Wavelet frame based surface reconstruction from unorganized points

The variational techniques (e.g. the total variation based method) are well established and effective for image restoration, as well as many other applications, while the wavelet frame based approach is relatively new and came from a different school. This paper is designed to establish a connection between these two major approaches for image restoration. The main result of this paper shows that when spline wavelet frames of are used, a special model of a wavelet frame method, called the analysis based approach, can be viewed as a discrete approximation at a given resolution to variational methods. A convergence analysis as image resolution increases is given in terms of objective functionals and their approximate minimizers. This analysis goes beyond the establishment of the connections between these two approaches, since it leads to new understandings for both approaches. First, it provides geometric interpretations to the wavelet frame based approach as well as its solutions. On the other hand, for any given variational model, wavelet frame based approaches provide various and flexible discretizations which immediately lead to fast numerical algorithms for both wavelet frame based approaches and the corresponding variational model. Furthermore, the built-in multiresolution structure of wavelet frames can be utilized to adaptively choose proper differential operators in different regions of a given image according to the order of the singularity of the underlying solutions. This is important when multiple orders of differential operators are used in various models that generalize the total variation based method. These observations will enable us to design new methods according to the problems at hand, hence, lead to wider applications of both the variational and wavelet frame based approaches. Links of wavelet frame based approaches to some more general variational methods developed recently will also be discussed.

A new mesh reconstruction scheme for approximating a surface from a set of unorganized 3D points is proposed. The proposed method, called a shrink-wrapped boundary face (SWBF) algorithm, produces the final surface by iteratively shrinking the initial mesh generated from the definition of the boundary faces. SWBF surmounts the genus-0 spherical topology restriction of previous shrink-wrapping-based mesh generation techniques and can be applied to any type of surface topology. Furthermore, SWBF is significantly faster than a related algorithm of Jeong and others, as SWBF requires only a local nearest-point-search in the shrinking process. Our experiments show that SWBF is very robust and efficient for surface reconstruction from an unorganized point cloud.

Digital Image Restoration is a technique used to recover original image information from degraded images. Digital images lose information because of factors like noise, blurring and camera misfocus. Image restoration plays a significant role in improving the quality of pictures obtained from unmanned aircrafts, poor quality video tape of a crime scene and blurred satellite images. In this paper we review how the field of image restoration has evolved over the years, and how the techniques of image restoration are making inroads into various applications, thus improving users perception. The article also portrays recent advances and patents in the field of image restoration. Keywords: Image deblurring, image restoration, classical approaches to image restoration, adaptive approaches, median filter, image inpainting, inverse filtering, patents on image restoration applications

The wavelet frame systems have been widely investigated and applied for image restoration and many other image processing problems over the past decades, attributing to their good capability of sparsely approximating piece-wise smooth functions such as images. Most wavelet frame based models exploit the $l_1$ norm of frame coefficients for a sparsity constraint in the past. The authors in \cite{ZhangY2013, Dong2013} proposed an $l_0$ minimization model, where the $l_0$ norm of wavelet frame coefficients is penalized instead, and have demonstrated that significant improvements can be achieved compared to the commonly used $l_1$ minimization model. Very recently, the authors in \cite{Chen2015} proposed $l_0$-$l_2$ minimization model, where the nonlocal prior of frame coefficients is incorporated. This model proved to outperform the single $l_0$ minimization based model in terms of better recovered image quality. In this paper, we propose a truncated $l_0$-$l_2$ minimization model which combines sparsity, nonlocal and support prior of the frame coefficients. The extensive experiments have shown that the recovery results from the proposed regularization method performs better than existing state-of-the-art wavelet frame based methods, in terms of edge enhancement and texture preserving performance.

