Harris(1988)Forstner(1986)和Harris and Stephens(198

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详细说明：Forstner(1986)和Harris and Stephens(1988)是第一个提出使用自相关矩阵导出的旋转不变标量测量的局部最大值来定位关键点以达到匹配系数特征目的的AUTO-CORRELATION DETECTOR The solution to this problem is to attempt to detect both edges and corners in the image: junctions would then The performance of Moravec's corner detector on a test consist of edges meeting at corners. To pursue this image is shown in Figure 4a; for comparison are shown approach, we shall start from Moravec's corner detector. the results of the Beaudet? and Kithen Rosenfeld& operators(Figures 4b and 4c respectively). The Moravec operator suffers from a number of problems; these are MORAVEC REVISITED listed below, together with appropriate corrective measures Moravecs comer detector functions by considering a local window in the image, and determining the average changes 1. The response is anisotropic because only a of image intensity that result from shifting the window by discrete set of shifts at every 45 degrees is a small amount in various directions. Three cases need to considered-all possible small shifts can be covered by be considered: performing an analytic expansion about the shift origin A. If the windowed image patch is flat (ie. approximately ∑ constant in intensity), then all shifts will result in only X, u,vLx+u, y+v-u,Y a small change B If the window straddles an edge, then a shift along the ∑ u,v XX+yY+O( (x,y edge will result in a small change, but a shift Uv perpendicular to the edge will result in a large change: where the first gradients are approximated by C If the windowed patch is a corner or isolated point, then all shifts will result in a large change. A corner can X=I②(-1,0,1)=IGx thus be detected by finding when the minimum change roduced by any of the shifts is large Y=I(1,0,1)1≈aIy We now give a mathematical specification of the above Denoting the image intensities by I, the change E produced Hence, for small shifts, E can be written by a shift(x, y)is given by E(x,y)=Ax2+2Cxy+By2 x,y ∑ u,vfU,y+vu,v where u, A=X2②w where w specifies the image window: it is unity within a B=y2 w specified rectangular region, and zero elsewhere. The shifts C=(XY)②w (x, y), that are considered comprise ((1,0),(1, 1),(0, 1), (-1, 1)). Thus Moravec's corner detector is simply this look for local maxima in min(E) above some threshold 2. The response is noisy because the window is binary and rectangular -use a smooth circular window, for example a Gaussian u y=exp-(u+v")/2 3. The operator responds too readily to edges because only the minimum of E is taken into account- reformulate the corner measure to make use of the variation of E with the direction of shift The change, E, for the small shift(x, y) can be concisely written as (x,y)=(x,y)M(x,y) where the 2x2 symmetric matrix M is A C M Note that E is closely related to the local autocorrelation function, with M describing its shape at the origin (explicitly, the quadratic terms in the Taylor expansion). Figure 4. Corner detection on a test image Let oz,B be the eigenvalues of M. a and B will be proportional to the principal curvatures of the local auto 149 correlation function, and form a rotationally invariant Consider the graph of (a, B) space. An ideal edge will have description of M. as before, there are three cases to be considered a large and B zero(this will be a surface of translation) but in reality B will merely be small in comparison to a A. If both curvatures are small. so that the local auto due to noise, pixellation and intensity quantisation. A correlation function is flat, then the windowed image comer will be indicated by both a and B being large, and a region is of approximately constant intensity (ie arbitrary shifts of the image patch cause little change in flat image region by both a and B being small, Since an E); increase of image contrast by a factor of p will increase C and p proportionately by p, then if (a, B)is deemed to B. If one curvature is high and the other low, so that the belong in an edge region, then so should(ap2 Bp2),for local auto-correlation function is ridge shaped, then only shifts along the ridge (ie. along the edge) cause positive values of p. Similar considerations apply to little change in E: this indicates an edge, corners. Thus(or, B)space needs to be divided as shown b the heavy lines in Figure 5 C. If both curvatures are high, so that the local auto correlation function is