generalizing the hough transform to detect arbitrary shapes abstract the hough transform is a method for detecting curves by exploiting the duality between points on a curve and parameters of that curve the initial work showed how to detect both analytic curves 1 2 and non analytic curves 3 but these methods were restricted to binary edge images this work was generalized to the detection of some analytic curves in grey level images specifically lines 4 circles 5 and parabolas 6 the line detection case is the best known of these and has been ingeniously exploited in several applications 7 8 9 we show how the boundaries of an arbitrary non analytic shape can be used to construct a mapping between image space and hough transform space such a mapping can be exploited to detect instances of that particular shape in an image furthermore variations in the shape such as rotations scale changes or figure ground reversals correspond to straightforward transformations of this mapping however the most remarkable property is that such mappings can be composed to build mappings for complex shapes from the mappings of simpler component shapes this makes the generalized hough transform a kind of universal transform which can be used to find arbitrarily complex shapes

fast and precise fourier transforms many applications of fast fourier transforms ffts such as computer tomography geophysical signal processing high resolution imaging radars and prediction filters require high precision output an error analysis reveals that the usual method of fixed point computation of ffts of vectors of length 2 sup l leads to an average loss of l 2 bits of precision this phenomenon often referred to as computational noise causes major problems for arithmetic units with limited precision which are often used for real time applications several researchers have noted that calculation of ffts with algebraic integers avoids computational noise entirely we combine a new algorithm for approximating complex numbers by cyclotomic integers with chinese remaindering strategies to give an efficient algorithm to compute b bit precision ffts of length l more precisely we approximate complex numbers by cyclotomic integers in z e 2 spl pi i 2 sup n whose coefficients when expressed as polynomials in e 2 spl pi i 2 sup n are bounded in absolute value by some integer m for fixed n our algorithm runs in time o log m and produces an approximation with worst case error of o 1 m 2 sup n 2 1 we prove that this algorithm has optimal worst case error by proving a corresponding lower bound on the worst case error of any approximation algorithm for this task the main tool for designing the algorithms is the use of the cyclotomic units a subgroup of finite index in the unit group of the cyclotomic field first implementations of our algorithms indicate that they are fast enough to be used for the design of low cost high speed high precision fft chips

tangent bundle elastica and computer vision visual curve completion an early visual process that completes the occluded parts between observed boundary fragments a k a inducers is a major problem in perceptual organization and a critical step toward higher level visual tasks in both biological and machine vision most computational contributions to solving this problem suggest desired perceptual properties that the completed contour should satisfy in the image plane and then seek the mathematical curves that provide them alternatively few studies including by the authors have suggested to frame the problem not in the image plane but rather in the unit tangent bundler 2 s 1 the space that abstracts the primary visual cortex where curve completion allegedly occurs combining both schools here we propose and develop a biologically plausible theory of elastica in the tangent bundle that provides not only perceptually superior completion results but also a rigorous computational prediction that inducer curvatures greatly affects the shape of the completed curve as indeed indicated by human perception

a fast algorithm for solving a toeplitz system of equations a fast algorithm for the solution of a toeplitz system of equations is presented the algorithm requires order n log n 2 computations where n is the number of equations for banded toeplitz matrices the order of computations is reduced to only n log n m log m 2 where 2m is the maximum number of nonzero principal subdiagonals of the toeplitz matrix the algorithm is in essence a fast implementation of the trench algorithm in reverse thus the algorithm involves imbedding of the given matrix in a cyclic matrix and a fast hd half divisor algorithm to compute the first row of the inverse matrix the desired solution is then obtained directly from the first row by applying fast fourier transform techniques in order n log n computations finally the extension of the algorithm to block toeplitz matrices is also presented

complex approximations using algebraic integers the problem of approximating complex numbers by elements of z omega the algebraic integers of q omega where omega is a primitive n th root of unity is considered the motivating application is to reduce the dynamic range requirements of residue number system implementations of the discrete fourier transform smallest error tolerances for the case of eighth roots of unity are derived using a geometric argument scale factors involved are reduced from alpha to sqrt alpha for this case with roughly the same percentage errors the case of sixteenth roots of unity gives even better range reductions and is considered only briefly