We present a multi-level partition of unity algebraic set surfaces (MPU-APSS) for surface reconstruction which can be represented by either a projection or in an implicit form. An algebraic point set surface (APSS) defines a smooth surface from a set of unorganized points using local moving least-squares (MLS) fitting of algebraic spheres. However, due to the local nature, APSS does not work well for geometry editing and modeling. Instead, our method builds an implicit approximation function for the scattered point set based on the partition of unity approach. By using an octree subdivision strategy, we first adaptively construct local algebraic spheres for the point set, and then apply weighting functions to blend together these local shape functions. Finally, we compute an error-controlled approximation of the signed distance function from the surface. In addition, we present an efficient projection operator which makes our representation suitable for point set filtering and dynamic point resampling. We demonstrate the effectiveness of our unified approach for both surface reconstruction and geometry modeling such as surface completion.

In the past few decades, mathematics based approaches have been widely adopted in various image restoration problems; the partial differential equation (PDE) based approach (e.g., the total variation model [L. Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259--268] and its generalizations, nonlinear diffusions [P. Perona and J. Malik, IEEE Trans. Pattern Anal. Mach. Intel., 12 (1990), pp. 629--639; F. Catté et al., SIAM J. Numer. Anal., 29 (1992), pp. 182--193], etc.) and wavelet frame based approach are some successful examples. These approaches were developed through different paths and generally provided understanding from different angles of the same problem. As shown in numerical simulations, implementations of the wavelet frame based approach and the PDE based approach quite often end up solving a similar numerical problem with similar numerical behaviors, even though different approaches have advantages in different applications. Since wavelet frame based and PDE based approaches have all been modeling the same types of problems with success, it is natural to ask whether the wavelet frame based approach is fundamentally connected with the PDE based approach when we trace them all the way back to their roots. A fundamental connection of a wavelet frame based approach with a total variation model and its generalizations was established in [J. Cai, B. Dong, S. Osher, and Z. Shen, J. Amer. Math. Soc., 25 (2012), pp. 1033--1089]. This connection gives the wavelet frame based approach a geometric explanation and, at the same time, it equips a PDE based approach with a time frequency analysis. Cai et al. showed that a special type of wavelet frame model using generic wavelet frame systems can be regarded as an approximation of a generic variational model (with the total variation model as a special case) in the discrete setting. A systematic convergence analysis, as the resolution of the image goes to infinity, which is the key step in linking the two approaches, is also given in Cai et al. Motivated by Cai et al. and [Q. Jiang, Appl. Numer. Math., 62 (2012), pp. 51--66], this paper establishes a fundamental connection between the wavelet frame based approach and nonlinear evolution PDEs, provides interpretations and analytical studies of such connections, and proposes new algorithms for image restoration based on the new understandings. Together with the results in [J. Cai et al., J. Amer. Math. Soc., 25 (2012), pp. 1033--1089], we now have a better picture of how the wavelet frame based approach can be used to interpret the general PDE based approach (e.g., the variational models or nonlinear evolution PDEs) and can be used as a new and useful tool in numerical analysis to discretize and solve various variational and PDE models. To be more precise, we shall establish the following: (1) The connections between wavelet frame shrinkage and nonlinear evolution PDEs provide new and inspiring interpretations of both approaches that enable us to derive new PDE models and (better) wavelet frame shrinkage algorithms for image restoration. (2) A generic nonlinear evolution PDE (of parabolic or hyperbolic type) can be approximated by wavelet frame shrinkage with properly chosen wavelet frame systems and carefully designed shrinkage functions. (3) The main idea of this work is beyond the scope of image restoration. Our analysis and discussions indicate that wavelet frame shrinkage is a new way of solving PDEs in general, which will provide a new insight that will enrich the existing theory and applications of numerical PDEs, as well as those of wavelet frames.

Vector field visualization is an important topic in scientific visualization. Its aim is to graphically represent field data in an intuitively understandable and precise way. Here a new approach based on anisotropic nonlinear diffusion is introduced. It enables an easy perception of flow data and serves as an appropriate scale space method for the visualization of complicated flow patterns. The approach is closely related to nonlinear diffusion methods in image analysis where images are smoothed while still retaining and enhancing edges. An initial noisy image is smoothed along streamlines, whereas the image is sharpened in the orthogonal direction. The method is based on a continuous model and requires the solution of a parabolic PDE problem. It is discretized only in the final implementational step. Therefore, many important qualitative aspects can already be discussed on a continuous level. Applications are shown in 2D and 3D and the provisions for flow segmentation are outlined.