sharply peaked, then shifts in CORNER/EDGE RESPONSE FUNCTION any direction will increase E: this indicates a comer Not only do we need corner and edge classification regions SO-response contours but also a measure of corner and edge quality or response. The size of the response will be used to select isolated corner pixels and to thin the edge pixels Let us first consider the measure of corner response, R which we require lo be a function of a and B alone, on grounds of rotational invariance. It is attractive to use Tr(M) and Det(M in the formulation, as this avoids the explicit eigenvalue decomposition of M, thus T(M)=O+B= A+R 0 2 DeM=c阝=AB-C Consider the following inspired formulation for the cormer response regon R=Det-k Tr C Contours of constant r are shown by the fine lines in Figure 5. Auto-correlation principal curvature space Figure 5. R is positive in the corner region, negative in heavy lines give corner//ffat classification, the edge regions, and small in the flat region. Note that fine lines are equi-response contours increasing the contrast(ie. moving radially away from the 飞 b Figure 6. Edge/corner classification for the outdoor images (grey corner regions, while= thinned edges) Figure 7. Compieted edges for the outdoor images (white corners, black edges origin) in all cases increases the magnitude of the response. The flat region is specified by Tr falling below (the comparison of comer operators, Figure 4) obtained under Mod contract. The grey-level images used in this some selected threshold papcr are subject to the following copyright: Copyright C Controller hmso london 1988 A corner region pixel (ie. one with a positive response)is selected as a nominated corner pixel if its response is an 8 way local maximum: corners so detected in the test image REFERENCES are shown in Figure 4d. Similarly, edge region pixels are deemed to be edgels if their responses are both negative and local minima in either the x or y directions, according to Harris, C G & J M Pike, 3D Positional Integration whether the magnitude of the first gradient in the x or y from Image Sequences, Proceedings third alvey vision direction respectively is the larger. This results in thin Conference(AVC87), pp 233-236, 1987; reproduced in edges. The raw edge/corner classification is shown in Image and Vision Computing, vol 6, no 2, pp. 87-90, May 1988 Figure 6, with black indicating corner regions, and grey, the thinned edges 2. Charnley, D&r j Blissett, Surface Reconstruction By applying low and high thresholds, edge hysteresis can from Outdoor Image Sequences, Proceedings fourth Alvey vision Club(AvC88, 1988 be carried out, and this can enhance the continuity of edges. These classifications thus result in a 5-level image 3. Stephens, M J C g Harris, 3D Wire-frame comprising: background, two carner classes and two edge classes. Further processing (similar to junction Integration from Image Sequences, Proceedings fourth completionwill delete edge spurs and short isolated ed Alvey Vision Club(AVC88), 1988 and bridge short breaks in edges. This results 4. Ayache, N F lustman, Fast and Reliable Passive continuous thin edges that generally terminate in the Trinocular Stereovision, Proccedings first ICCV, 1987 corner regions. The edge terminators are then linked to the corner pixels residing within the corner regions, to form a connected edge-vertex graph, as shown in Figure 7. Note 5. Canny, J F, Finding Edges and Lines in Images, MIT technical report Al-TR-720, 1983 that many of the corners in the bush are unconnected to edges, as they reside in essentially textural regions. 6. Moravcc, H, Obstacle Avoidance and Navigation in the Although not readily apparent from the Figure, many of Real World by a Seeing Robot Rover, Tech Report the corners and edges arc directly matchable. Further work remains to be undertaken concerning the junction CMU-RI-TR-3, Carnegie-Mellon University, Robotics Institute, September 198 completion algorithm, which is currently quite rudimentary, and in the area of adaptive thresholding 7. Beaudet, PR, Rotationally invariant Image Operators International Joint Conference on Pattern Recognition ACKNOWLEDGMENTS pp.579583(1987) The authors gratefully acknowledge the use of imagery 8. Kitchen, L, and A. Rosenfeld, Grey-level Corner supplied by Mr J Sherlock of RSRE, and of the results Detection, Pattern Recognition Letters, 1, pp. 95-102 (1982) 151 152

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