APPLICATION OF SPLINE SURFACE PROFILE FILTERS TO SUBPIXEL CONTOUR DECOMPOSITION PROBLEMS

Transcription

1 1 APPLICATION OF SPLINE SURFACE PROFILE FILTERS TO SUBPIXEL CONTOUR DECOMPOSITION PROBLEMS Edgar Reetz and Alexander Schlegel and Maik Schumann and Jörg Bargenda and André Göpfert and Matthias Rückwardt and Gerhard Linß Department of Quality Management, Ilmenau University of Technology, Ilmenau, 98693/Thüringen, Germany This article explains an approach to further enhance detection of straight lines and arcs, using subpixel-precision methods, based on the contour points approach by Schumann 1 using polynomial based filtering methods. Keywords: Digital filter; Geometric primitives; Contour decomposition; Subpixeling 1. Introduction Modern measurement devices aim to reduce user influence by increasing the automation level of their measurement tasks to improve objectivity and accuracy of the measurement results. Targeting automized optical measurement tasks using two dimensional grayscale pictures, image segmentation is a mandatory preliminiary step before the actual measurement operation can be executed. In computer vision and image processing using a constant curvature criterion for contur decomposition is a common approach. 1,2 The idea aims to split complex features into simple primitives. The contour decomposition is necessary for further structural analysis like spatial position, distance and angles between the shape primitives. Many ideas and aproaches were published and implemented in the past, mostly targeting shape analysis and object recognition problems where pixel-scaled precision is sufficient.

2 2 2. State Of The Art The method proposed in this article is based on the segmentation of an objects contour by the contour curvature κ only. Extracting features from a set of contour points, respectively lines and arcs, is based on a simple curvature criterion, where: κ(t) = const. κ(t) = 0 for line segments (1) κ(t) = const. κ(t) 0 for arcs (2) is used to differ between these basic features. Since the input data for the method proposed is not an image but a set of contour points, the curvature κ along the contourlength for discrete contour points t is calculated using: κ(t) = φ t+1 φ t 1. (3) The computation of differences in contour direction utilizes the atan2- function, which advantageously considers the perpendicular case of the arctan-function: φ t = atan2(y,x) (4) arctan( y x ) x > 0 arctan( y x )+π x < 0,y 0 arctan( y x ) π x < 0,y 0 atan2(y, x) = (5) + π 2 x = 0,y 0 π 2 x = 0,y < 0 0 x = 0,y = 0. The data retrieved from images using subpixeling methods based on treshold values or gradients as well as integral methods 3 delivers usually a very noisy signal as could be seen in the computed curvature of the image in figure 1. Robust feature detection needs further pre-processing steps to condition the curve. The approach by Wuescher/Boyer 2 uses the curvature property of the contour as criterion to decompose straight lines and arcs as primitive features. For signal smoothing Wuescher/Boyer proposed a one dimensional gaussian smoothing kernel (equation 6) to decrease the influence of quantization noise and rough object contours (see Wuescher/Boyer 2 ): g(t,σ) = 1 σ 2 2π e t 2σ 2. (6) For measurement purposes this approach is not sufficient, considering that the curvature smoothing has no fixed limits, the process could erase impor-

3 3 Raw Contour Curvature Contour Curvature k(t) Contour Length t Fig. 1. Object contour (left) and the corresponding contour curvature κ based on subpixel precision edge detection methods tant contour information, which could be a considerable element of the measurement task to be executed. Another step in the smoothing process is the pixel correction called blip filtering which brings additional uncertainty in the measuring process. Pixelbased smoothing methods, respectively methods using the pixels arrangement to e.g. normalize the curvature on the basis of π 4-clockwise-smoothing according to the pixelgrid, as Thiemann4 proposed, highly depend on the profiles, could not be applied either, due to the non-equidistantly positioned subpixel-coordinates. Considering nonthreshold based segmentation, significance-measure based methods, proposed by Lowe, 5 West/Rosin 6 and Faber, 7 already have shown weaknesses in segmentation of profiles where the significance measure causes deletion of importantprofilepoints 6 (curvaturepeaks).enhancinginitialsegmentation results of the methods given by Schumann 1 causes further improvements of subpixel data preprocessing, respectively data smoothing, to reduce sensitivity of subsequent processing stages when peak point detection is disturbed by profile roughness and waviness. The sequential method takes a set of contour points as its input data, followed by low pass filtering using moving triangular window averaging, displayed in figure 2. The Problem Setting The contour segmentation by Schumann 1 is based on a signal smoothing procedure using a moving triangular window averaging. Determining the correct window length is a sensitive process. The results generated still are superimposed by certain amount of noise caused by subpixeling and the profiles roughness and waviness. Setting appropriate treshold values for the contour peak point detection itself is another difficult step, solved by using

