Posteriori Reconstruction and Denoising for Path Tracing

<p>Path tracing is a well-established technique for photo-realistic rendering to simulate light path transport. This method has been widely adopted in visual effects industries to generate high quality synthetic images requiring a large number of samples and a long computation time. Due to the high cost to produce the final output, intermediate previsualization of path tracing is in high demand from production artists to detect errors in the early stage of rendering. But visualizing intermediate results of path tracing is also challenging since the synthesized image with limited samples or improper sampling usually suffers from distracting noise. The ideal solution would be to provide a highly plausible intermediate result in the early stages of rendering, using a small fraction of samples, and apply a posteriori manner to approximate the ground truth. In this thesis, this issue is addressed by providing several efficient posteriori reconstructions and denoising technique for previsualization of pa-th tracing. Firstly, we address the problem for the recovery of the missing values to construct low rank matrices for incomplete images including missing pixel, missing sub-pixel, and multi-frame scenarios. A novel approach utilizing a convolutional neural network which provides fast precompletion for initializing missing values, and subsequent weighted nuclear norm minimization with a parameter adjustment strategy efficiently recovers missing values even in high frequency details. The result shows better visual quality compared to the recent methods including compressed sensing based reconstruction. Furthermore, to mitigate the computation budgets of our new approac-h, we extend our method by applying a block Toeplitz structure forming a low-rank matrix for pixel recovery, and tensor structure for multi-frame recovery. In this manner, the reconstruction time can be significantly reduced. Besides that, by exploiting temporal coherence of multi-frame with a tensor structure, we demonstrate an improvement in the overall recovery quality compared to our previous approach. Our recovery methods provide satisfying solution but still require plen-ty of rendering time at prior stage compared with denoising solutions. Finally, we introduce a novel filter for denoising based on convolutional neural network, to address the problem as conventional denoising approach for rendered images. Unlike a plain CNN that applies fixed kernel size in each layer, we propose a multi-scale residual network with various auxiliary scene features to leverage a new efficient denoising filter for path tracing. Our experimental results show on par or better denoising quality compare to state-of-the-art path tracing denoiser.</p>

Posteriori Reconstruction and Denoising for Path Tracing

<p>Path tracing is a well-established technique for photo-realistic rendering to simulate light path transport. This method has been widely adopted in visual effects industries to generate high quality synthetic images requiring a large number of samples and a long computation time. Due to the high cost to produce the final output, intermediate previsualization of path tracing is in high demand from production artists to detect errors in the early stage of rendering. But visualizing intermediate results of path tracing is also challenging since the synthesized image with limited samples or improper sampling usually suffers from distracting noise. The ideal solution would be to provide a highly plausible intermediate result in the early stages of rendering, using a small fraction of samples, and apply a posteriori manner to approximate the ground truth. In this thesis, this issue is addressed by providing several efficient posteriori reconstructions and denoising technique for previsualization of pa-th tracing. Firstly, we address the problem for the recovery of the missing values to construct low rank matrices for incomplete images including missing pixel, missing sub-pixel, and multi-frame scenarios. A novel approach utilizing a convolutional neural network which provides fast precompletion for initializing missing values, and subsequent weighted nuclear norm minimization with a parameter adjustment strategy efficiently recovers missing values even in high frequency details. The result shows better visual quality compared to the recent methods including compressed sensing based reconstruction. Furthermore, to mitigate the computation budgets of our new approac-h, we extend our method by applying a block Toeplitz structure forming a low-rank matrix for pixel recovery, and tensor structure for multi-frame recovery. In this manner, the reconstruction time can be significantly reduced. Besides that, by exploiting temporal coherence of multi-frame with a tensor structure, we demonstrate an improvement in the overall recovery quality compared to our previous approach. Our recovery methods provide satisfying solution but still require plen-ty of rendering time at prior stage compared with denoising solutions. Finally, we introduce a novel filter for denoising based on convolutional neural network, to address the problem as conventional denoising approach for rendered images. Unlike a plain CNN that applies fixed kernel size in each layer, we propose a multi-scale residual network with various auxiliary scene features to leverage a new efficient denoising filter for path tracing. Our experimental results show on par or better denoising quality compare to state-of-the-art path tracing denoiser.</p>

Learning Predictors from Multidimensional Data with Tensor Factorizations

Statistical machine learning algorithms often involve learning a linear relationship between dependent and independent variables. This relationship is modeled as a vector of numerical values, commonly referred to as weights or predictors. These weights allow us to make predictions, and the quality of these weights influence the accuracy of our predictions. However, when the dependent variable inherently possesses a more complex, multidimensional structure, it becomes increasingly difficult to model the relationship with a vector. In this paper, we address this issue by investigating machine learning classification algorithms with multidimensional (tensor) structure. By imposing tensor factorizations on the predictors, we can better model the relationship, as the predictors would take the form of the data in question. We empirically show that our approach works more efficiently than the traditional machine learning method when the data possesses both an exact and an approximate tensor structure. Additionally, we show that estimating predictors with these factorizations also allow us to solve for fewer parameters, making computation more feasible for multidimensional data.

