An introduction to the Kalman filter

In 1960, R.E. Kalman published his famous paper describing a recursive solution to the discrete-data linear filtering problem. Since that time, due in large part to advances in digital computing, the Kalman filter has been the subject of extensive research and application, particularly in the area of autonomous or assisted navigation. The Kalman filter is a set of mathematical equations that provides an efficient computational (recursive) means to estimate the state of a process, in a way that minimizes the mean of the squared error. The filter is very powerful in several aspects: it supports estimations of past, present, and even future states, and it can do so even when the precise nature of the modeled system is unknown. The purpose of this paper is to provide a practical introduction to the discrete Kalman filter. This introduction includes a description and some discussion of the basic discrete Kalman filter, a derivation, description and some discussion of the extended Kalman filter, and a relatively simple (tangible) example with real numbers & results.

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An Introduction to the Kalman Filter

We present a simple image-based method of generating novel visual appearance in which a new image is synthesized by stitching together small patches of existing images. We call this process image quilting. First, we use quilting as a fast and very simple texture synthesis algorithm which produces surprisingly good results for a wide range of textures. Second, we extend the algorithm to perform texture transfer — rendering an object with a texture taken from a different object. More generally, we demonstrate how an image can be re-rendered in the style of a different image. The method works directly on the images and does not require 3D information.

Image quilting for texture synthesis and transfer

We present a new tool, named DART, for automatically testing software that combines three main techniques: (1) automated extraction of the interface of a program with its external environment using static source-code parsing; (2) automatic generation of a test driver for this interface that performs random testing to simulate the most general environment the program can operate in; and (3) dynamic analysis of how the program behaves under random testing and automatic generation of new test inputs to direct systematically the execution along alternative program paths. Together, these three techniques constitute Directed Automated Random Testing, or DART for short. The main strength of DART is thus that testing can be performed completely automatically on any program that compiles -- there is no need to write any test driver or harness code. During testing, DART detects standard errors such as program crashes, assertion violations, and non-termination. Preliminary experiments to unit test several examples of C programs are very encouraging.

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DART: directed automated random testing

In this paper we describe a new tool for interactive free-form fair surface design. By generalizing classical discrete Fourier analysis to two-dimensional discrete surface signals – functions defined on polyhedral surfaces of arbitrary topology –, we reduce the problem of surface smoothing, or fairing, to low-pass filtering. We describe a very simple surface signal low-pass filter algorithm that applies to surfaces of arbitrary topology. As opposed to other existing optimization-based fairing methods, which are computationally more expensive, this is a linear time and space complexity algorithm. With this algorithm, fairing very large surfaces, such as those obtained from volumetric medical data, becomes affordable. By combining this algorithm with surface subdivision methods we obtain a very effective fair surface design technique. We then extend the analysis, and modify the algorithm accordingly, to accommodate different types of constraints. Some constraints can be imposed without any modification of the algorithm, while others require the solution of a small associated linear system of equations. In particular, vertex location constraints, vertex normal constraints, and surface normal discontinuities across curves embedded in the surface, can be imposed with this technique. CR

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A signal processing approach to fair surface design

\"Some half dozen persons have written technically on combinatory logic, and most of these, including ourselves, have published something erroneous. Since some of our fellow sinners are among the most careful and competent logicians on the contemporary scene, we regard this as evidence that the subject is refractory. Thus fullness of exposition is necessory for accurary; and excessive condensation would be false economy here, even more than it is ordinarily.\

Report on the Programming Language Haskell, A Non-strict, Purely Functional Language.

Quick Check is a tool which aids the Haskell programmer in formulating and testing properties of programs. Properties are described as Haskell functions, and can be automatically tested on random input, but it is also possible to define custom test data generators. We present a number of case studies, in which the tool was successfully used, and also point out some pitfalls to avoid. Random testing is especially suitable for functional programs because properties can be stated at a fine grain. When a function is built from separately tested components, then random testing suffices to obtain good coverage of the definition under test.

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QuickCheck: a lightweight tool for random testing of Haskell programs

"Some half dozen persons have written technically on combinatory logic, and most of these, including ourselves, have published something erroneous. Since some of our fellow sinners are among the most careful and competent logicians on the contemporary scene, we regard this as evidence that the subject is refractory. Thus fullness of exposition is necessory for accurary; and excessive condensation would be false economy here, even more than it is ordinarily."

Report on the programming language Haskell: a non-strict, purely functional language version 1.2

Computer graphics: principles and practice (2nd ed.)

As software becomes more and more complex, it is more and more important to structure it well. Well-structured software is easy to write and to debug, and provides a collection of modules that can be reused to reduce future programming costs. In this paper we show that two features of functional languages in particular, higher-order functions and lazy evaluation, can contribute significantly to modularity. As examples, we manipulate lists and trees, program several numerical algorithms, and implement the alpha-beta heuristic (an algorithm from Artificial Intelligence used in game-playing programs). We conclude that since modularity is the key to successful programming, functional programming oers important advantages for software development.

John Hughes

Papers

Report on the Programming Language Haskell, A Non-strict, Purely Functional Language.

QuickCheck: a lightweight tool for random testing of Haskell programs

Report on the programming language Haskell: a non-strict, purely functional language version 1.2

Computer graphics: principles and practice (2nd ed.)

Why functional programming matters