4 4 Sensor Data Edge Detection and Subpixeling Set of Contour Points with Subpixel Precision Calculate Discrete Contour Curvature Smoothing Data by Low Pass Filtering Further processing stages... Fig. 2. Pre-processing stages of contour processing according to the workflow proposed by Schumann 1 genetic algorithms trained on the basis of representative image material. a Applying that algorithm to image data will lead to fragmented initial segmentation results. The contour fragmentation is displayed in figure 4 on the left hand side. Several postprocessing steps manage recombining contour fragments into longer continous pieces. In the following section different methods for data smoothing are introduced, aiming to enhance first pass segmentation results(initial segmentation results) to initially achieve longer contour segments while taking contour corner points into consideration. 3. Novell aproach The two methods presented here basically differ in their type of input data. To further smoothen the curvature, one approach is to enhance the signal smoothing by combining several filter within a filter cascade (section 3.1). Taking the basic input data (discrete contour points) and smoothening their spatial position according to a filtering criterion, will deliver already smoothened input data for further processing, like curvature computation Curvature Smoothing Since treshold based dection methods were used, attenuating curvature peak points in contrast to the profiles waviness is important. The Savitzky- Golay smoothing filter, in contrast to common moving averaging window a Further modifications and enhancements on curvature peak point detection, contour corner finding and determining appropriate treshold values are not content of this paper.

5 5 methods, is used when higher moments of the curve to be smoothened are the relevant field of interest. When using moving averaging window methods, single outstanding peaks in the curve might be heavily smoothened out by evaluating further data points within the window length for the current data point to be smoothened. The idea of the Savitzky-Golay smoothing filter is to fit a polynomial of degree k to the data points, which is helpful to preserve the curve its characteristic shape. The polynomial coefficients of the filter are choosen according to the best least-squares fit to the points within the window of the window length n. Since the Savitzky-Golay smoothing filter is carried out in the time domain, no additional transformation is necessary. Avoiding curve bias caused by end effects could be done by zero-padding. In figure 3 the Savitzky-Golay filter uses 3rd or ,000 Savitzky Golay f ilter Savitzky Golay f ilter + Moving Triangular Window Averaging Moving Triangular Window Averaging Fig. 3. Curvature smoothing methods in comparison der polynomials (k = 3) at a window length of n = 79, while the moving averaging triangular window uses a window length of n = 59. For both filter, finding an appropriate setup is time consuming and of course higly depends on the number of contour points extracted as well as the contour its state. In figure 3 is shown that the filter cascade of Savitzky-Golay filter and moving triangular window averaging enhances curvature smoothing furthermore in contrast to the averaging window method only. Wavy curve segments caused by rough contour pieces were further flatenend while the

6 6 curve its downturn at the beginning and at the end of the curvature peak were also smoothened, but with the side effect to stretch the peak the peak a little in horizontal direction. The important information of the curvature peak is still in the curve, at its exact location. Applying Savitzky-Golay smoothing filters to raw contour curvature data prior using adjusted triangular smoothing filters could deliver quite satisfyingly smoothened curvature plots (continous straight line segments), respecting curvature peaks and smoothing out profile roughness. Initially Detected Straight Line Segments Missing Line Segments (Gaps) Initially Detected Straight Line Segments Fig. 4. Initial segmentation results: Fragmented straigt line segments by using moving averaging window method only (left picture); Continous straight line segment resulting after applying the smoothing filter cascade of Savitzky-Golay filter and moving averaging triangular window (right picture) 3.2. Contour Smoothing Surface profile filters are used to distinguish the actual surface profile, superimposed by waviness and roughness, as given by Seewig: 8 z(x) = f(x)+w(x)+r(x), (7) where f represents the shape, w stands for waviness and r for roughness, resulting in the surface profile z for each spatial coordinate x. Krystek 9 proposed linear filters as mapping operators, where an input function z(x) will be mapped to an output function w(x) by applying a filter kernel or so called weighting function for smoothing or averaging purpose. Using spline filters, the filtered output is determined by spline functions which consists of piecewise polynomials smoothing the input data along supporting input points. Data representing closed profiles could be smoothened using periodic splines while open profiles will be filtered using non-periodic spline filters. For non-periodic spline filters the filter equation: (1+α 4 Q)w = z, (8)