Analytic tadpole coefficients of one-loop integrals

One remaining problem of unitarity cut method for one-loop integral reduction is that tadpole coefficients can not be straightforward obtained through this way. In this paper, we reconsider the problem by applying differential operators over an auxiliary vector R. Using differential operators, we establish the corresponding differential equations for tadpole coefficients at the first step. Then using the tensor structure of tadpole coefficients, we transform the differential equations to the recurrence relations for undetermined tensor coefficients. These recurrence relations can be solved easily by iteration and we can obtain analytic expressions of tadpole coefficients for arbitrary one-loop integrals.

Fast Prediction of Thermal Data Stream for Direct Laser Deposition Processes Using Network-based Tensor Regression

The objective of this research is to study an effective thermal history prediction method for additive manufacturing (AM) processes using thermal image streams in a layer-wise manner. The need for immaculate integration of in-process sensing and data-driven approaches to monitor process dynamics in AM has been clearly stated in blueprint reports released by various U.S. agencies such as NIST and DoD over the past five years. Reliable physics-based models have been developed to delineate the underlying thermo-mechanical dynamics of AM processes; however, the computational cost is extremely high. We propose a tensor-based surrogate modeling methodology to predict the layer-wise relationship in the thermal history of the AM parts, which is time-efficient compared to available physics-based prediction models. We construct a network-tensor structure for freeform shapes based on thermal image streams obtained in metal-based AM process. Subsequently, we simplify the network-tensor structure by concatenating images to reach layer-wise structure. Succeeding layers are predicted based on antecedent layer using the tensor regression model. Generalized multilinear structure, called the higher-order partial least squares (HOPLS) is used to estimate the tensor regression model parameters. Through proposed method, high-dimensional thermal history of AM components can be predicted accurately in a computationally efficient manner. The proposed thermal history prediction is applied on simulated thermal images from finite element method (FEM) simulations. This shows that the proposed model can be used to enhance their performance alongside simulation-based models.

Numerical Approximation of Poisson Problems in Long Domains

In this paper, we consider the Poisson equation on a “long” domain which is the Cartesian product of a one-dimensional long interval with a (d − 1)-dimensional domain. The right-hand side is assumed to have a rank-1 tensor structure. We will present and compare methods to construct approximations of the solution which have tensor structure and the computational effort is governed by only solving elliptic problems on lower-dimensional domains. A zero-th order tensor approximation is derived by using tools from asymptotic analysis (method 1). The resulting approximation is an elementary tensor and, hence has a fixed error which turns out to be very close to the best possible approximation of zero-th order. This approximation can be used as a starting guess for the derivation of higher-order tensor approximations by a greedy-type method (method 2). Numerical experiments show that this method is converging towards the exact solution. Method 3 is based on the derivation of a tensor approximation via exponential sums applied to discretized differential operators and their inverses. It can be proved that this method converges exponentially with respect to the tensor rank. We present numerical experiments which compare the performance and sensitivity of these three methods.

Multi-view feature selection via sparse tensor regression

In this paper, we propose a sparse tensor regression model for multi-view feature selection. Apart from the most of existing methods, our model adopts a tensor structure to represent multi-view data, which aims to explore their underlying high-order correlations. Based on this tensor structure, our model can effectively select the meaningful feature set for each view. We also develop an iterative optimization algorithm to solve our model, together with analysis about the convergence and computational complexity. Experimental results on several popular multi-view data sets confirm the effectiveness of our model.

Eigenfunction Expansions of Ultradifferentiable Functions and Ultradistributions. III. Hilbert Spaces and Universality

In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our papers (Dasgupta and Ruzhansky in Trans Am Math Soc 368(12):8481–8498, 2016) and (Dasgupta and Ruzhansky in Trans Am Math Soc Ser B 5:81–101, 2018). We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary.

Extension Rules of Newman–Janis Algorithm for Rotation Metrics in General Relativity

Aims: The aim of this study is to extend the formula of Newman–Janis algorithm (NJA) and introduce the rules of the complexifying seed metric. The extension of NJA can help determine more generalized axisymmetric solutions in general relativity.Methodology: We perform the extended NJA in two parts: the tensor structure and the seed metric function. Regarding the tensor structure, there are two prescriptions, the Newman–Penrose null tetrad and the Giampieri prescription. Both are mathematically equivalent; however, the latter is more concise. Regarding the seed metric function, we propose the extended rules of a complex transformation by r2/Σ and combine the mass, charge, and cosmologic constant into a polynomial function of r. Results: We obtain a family of axisymmetric exact solutions to Einstein’s field equations, including the Kerr metric, Kerr–Newman metric, rotating–de Sitter, rotating Hayward metric, Kerr–de Sitter metric and Kerr–Newman–de Sitter metric. All the above solutions are embedded in ellipsoid- symmetric spacetime, and the energy-momentum tensors of all the above metrics satisfy the energy conservation equations. Conclusion: The extension rules of the NJA in this research avoid ambiguity during complexifying the transformation and successfully generate a family of axisymmetric exact solutions to Einsteins field equations in general relativity, which deserves further study.