pattern recognition by affine moment invariants abstract the paper deals with moment invariants which are invariant under general affine transformation and may be used for recognition of affine deformed objects our approach is based on the theory of algebraic invariants the invariants from second and third order moments are derived and shown to be complete the paper is a significant extension and generalization of recent works several numerical experiments dealing with pattern recognition by means of the affine moment invariants as the features are described

using dynamic programming for solving variational problems in vision dynamic programming is discussed as an approach to solving variational problems in vision dynamic programming ensures global optimality of the solution is numerically stable and allows for hard constraints to be enforced on the behavior of the solution within a natural and straightforward structure as a specific example of the approach s efficacy applying dynamic programming to the energy minimizing active contours is described the optimization problem is set up as a discrete multistage decision process and is solved by a time delayed discrete dynamic programming algorithm a parallel procedure for decreasing computational costs is discussed

fast computation of moment invariants abstract the subject of moment invariants has found wide application in pattern recognition since it was proposed in the present paper we propose a fast algorithm for the calculation of moment invariants firstly we propose using green s theorem for double integral to transform moment calculation to a simple integration along the boundary which reduces the complexity from o n 2 to o n secondly the relation between the monomials for successive pixels along a curve is analysed and the pascal triangle matrix transform is proposed to calculate the monomials for a pixel from those for the preceding one an iterative algorithm is thus proposed for moment calculation which needs no multiplications and the number of additions needed is reduced to o n moreover a very simple systolic structure is proposed to implement the pascal triangle transformation and the proposed algorithm can thus be realized by simple hardware to further accelerate the calculation comparison of the computational complexity with some known methods is also given which shows that our algorithm significantly reduces the complexity

improved moment invariants for shape discrimination abstract moment invariants have been frequently used as features for shape recognition they are computed based on the information provided by both the shape boundary and its interior region although several fast algorithms for computing traditional moment invariants have been proposed none has ever shown the theoretical results of moment invariants computed based on the shape boundary only this paper proposes improved moment invariants computed using the shape boundary only which tremendously reduces computations the new moment invariants called improved moment invariants are mathematically proved to be invariant to scaling translation and rotation graphical plots of the first two improved moment invariants for six country maps and four industrial tools using improved moment invariants are also given the results suggest that improved moment invariants can be used as effective features for shape discrimination or recognition

exact maximum likelihood parameter estimation of superimposed exponential signals in noise a unified framework for the exact maximum likelihood estimation of the parameters of superimposed exponential signals in noise encompassing both the time series and the array problems is presented an exact expression for the ml criterion is derived in terms of the linear prediction polynomial of the signal and an iterative algorithm for the maximization of this criterion is presented the algorithm is equally applicable in the case of signal coherence in the array problem simulation shows the estimator to be capable of providing more accurate frequency estimates than currently existing techniques the algorithm is similar to those independently derived by kumaresan et al in addition to its practical value the present formulation is used to interpret previous methods such as prony s pisarenko s and modifications thereof

simple and fast computation of moments abstract in this paper we address the problem of efficient computation of moments from the boundary of a digital area boundary based computation is superior to usual region based approaches as the data dimension of boundary representations is substantially smaller than that of region representations we investigate the inter order relationship of moments one of our results is that moments of higher order can be computed from moments of lower order based on this relationship a simple iterative algorithm is proposed for the computation of moments from a polygonal approximation of the boundary in comparison with a direct computation method our algorithm is simpler to program the memory requirement is minimum simulation results show that a speed up of factor 8 can be achieved using our algorithm a special version of the algorithm can be utilized to compute moments from the run length chain code of the boundary our algorithm can be applied to compute the most popular geometric moments as well as other types of moments like legendre zernike rotational and complex moments

an algorithm for complex approximations in z e 2 pi i 8 corresp an algorithm is described that approximates complex numbers by elements of the algebraic integers of z e 2 pi i 8 with integer coordinates of at most a prescribed size the motivating application is to reduce the dynamic range requirements of residue number system implementations of the discrete fourier transform the closest points to zero of z e 2 pi i 8 m gor any integer m are determined a particular sequence of such points forms the basis of the algorithm an example of 8 bit z omega m approximations of the 128th roots of unity is considered the algorithm yields m 186 with scaling m is reduced to 18

