard ones with the same model complexity. Our considerations serve as a basis for explaining the success of popular neural architectures and provide a blueprint for developing new mathematically well-founded neural building blocks