Stochastic Partial Differential Equations for Computer Vision with Uncertain Data

The goal of this book was to introduce the reader to the recent advances in the field of uncertainty quantification and error propagation for computer vision, image processing, and image analysis that are based on partial differential equations (PDEs). Our original motivation was the observation that more and more often, measured data in the form of images are being used as part of the simulation pipeline, and that error propagation and uncertainty quantification—starting from images but then moving through the pipeline—are important to understanding the ending result of this scientific or engineering process. To analyze the image processing pipeline, we have presented concepts that enable error propagation to be formulated with a set of basic operations: the idea of stochastic images and corresponding (numerical PDE) operations on those images. From this perspective, one approach commonly used within the engineering literature is to employ Monte Carlo sampling techniques to attempt to quantify the impact of errors and variability on an engineering pipeline. Within the uncertainty quantification world, alternative approaches such as the class of generalized polynomial chaos (gPC) methods, which make assumptions about the ability to approximate the process of interest through polynomial approximations, have gained traction as a way of accelerating the convergence of the error and uncertainty quantification process in a computationally tractable way. In this book, we have shown the results of our exploration of the use of the gPC methodology for image processing and computer vision problems. We have relied on the fact that all the prerequisites of the gPC framework (e.g., finite variance, smoothness in the stochastic space, etc.) are intrinsically satisfied in the image processing context. In particular, we have shown how the generalized polynomial chaos (gPC) approach and its corresponding rules for computation (e.g., sums, products, projections on the gPC space, etc.) lead to straightforward generalizations to image processing and computer vision techniques. We have shown that the gPC methodology, when combined with image processing and computer vision techniques, provides a powerful approach to error propagation and uncertainty quantification, and to accessing system sensitivity.

Error propagation analysis is a consolidated practice to gain insights into error modes and effects that pertain to the activation of faults in software systems. A variety of approaches, such as architecture-based, source code instrumentation and variable tracing, have been proposed so far to address software error propagation analysis. Although valuable, existing approaches entail a substantial degree of system internals’ knowledge, visibility and code manipulation that is not well-suited for real-life production environments. This paper proposes an empirical analysis of error propagation. We specifically address the challenges in using fault data and error events in the logs, which are a convenient byproduct of the system’s execution. The approach puts forth the construction of error reporting graphs. We apply the approach to 2,042 failure data points from two real-world critical systems from the Air Traffic Control domain by a top industry provider. The approach contributes to develop a deep understanding on error modes and propagation paths, which can be leveraged by practitioners to make informed decisions on the placement of error detection mechanisms.

Abstract. This paper describes the analysis of the propagation of positional uncertainty in 3D city models to the uncertainty in the computation of their volumes. Current work related to error propagation in GIS is limited to 2D data and 2D GIS operations, especially of rasters. In this research we have (1) developed two engines, one that generates random 3D buildings in CityGML in multiple LODs, and one that simulates acquisition errors to the geometry; (2) performed an error propagation analysis on volume computation based on the Monte Carlo method; and (3) worked towards establishing a framework for investigating error propagation in 3D GIS. The results of the experiments show that a comparatively small error in the geometry of a 3D city model may cause significant discrepancies in the computation of its volume. This has consequences for several applications, such as in estimation of energy demand and property taxes. The contribution of this work is twofold: this is the first error propagation analysis in 3D city modelling, and the novel approach and the engines that we have created can be used for analysing most of 3D GIS operations, supporting related research efforts in the future.

ABSTRACTSpace–time prism (STP) is an important concept for the modeling of object movements in space and time. An STP can be conceptualized as the result of the potential path of a moving object revolving around in the three-dimensional space. Though the concept has found applications in time geography, research on the analysis and propagation of uncertainty in STPs, particularly under high degree of nonlinearity, is scanty. Based on the efficiency and effectiveness of the moment-design (M-D) method, this paper proposes an approach to deal with nonlinear error propagation problems in the potential path areas (PPAs) of STPs and their intersections. Propagation of errors to the PPA and its boundary, and to the intersection of two PPAs is investigated. Performance of the proposed method is evaluated via a series of experimental studies. In comparison with the Monte Carlo method and the implicit function method, simulation results show the advantages of the M-D method in the analysis of error propagation in STPs.