boundary based corner detection using neural networks abstract in this paper we present a novel boundary based corner detection approach using artificial neural networks anns two neural networks are proposed one for detecting corner points with high curvature and the other for detecting tangent points and inflection points that generally have low curvature for a given boundary point p i the first ann uses the normalized coordinates of points on the forward arm neighboring points succeeding p i or on the backward arm neighboring points preceding p i of the point p i as the input vector the output feature of the network is the angle of the forward arm or backward arm with respect to the x axis the boundary point with sufficiently small angle between the forward and backward arms is identified as a corner since the feature points of tangency and inflection have relatively low curvature the signs of curvature rather than the magnitude of curvature for points in the neighborhood of p i are used as the input vector to the second ann the curvature sign at each boundary point is derived from the outputs of the first ann the outputs of the second ann only respond to the sign patterns of tangent points and inflection points by using both anns all features of corners tangent points and inflection points can be extracted from the boundary of any arbitrary shape experimental results have shown that the proposed anns have good detection and localization for objects in random orientations and with moderate scale changes

an algorithm for pole zero modeling and spectral analysis an explicit connection between fitting exponential models and pole zero models to observed data is made the fitting problem is formulated as a constrained nonlinear minimization problem this problem is then solved using a simplified iterative algorithm the algorithm is applied to simulated data and the performance of the algorithm is compared to previous results

robust automatic selection of optimal views in multi view free form object recognition this paper addresses the issue of automatic selection of the best and the optimum number of representative views for each object in a database that can enable the accurate recognition of that object from any single arbitrary view of the object the object boundary in each view is represented by its curvature scale space css image the css representation has been selected for mpeg 7 standardisation as a contour shape descriptor the paper also presents a novel method for fusion of results from combined shape descriptors the utilisation of this method for multi view three dimensional 3 d object recognition has been explored the object boundary of each view is represented effectively using the css technique moment invariants and fourier descriptors it has been shown that the results obtained from the fusion method are superior to the results obtained from any single technique the method has been tested on a collection of free form 3 d objects each object has been modelled using an optimal number of silhouette contours obtained from different viewpoints this number varies depending on the complexity of the object and the measure of expected accuracy a comprehensive analysis of the performance of the system has been given

wavelet approximation based affine invariant shape representation functions in this paper new wavelet based affine invariant functions for shape representation are presented unlike the previous representation functions only the approximation coefficients are used to obtain the proposed functions one of the derived functions is computed by applying a single wavelet transform the other function is calculated by applying two different wavelet transforms with two different wavelet families one drawback of the previously derived detail based invariant representation functions is that they are sensitive to noise at the finer scale levels which limits the number of scale levels that can be used the experimental results in this paper demonstrate that the proposed functions are more stable and less sensitive to noise than the detail based functions

a new algorithm for line image vectorization in this paper a new approach which differentiates from the conventional thinning algorithms in vectorizing a raster line image called maximal inscribing circle mic is used this region based vectorization algorithm takes the ensemble of pixels within the line segments collectively as legitimate candidates in deciding the vectorized representation an iterative procedure is outlined and the criteria for merging short straight line segments into the corresponding curve representation are described this method cannot only segment the lines and junctions construct their spatial relationships in a computation efficient manner but also retain their line width

computing the discrete fourier transform using residue number systems in a ring of algebraic integers a new method is described for computing an n r m 2 upsilon m point complex discrete fourier transform that uses quantization within a dense ring of algebraic integers in conjunction with a residue number system over this ring the algebraic and analytic foundations for the technique are derived and discussed the architecture for a radix r fast fourier transform algorithm using a residue number system over z omega where omega is a primitive r th root of unity is developed and range and error estimates for this algorithm are derived

classification of nonstationary narrowband signals using segmented chirp features and hidden gauss markov models a method is provided for classifying finite duration signals with narrow instantaneous bandwidth and dynamic instantaneous frequency if in this method events are partitioned into nonoverlapping segments and each segment is modeled as a linear chirp forming a piecewise linear if model the start frequency chirp rate signal energy and noise energy are estimated in each segment the resulting sequences of frequency and rate features for each event are classified by evaluating their likelihood under the probability density function pdf corresponding to each narrowband class hypothesis the class conditional pdfs are approximated using continuous state hidden gauss markov models hgmms whose parameters are estimated from labeled training data previous hgmm algorithms are extended by dynamically weighting the output covariance matrix by the ratio of the estimated signal and noise energies from each segment this covariance weighting discounts spurious features from segments with low signal to noise ratio snr making the algorithm more robust in the presence of dynamic noise levels and fading signals the classification algorithm is applied in a simulated three class cross validation experiment for which the algorithm exhibits percent correct classification greater than 97 as low as 7 db snr