Score-based diffusion methods provide a powerful strategy to solve image restoration tasks by flexibly combining a pre-trained foundational prior model with a likelihood function specified during test time . Such methods are predominantly derived from two stochastic processes: reversing Ornstein–Uhlenbeck, which underpins the celebrated denoising diffusion probabilistic models (DDPM) and denoising diffusion implicit models (DDIM), and the Langevin diffusion process. The solutions delivered by DDPM and DDIM are often remarkably realistic, but they are not always consistent with measurements because of likelihood intractability issues and the associated required approximations. Alternatively, using a Langevin process circumvents the intractable likelihood issue, but usually leads to restoration results of inferior quality. This paper presents a novel and highly computationally efficient image restoration method that carefully embeds a foundational DDPM denoiser within an empirical Bayesian Langevin algorithm, which jointly calibrates key model hyper-parameters as it estimates the model’s posterior mean. Extensive experimental results on three canonical tasks (image deblurring, super-resolution, and inpainting) demonstrate that the proposed approach improves on state-of-the-art strategies both in image estimation accuracy and computing time.

In this paper, we present a new greedy algorithm for surface reconstruction from unorganized point sets. Starting from a seed facet, a piecewise linear surface is grown by adding Delaunay triangles one by one. The most plausible triangles are added first and in such a way as to prevent the appearance of topological singularities. The output is thus guaranteed to be a piecewise linear orientable manifold, possibly with boundary. Experiments show that this method is very fast and achieves topologically correct reconstruction in most cases. Moreover, it can handle surfaces with complex topology, boundaries, and nonuniform sampling.

Compressive total variation for image reconstruction and restoration

A practical generative adversarial network architecture for restoring damaged character photographs

In this paper, we address the problem of curve and surface reconstruction from sets of points. We introduce regular interpolants which are polygonal approximations of planar curves and surfaces verifying a local sampling criterion. Properties of regular interpolants lead to new polygonal reconstruction methods from sets of organized and unorganized points. These methods do not need any parameter of additional information apart from the original points and allow unorganized sets of points to be easily handled.

Variational models with second order regularizers can efficiently overcome the problems of staircasing effects caused by first order models. However, different types of second order regularizers may lead to different properties of feature preserving in restored images. In this paper, we show two variational models with second order regularizers. The first one is bounded Hessian model with Jacobian of normals, which uses bounded Jacobian of image intensity normals as regularizer, it is an extension of classical bounded Hessian (BH) model. The second one is total generalized variation model with Jacobian of normals, which is an extension to total generalized variation (TGV) model by replacing gradients in TGV with normals. The common objective is to improve feature preserving, such as edge, contrast and smoothness preservation. Additionally, their Alternating Direction Method of Multipliers (ADMM) are designed by introducing some proper auxiliary variables, Lagrange multipliers and penalty parameters to decompose the original models into some simple minimization sub-problems to solve. Extensive comparisons demonstrate that the proposed models are superior to the classical models with Hessian and TGV regularizers, especially in edge and corner preservation, smoothness, contrast enhancement. Moreover, the proposed models can be also extended to image inpainting, deblurring, and image enhancement.

This paper introduces a new variational model on CUDA platform for the restoration (deblurring and denoising) of ultrasound image degraded by additive Gaussian noise and blur effect. In the deblurring step, we apply an inverse algorithm with the fast transform approach. In the denoising step, a total variational model (TVM) using second-order partial anisotropic diffusion equations is used. A unique and stable solution for the proposed model is presented in terms of the Euler–Lagrange equation. Later, an accurate numerical approximation is constituted by the finite-difference-based discretization technique and the parameter dependence of the proposed model is also described. To achieve better acceleration with satisfactory performance, the proposed algorithm is properly devised on the CUDA GPU and compared with a sequential execution of the multicore CPU system. Experimental results and quantitative analysis show that our algorithm is efficient to restore the ultrasound image compared to the state-of-the-art restoration methods.

3D surface modeling from curves