The central theme is the study of error propagation effects in decision feedback equalizers (DFEs). The thesis contains: a stochastic analysis of error propagation in a tuned DFE; an analysis of the effects of error propagation in a blindly adapted DFE; a deterministic analysis of error propagation through input-output stability ideas; and testing procedures for establishing correct tap convergence in blind adaptation. To a lesser extent, the decision directed equalizer (DDE) is also treated. Characterizing error propagation using finite state Markov process (FSMP) techniques is first considered. We classify how the channel and DFE parameters affect the FSMP model and establish tight bounds on the error probability and mean error recovery time of a tuned DFE. These bounds are shown to be too conservative for practical use and highlight the need for imposing stronger hypotheses on the class of channels for which a DFE may be effectively used. In blind DFE adaptation we show the effect of decision errors is to distort the adaptation relative to the use of a training sequence. The mean square error surface in a LMS type setting is shown to be a concatenation of quadratic functions exposing the possibility of false tap convergence to undesirable DFE parameter settings. Averaging analysis and simulation are used to verify this behaviour on some examples. Error propagation in a tuned DFE is also examined in a deterministic setting. A finite error recovery time problem is set up as an input-output stability problem. Passivity theory is invoked to prove that a DFE can be effectively used on a channel satisfying a simple frequency domain condition. These results give performance bounds which relate well with practice. Testing for false tap convergence in blind adaptation concludes our study. Simple statistic output tests are shown to be capable of discerning correct operation of a DDE. Similar tests are conjectured for the DFE, supported by proofs for the low dimensional cases.

With dependability-, and specifically safety-critical systems becoming more and more open, the importance of the ability to reason about error propagation in an exploratory style is becoming increasingly important. This paper proposes an initial theoretical framework for Error Propagation Space Exploration (EPSE) and outlines the activities it is able to support. The key difference from classic Error Propagation Analysis (EPA) is that all error propagation hypotheses are handled together as a hypothesis set relation. Key aspects of operationalizing the framework are also discussed.

Chapter Eighteen Uncertainty and Sensitivity Issues in Process-based Models of Carbon and Nitrogen Cycles in Terrestrial Ecosystems

Simultaneous localization and tracking (SLAT) in wireless sensor networks (WSNs) involves tracking the mobile target while calibrating the nearby sensor node locations. In practice, localization error propagation (EP) phenomenon will arise, due to the existence of the latest tracking error, target mobility, measurement error, and reference node location errors. In this case, the SLAT performance limits are crucial for the SLAT algorithm design and WSN deployment, and the study of localization EP principle is desirable. In this paper, we focus on the EP issues for the received signal strength-based SLAT scheme, where the measurement accuracy is assumed to be spatial-temporal-domain doubly random due to the target mobility, environment dynamics, and different surroundings at different reference nodes. First, the Cramer-Rao lower bound (CRLB) is derived to unveil both the target tracking EP and the node location calibration EP. In both cases, the EP principles turn out to be in a consistent form of the Ohm's Law in circuit theory. Second, the asymptotic CRLB analysis is then presented to reveal that both EP principles scale with the inverse of sensor node density. Meanwhile, it is shown that, the tracking and calibration accuracy only depends on the expectation of the measurement precision. Third, the convergence conditions, the convergence properties, and the balance state of the target tracking EP and the location calibration EP are examined to shed light on the EP characteristics of the SLAT scheme Finally, numerical simulations are presented to corroborate the EP analysis.

Wavefront sensing and reconstruction finds numerous applications in the field of optical technology. Zonal estimation from the wavefront difference or slope data is an important wavefront reconstruction approach. In this reconstruction method, the wavefront is estimated at specific grid points directly from the wavefront differences by using the least-square method. One of the important sources of error in wavefront estimation process is the detector or CCD centroiding error which may propagate in a basic wavefront estimation process, thereby degrading the performance of the wavefront sensor. Hence, quantification of this error is important as this may be considered as one of the selection parameter of a particular estimation geometry. In the present work, we compute the wavefront difference based (WFDB) error propagation coefficient due to this centroiding error for an improved zonal phase-gradient model which is formally applicable for a Shack-Hartmann (S-H) type sensor and show that the improved model offers a substantial reduction of error propagation. The theoretical error propagation coefficient is shown to have a strong correlation with the experimentally obtained RMS errors for the same model.

The Jacobian–torsor theory is used to provide mathematical models of tolerance analysis, establishing a mathematical relationship between the open loop and closed loop of tolerances. The model allows additional chain of torsors joining them. So it is easy to model new ones. Therefore, the ability to calculate the end tolerance of assembly is useful in error propagation analysis. This paper formulates the error propagation model based on the Jacobian–torsor theory. With this theory, workpiece, fixtures, machine tool, and tool are regarded as an assembly. As a result, error sources can be grouped in a sequential manner and described by the parameters of the small displacement torsor. Moreover, the corresponding error propagation model can consider nonpunctual contacts of the fixture assembly, overcoming the limitation of the 3-2-1 layout. An experiment was conducted to demonstrate the model application and verify the effectiveness of the proposed methodology.