matching shapes with self intersections application to leaf classification we address the problem of two dimensional 2 d shape representation and matching in presence of self intersection for large image databases this may occur when part of an object is hidden behind another part and results in a darker section in the gray level image of the object the boundary contour of the object must include the boundary of this part which is entirely inside the outline of the object the curvature scale space css image of a shape is a multiscale organization of its inflection points as it is smoothed the css based shape representation method has been selected for mpeg 7 standardization we study the effects of contour self intersection on the curvature scale space image when there is no self intersection the css image contains several arch shape contours each related to a concavity or a convexity of the shape self intersections create contours with minima as well as maxima in the css image an efficient shape representation method has been introduced in this paper which describes a shape using the maxima as well as the minima of its css contours this is a natural generalization of the conventional method which only includes the maxima of the css image contours the conventional matching algorithm has also been modified to accommodate the new information about the minima the method has been successfully used in a real world application to find for an unknown leaf similar classes from a database of classified leaf images representing different varieties of chrysanthemum for many classes of leaves self intersection is inevitable during the scanning of the image therefore the original contributions of this paper is the generalization of the curvature scale space representation to the class of 2 d contours with self intersection and its application to the classification of chrysanthemum leaves

full newton and constraint methods for semilinear signal modeling the authors derive an expression for the hessian matrix of the variable projection functional vpf and implement the hessian using qr factorization this is incorporated into a full newton variable projection fnvp algorithm for estimating parameters in semilinear signals they introduce a deflation technique for constraining the vpf to contain known basis vectors for modeling exponential signals the computational cost of the fnvp algorithm is shown to vary linearly with n the size of the data vector while other algorithms vary as n sup 2 or n log sub 2 n

curve fitting approach for tangent angle and curvature measurements abstract the measures of tangent angle and curvature of a digital curve play an important role in image analysis such as corner detection 1d shape representation and shape signature in the generalized hough transform instead of using the discrete measurement approach the least squares method is employed to fit known digital points to two cubic polynomial functions one with y f x that minimizes the sum of the vertical distances and the other with x g y that minimizes the sum of the horizontal distances from the known points to the approximated curve the tangent angle and curvature are therefore directly computable from the first and second order derivatives of the continuous functions hybrid procedures are also proposed to select the better curve from f and g for accurate evaluation of tangent angle and curvature experimented results on both analytic curves and real world images are included

on the detection of dominant points on digital curves a parallel algorithm is presented for detecting dominant points on a digital closed curve the procedure requires no input parameter and remains reliable even when features of multiple sizes are present on the digital curve the procedure first determines the region of support for each point based on its local properties then computes measures of relative significance e g curvature of each point and finally detects dominant points by a process of nonmaximum suppression this procedure leads to the observation that the performance of dominant points detection depends not only on the accuracy of the measure of significance but also on the precise determination of the region of support this solves the fundamental problem of scale factor selection encountered in various dominant point detection algorithms the inherent nature of scale space filtering in the procedure is addressed and the performance of the procedure is compared to those of several other dominant point detection algorithms using a number of examples

fast algorithm for the computation of moment invariants abstract moment invariants have been used as feature descriptors in a variety of object recognition applications when assuming a continuous image function moments calculated using a double integral formulation are invariant to variations in translation rotation and size of the object however due to the recursive nature of the calculations and the limited speed of microprocessors the moments were not computable in real time in this paper we present fast invariant moment computations using the delta method as a means of scene representation this method computes moments of contiguous images by using the contribution of each line rather than individual pixel

affine invariant b spline moments for curve matching the article deals with the problem of matching and recognizing planar curves that are modeled by b splines independently of possible affine transformations to which the original curve has been subjected for example rotation translation scaling orthographic and semiperspective projections and possible occlusion it presents a fast algorithm for estimating the b spline control points that is robust to nonuniform sampling noise and local deformations curve matching is achieved by using a similarity measure based on the b spline knot points introduced by cohen et al 1991 this method however can neither handle the affine transformation between the curves nor the occlusion solutions to these two problems are presented through the use of a new class of weighted b spline curve moments that are well defined for both open and closed curves the method has been applied to classifying affine transformed aircraft silhouettes and appears to perform well