7 7 where z is the profiles contour points and w is the waviness profile filtered, isused.theparameterqrepresentsan nmatrixwherenisthenumberof discrete points equally to the number of raw profile points. The parameter α is given by: α = 1 2sin( π x λ c ). (9) Applying the spline filter to raw contour data results in relocated spatial coordinates for horizontal and vertical direction, smoothened by the spline filter. The smoothing parameter λ c declares the cutoff-wavelength for the filter to seperate waviness from input profile data. b To avoid loosing corner peaks by spline filtering due to improperly choosen parameter, the average contour point distance: d j = (x i+1 x i ) 2 +(y i+1 y i ) 2 i = 1...n (10) is used to adjust the scale space for smoothing. The point spacing x is determined according to the average contour point distance d using: x = d = 1 n 1 d j. (11) n 1 The recommended ratio for λc x is an interval between , where a good compromise between contour smoothing and shape preservation is achieved. j=1 4. Conclusion Both methods presented above deliver further improvements to the state of the art method. 1 The setup for the filter cascade is a sensitive issue if the target is not only a smoothened curvature curve but also to preserve shape information. The spline filter instead has a higher objectivity since its parameter could be determined by the object to be inspected (produced by milling or turning) or by the parameter scale ratio. Is the scale ratio choosen above 100, the contour will be smoothened to strong and also contains bias, comparing to the original contour data. Scale ratios below 40 perfectly resembles the original contour but are not helpful for curvature computations. b An implementation for computation using MATLAB, based on Krystek, 9 not only for non-periodic spline filters but also for periodic spline filters for closed profiles, could be found in Muralikrishnan/Raja. 10

8 8 Vertical Coordinates y Spatial Coordinates Raw Contour Spline Smoothened Contour Horizontal Coordinates x Vertical Coordinates y Spatial Coordinates Corner Point Raw Contour Spline Smoothened Contour Horizontal Coordinates x Fig. 5. On the left picture it could be seen that the spline filtered profile is nearly identical with the original one. The picture on the right shows the corner 5. Acknowledgement The support of the graduate school on image processing and image interpretation at the University of Technology Ilmenau is greatfully acknowledged. References 1. M. Schumann et al., Extraction of geometrical primitives from a set of contour points, in Proceedings of the 10th International Symposium on Measurement and Quality Control, (JSPE Technical Committee for Intelligent Nano-Measure, Osaka, Japan, September 2010). 2. D. M. Wuescher and K. L. Boyer, Robust contour decomposition using a constant curvature criterion, in IEEE Transactions On Pattern And Machine Intelligence, January O. Kühn, Ein beitrag zur hochauflsenden geometriemessung mit ccdzeilensensoren, dissertation, Technische Universitt Ilmenau, (Ilmenau, 1997). 4. I. Thiemann, Minimierung der Fehlereinflüsse auf die Parameterschätzung von Bildkonturen (Shaker, Aachen, 1992). Phd. thesis, University of Technology Clausthal. 5. D. G. Lowe, Artificial Intelligence 31, 355 (1987). 6. G. L. West and P. L. Rosin, Techniques for segmenting image curves into meaningful descriptions, in IEEE Transactions On Pattern And Machine Intelligence, (Pergamon Press, January 1991). 7. P. Faber, Parameterlose kontursegmentierung, in DAGM-Symposium, J. Seewig, Praxisgerechte Signalverarbeitung zur Trennung der Gestaltabweichungen technischer Oberflächen (Shaker, Aachen, 2000). Phd. thesis, University Hannover. 9. M. Krystek, Discrete linear profile filters, in X. International Colloquium on Surfaces Chemnitz (Germany), eds. M. Dietsch and H. Trumpold (Shaker Verlag, Aachen, 2000). 10. B. Muralikrishnan and J. Raya, Computational Surface and Roundness Metrology (Springer, London, 2009).