This paper presents an approach and Proof-of-Concept implementation for combined design time error propagation analysis and runtime diagnosis in business process driven systems. We show how error propagation analysis can be made practical in this context with qualitative error propagation notation that is approachable for the domain expert. The method uses models of business processes and their supporting IT infrastructure captured in industry-standard tools. Finite domain constraint solving is used to evaluate system alternatives from a dependability point of view. The systematic generation of event detection rules for runtime diagnosis is also supported. A real life example from the banking domain is used to demonstrate the approach.

Partial differential equation(PDE) plays an important role in the field of image segmentation in biological detection. According to the analysis of some problems existing in the process of current biological detection image segmentation, this research set up a kind of image analysis and processing of biological detection algorithm based on PDE.Through the analysis of PDE in image processing and application situation of development methods, this study constructed a biological detection image segmentation application model based on PDE. The results show that the PDE is a mathematical operation as the foundation, and with an open framework for development mode. PDE is a biological detection very effective image segmentation analysis tool that can solve the image topology in biology of high complexity and noise problems.

Propagation of Uncertainty in Nasal Spray In Vitro Performance Models Using Monte Carlo Simulation: Part II. Error Propagation During Product Performance Modeling

Image processing is one of the rapidly growing technologies with operations that can be applied to image data which may be in the form of a 2D, 3D or 4D signals. It has created tremendous opportunities for mathematical modeling, analysis, and computation. A novel approach in image processing has been developed, known as PDE (Partial Differential Equations) based image processing. The PDE approach is very promising for solving many problems in image processing because it provides new and more intuitive mathematical models, gives better approximations to the Euclidean geometry of the problem, and is supported by efficient discrete numerical algorithms based on difference approximations. In this paper, we have discussed how Euler Lagrange Equation plays a significant role in solving PDE based image processing problems. The typical image processing problems such as denoising and inpainting can be solved using EL equation. The ROF and Tikhnov models of denoising and Total Variation based inpainting has been implemented and a comparative study is done with the conventional methods. We have considered various well-known objective quality evaluation metrics used in image processing in this comparison. The results show that PDE based image processing techniques outperforms the conventional approaches.

In homogeneous charge compression ignition engines, the ignition takes place in the absence of any external source, initiated by the autoignition of the well-mixed in-cylinder charge. The heat loss from the hot combustion gases to the surroundings causes the thermal conditions in the cylinder to deteriorate. These conditions are governed by the interaction of the chemical processes with the temperature and pressure changes in the cylinder. As a result, the heat release rate and the heat transfer inside the combustion chamber play significant roles in the homogeneous charge compression ignition combustion mode. The use of detailed chemical kinetics mechanisms in predictive combustion models increases the simulation time, which makes the utilization of these mechanisms questionable. As a result, reduced mechanisms of smaller sizes are required. However, in developing a reduced chemical kinetics mechanism from the detailed chemical kinetics mechanism, it is not common to consider the role of heat transfer, and this is needed to focus on only the kinetics aspects of homogeneous charge compression ignition. The objective of this work is to investigate the effect of heat transfer (through the boundaries of the combustion chamber) on the development of the reduced mechanisms from the detailed mechanisms. A single-zone combustion model is used to simulate the homogeneous charge compression ignition engine. Insignificant species and reactions of the detailed GRI-Mech 3.0 mechanism are eliminated by employing a two-stage reduction process based on a directed relation graph with error propagation and principal-component analysis methods respectively. Several different operating conditions are considered. The results demonstrate that the reduction process based on the directed relation graph with error propagation is hardly affected by considering the heat transfer. However, taking into account the heat transfer slightly influences the reduction process based on the principal-component analysis. Simulation outcomes from the generated mechanisms under adiabatic conditions and non-adiabatic conditions agree closely with the results obtained from the detailed mechanism. The differences in the crank angle when 50% of heat is released, the peak pressure and the maximum heat release between the original mechanisms and the reduced mechanisms are less than 1° crank angle, 1% and 1% respectively. Therefore, for a chemical mechanism reduction strategy, considering the heat transfer component will lead to only a minor effect on the generated reduced mechanism.