affine curve moment invariants for shape recognition this paper presents a method of affine curve moment invariants for shape recognition the proposed method extends affine moment invariants from an area domain to a curve domain first a new type of curve moments is defined on a parameterized boundary description of an object a set of affine moment invariants are then derived based on the theory of algebraic invariants the proposed moment invariants only deal with pixels on a shape boundary and are mathematically proven to be invariant under arbitrary affine transformations furthermore after a parameterized object boundary description is acquired through curve fitting the newly defined curve moments can be represented by an integration of the fitting function so that the computation of moment invariants can be obtained from either one by one computation of boundary pixels or a direct integration of the fitting function which makes the computation efficient while at the same time reducing noise effect affine curve moment invariants work well in recognition of affine deformed objects which is demonstrated by processing various 2 d shapes under affine transformations

deb definite error bounded tangent estimator for digital curves we propose a simple and fast method for tangent estimation of digital curves this geometric based method uses a small local region for tangent estimation and has a definite upper bound error for continuous as well as digital conics i e circles ellipses parabolas and hyperbolas explicit expressions of the upper bounds for continuous and digitized curves are derived which can also be applied to nonconic curves our approach is benchmarked against 72 contemporary tangent estimation methods and demonstrates good performance for conic nonconic and noisy curves in addition we demonstrate a good multigrid and isotropic performance and low computational complexity of o 1 and better performance than most methods in terms of maximum and average errors in tangent computation for a large variety of digital curves

encoding algorithms for complex approximations in z e sup 2 pi i 8 two algorithms are presented that approximate complex numbers by elements of the algebraic integers of q w where w e sup 2 pi i 8 these algorithms substantially reduce the computational burden range and memory requirements are given as a function of the desired accuracy and expressions are obtained for the number of computations required by each algorithm

a theory of multiscale curvature based shape representation for planar curves a shape representation technique suitable for tasks that call for recognition of a noisy curve of arbitrary shape at an arbitrary scale or orientation is presented the method rests on the describing a curve at varying levels of detail using features that are invariant with respect to transformations that do not change the shape of the curve three different ways of computing the representation are described they result in three different representations the curvature scale space image the renormalized curvature scale space image and the resampled curvature scale space image the process of describing a curve at increasing levels of abstraction is referred to as the evolution or arc length evolution of that curve several evolution and arc length evolution properties of planar curves are discussed

sensor array processing based on subspace fitting algorithms for estimating unknown signal parameters from the measured output of a sensor array are considered in connection with the subspace fitting problem the methods considered are the deterministic maximum likelihood method ml esprit and a recently proposed multidimensional signal subspace method these methods are formulated in a subspace fitting based framework which provides insight into their algebraic and asymptotic relations it is shown that by introducing a specific weighting matrix the multidimensional signal subspace method can achieve the same asymptotic properties as the ml method the asymptotic distribution of the estimation error is derived for a general subspace weighting and the weighting that provides minimum variance estimates is identified the resulting optimal technique is termed the weighted subspace fitting wsf method numerical examples indicate that the asymptotic variance of the wsf estimates coincides with the cramer rao bound the performance improvement compared to the other techniques is found to be most prominent for highly correlated signals

affine invariants of convex polygons in this correspondence we prove that the affine invariants for image registration and object recognition proposed recently by yang and cohen see ibid vol 8 no 7 p 934 46 july 1999 are algebraically dependent we show how to select an independent and complete set of the invariants the use of this new set leads to a significant reduction of the computing complexity without decreasing the discrimination power

polygonal representation of digital planar curves through dominant point detection a nonparametric algorithm we describe a new algorithm that detects a set of feature points on the boundary of an 8 connected shape that constitute the vertices of a polygonal approximation of the shape itself the set of feature points nodes is a ranked subset of the original shape points whose connected left and right arm extents cover the entire shape nodes are ranked based on their strength in terms of their importance to other boundary points length of support region and distance from the centroid the polygon obtained by linking the detected nodes approximates the contour in an intuitive way the proposed algorithm does not require an input parameter and works well for shapes with features of multiple sizes

