Algorithms for shape analysis are reviewed and classified under various criteria, whether they examine the boundary only or the whole area and whether they describe the original picture in terms of scalar measurements or through structural descriptions. The emphasis is on methodologies which have been popular during the last five years and among them, those which are information preserving.

Algorithms for Shape Analysis of Contours and Waveforms

Parametric piecewise-cubic functions are used throughout the computer graphics industry to represent curved shapes. For many applications, it would be useful to be able to reliably derive this representation from a closely spaced set of points that approximate the desired curve, such as the input from a digitizing tablet or a scanner. This paper presents a solution to the problem of automatically generating efficient piecewise parametric cubic polynomial approximations to shapes from sampled data. We have developed an algorithm that takes a set of sample points, plus optional endpoint and tangent vector specifications, and iteratively derives a single parametric cubic polynomial that lies close to the data points as defined by an error metric based on least-squares. Combining this algorithm with dynamic programming techniques to determine the knot placement gives good results over a range of shapes and applications.

https://www.cs.bgu.ac.il/~ben-shahar/Teaching/Computational-Vision/Readings/1983-Plass_and_Stone-Curve_Fitting_with_Piecewise_Cubics.pdf

Curve-fitting with piecewise parametric cubics

A method is developed for surface-fitting from sampled data Surface-fitting is the process of constructing a compact representation to model the surface of an object based on a fairly large number of given data points In our case, the data is obtained from a real object using an automatic three-dimensional digitizing system The method is based on an adaptive subdivision approach, a technique previously used for the design and display of free-form curved surface objects Our approach begins with a rough approximating surface and progressively refines it in successive steps in regions where the data is poorly approximated The method has been implemented using a parametric piecewise bicubic Bernstein-Bezier surface possessing G1 geometric continuity An advantage of this approach is that the refinement is essentially local reducing the computational requirements which permits the processing of large databases Furthermore, the method is simple in concept, yet realizes efficient data compression Some experimental results are given which show that the representation constructed by this method is faithful to the original database

An adaptive subdivision method for surface-fitting from sampled data

To obtain a good approximation for data fitting with a spline, frequently we have to deal with knots as variables. The problem to be solved then becomes a continuous nonlinear and multivariate optimization problem with many local optima. Therefore, it is difficult to obtain the global optimum. In this paper, we propose a method for solving this problem by using a real-coded genetic algorithm. Our method can treat not only data with a smooth underlying function, but also data with an underlying function having discontinuous points and/or cusps. We search for the best model among candidate models by using the Bayes Information Criterion (BIC). With this, we can appropriately determine the number and locations of knots automatically and simultaneously. Five examples of data fitting are given to show the performance of our method.

Data fitting with a spline using a real-coded genetic algorithm

Drill bits and methods for sampling sensor data associated with the state of a drill bit are disclosed. A drill bit (200) for drilling a subterranean formation comprises a bit body (240) and a shank (210). The shank further includes a central bore (280) formed through an inside diameter of the shank and configured for receiving a data analysis module (290). The data analysis module comprises a plurality of sensors, a memory, and a processor (300). The processor is configured for executing computer instructions to collect the sensor data by sampling the plurality of sensors, analyze the sensor data to develop a severity index, compare the sensor data to at least one adaptive threshold, and modify a data sampling mode responsive to the comparison. A method comprises collecting sensor data by sampling a plurality of physical parameters associated with a drill bit state while in various sampling modes and transitioning between those sampling modes.

Method and apparatus for collecting drill bit performance data

A method is described for fitting a piecewise cubic polynomial to a sequence of data by a onepass method. The polynomial pieces are calculated as the data is scanned only once from left to right. The algorithm is shown to be stable when a piecewise cubic polynomial is used which is continuous with its first derivative, while it becomes unstable if the polynomial is made continuous up to the second derivative. The knots of the approximating function are determined successively using a criterion by Powell.

Curve Fitting by a One-Pass Method With a Piecewise Cubic Polynomial

A new Monte Carlo technique is applied to solve difference equations of elliptic and parabolic partial differential equations with given boundary values. Fixed random walk is extended to modified random walk, whereby a random walk is made on a maximum square. The average number of steps and the computational time in a modified random walk is much less than in a fixed random walk. Numerical examples support the utility of this method.

Solution of partial differential equations by a modified random walk

This paper deals with curve fitting by a piecewise cubic polynomial which is continuous with its first derivative. A knot is inserted successively until a certain criterion is satisfied. Then a suboptimal algorithm is applied to minimize the sum of squares of residuals.

curve fitting by a piecewise cubic polynomial

A computational technique based on the method of path integral is studied with a view to finding approximate solutions of a class of two-point boundary-value problems. These solutions are "rough" solutions by Monte Carlo sampling. From the computational point of view, however, once these rough solutions are obtained for any nonlinear cases, they serve as good starting approximations for improving the solutions to higher accuracy. Numerical results of a few examples are also shown.

Monte Carlo path-integral calculations for two-point boundary-value problems

The nonlinear interpolation of functions of very many variables is discussed. Deterministic termwise assessment of a prohibitively large number of terms naturally leads to a choice of random sampling from these numerous terms. After introduction of an appropriate higher order interpolation formula, a working algorithm is established by the Monte Carlo method. Numerical examples are also given.

Kozo Ichida

Papers

Curve Fitting by a One-Pass Method With a Piecewise Cubic Polynomial

Solution of partial differential equations by a modified random walk

curve fitting by a piecewise cubic polynomial

Monte Carlo path-integral calculations for two-point boundary-value problems

Nonlinear Interpolation of Multivariable Functions by the Monte Carlo Method