affine similar shape retrieval application to multiview 3 d object recognition the feasibility of representing a three dimensional 3 d object with a small number of standard views is studied the object boundary of each view is considered as a two dimensional 2 d shape and is represented by the locations of the maxima of its curvature scale space css image contours the idea is to identify an unknown object from an image taken from a random view by using the stored descriptions of the standard views the css image has been selected for mpeg 7 standardization the maxima of css image have already been used to represent 2 d shapes in different applications under similarity transforms since the new application involves affine transforms we first examine the effects of general affine transforms on the representation and show that the locations of the maxima of the css image do not move dramatically even under large affine transformations our system for shape based retrieval from large image databases is then applied to multiview 3 d object representation and recognition our collection of 3 d objects consists of 18 aircraft of different shapes three silhouette contours corresponding to random views are separately used as input for each object results indicate that robust and efficient 3 d free form object recognition through multiview representation can be achieved using the css representation

partial shape classification using contour matching in distance transformation an algorithm is presented to recognize and locate partially distorted 2d shapes without regard to their orientation location and size the algorithm first calculates the curvature function from the digitized image of an object the points of local maxima and minima extracted from the smooth curvature are used as control points to segment the boundary and to guide the boundary matching procedure the boundary matching procedure considers two shapes at a time one shape from the template databank and the other from the object being classified the procedure tries to match the control points in the unknown shape to those of a shape from the template databank and estimates the translation rotation and scaling factors to be used to normalize the boundary of the unknown shape the chamfer 3 4 distance transformation and a partial distance measurement scheme constitute the final step in measuring the similarity between the two shapes the unknown shape is assigned to the class corresponding to the minimum distance the algorithm has been successfully tested on partial shapes using two sets of data one with sharp corners and the other with curve segments this algorithm not only is computationally simple but also works reasonably well in the presence of a moderate amount of noise

observations on oblique projectors and pseudoinverses oblique projectors which were examined by behrens and scharf see ibid vol 42 no 6 p 1413 24 1994 are related to earlier work on oblique pseudoinverses and constrained least squares methods and are reconsidered from the standpoint of orthogonal basis vectors and qr factorizations construction algorithms are presented that are numerically more stable than the normal equations constructions previously used

application of affine invariant fourier descriptors to recognition of 3 d objects the method of fourier descriptors is extended to produce a set of normalized coefficients which are invariant under any affine transformation translation rotation scaling and shearing the method is based on a parameterized boundary description which is transformed to the fourier domain and normalized there to eliminate dependencies on the affine transformation and on the starting point invariance to affine transforms allows considerable robustness when applied to images of objects which rotate in all three dimensions as is demonstrated by processing silhouettes of aircraft maneuvering in three space

a new algorithm for dominant points detection and polygonization of digital curves a new algorithm for detecting dominant points and polygonal approximation of digitized closed curves is presented it uses an optimal criterion for determining the region of support of each boundary point and a new mechanism for selecting the dominant points the algorithm does not require an input parameter and can handle shapes that contain features of multiple sizes efficiently in addition the approximating polygon preserves the symmetry of the shape

on the sensitivity of the hough transform for object recognition object recognition from sensory data involves in part determining the pose of a model with respect to a scene a common method for finding an object s pose is the generalized hough transform which accumulates evidence for possible coordinate transformations in a parameter space whose axes are the quantized transformation parameters large clusters of similar transformations in that space are taken as evidence of a correct match a theoretical analysis of the behavior of such methods is presented the authors derive bounds on the set of transformations consistent with each pairing of data and model features in the presence of noise and occlusion in the image bounds are provided on the likelihood of false peaks in the parameter space as a function of noise occlusion and tessellation effects it is argued that haphazardly applying such methods to complex recognition tasks is risky as the probability of false positives can be very high

scale based description and recognition of planar curves and two dimensional shapes the problem of finding a description at varying levels of detail for planar curves and matching two such descriptions is posed and solved in this paper a number of necessary criteria are imposed on any candidate solution method path based gaussian smoothing techniques are applied to the curve to find zeros of curvature at varying levels of detail the result is the generalized scale space image of a planar curve which is invariant under rotation uniform scaling and translation of the curve these properties make the scale space image suitable for matching the matching algorithm is a modification of the uniform cost algorithm and finds the lowest cost match of contours in the scale space images it is argued that this is preferable to matching in a so called stable scale of the curve because no such scale may exist for a given curve this technique is applied to register a landsat satellite image of the strait of georgia b c manually corrected for skew to a map containing the shorelines of an overlapping area