@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix dc: . @prefix skos: . vivo:departmentOrSchool "Science, Faculty of"@en, "Computer Science, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Ando, Yoko"@en ; dcterms:issued "2009-02-26T00:00:00"@en, "1994"@en ; vivo:relatedDegree "Master of Science - MSc"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """Subjective contours are physically invisible borders drawn on certain images that can nevertheless be seen by humans. This is because the human vision system makes assumptions on the occlusion of objects. The study of subjective contours is important for helping us understand more about the human visual perception. The purpose of this thesis is to understand the perception of subjective contours and to detect subjective contours by computer. The previous subjective contour detection systems limit the subjective contours they can detect by restricting the locations on the figures where the subjective contours can be seen and by using the consistent subjective surface orientation. In this thesis, we consider the overall organization of subjective contours. We do not put the restriction on the subjective surface orientation because we view the subjective contour as a boundary separating the two regions locally. A model for subjective contour detection is presented based on four criteria: no prior knowledge is necessary to detect a subjective contour; a subjective contour is a special type of occluding contour; the shape of a subjective contour is determined by the viewing condition; and it is possible to have many subjective contour organizations from one image. The rules for subjective contour organization are described and the model explains different types of subjective contour organizations. There are three stages in the computer implementation of subjective contour detection. The first stage is preprocessing of figures where the real contours are segmented according to their curvature discontinuities by Lowe's curve partition method. The next stage is local processing in which each real contour segment selects all the potential subjective contours and their connecting real contour segments. The final stage is global processing to organize the real and subjective contours which can be seen at the same time. Many subjective contour images are tested and good results are produced."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/5110?expand=metadata"@en ; dcterms:extent "4227175 bytes"@en ; dc:format "application/pdf"@en ; skos:note "Selection and Organization of Subjective Contours By YOKO ANDO B.Sc, University of Guelph, Canada, 1989 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF COMPUTER SCIENCE We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1994 © Yoko Ando, 1994 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Co/'ipufc 6^> / ' ence The University of British Columbia Vancouver, Canada Date l\\r;L 27. /ff4-DE-6 (2/88) Abstract Subjective contours are physically invisible borders drawn on certain images that can nev-ertheless be seen by humans. This is because the human vision system makes assumptions on the occlusion of objects. The study of subjective contours is important for helping us under-stand more about the human visual perception. The purpose of this thesis is to understand the perception of subjective contours and to detect subjective contours by computer. The previous subjective contour detection systems limit the subjective contours they can detect by restrict-ing the locations on the figures where the subjective contours can be seen and by using the consistent subjective surface orientation. In this thesis, we consider the overall organization of subjective contours. We do not put the restriction on the subjective surface orientation because we view the subjective contour as a boundary separating the two regions locally. A model for subjective contour detection is presented based on four criteria: no prior knowl-edge is necessary to detect a subjective contour; a subjective contour is a special type of oc-cluding contour; the shape of a subjective contour is determined by the viewing condition; and it is possible to have many subjective contour organizations from one image. The rules for sub-jective contour organization are described and the model explains different types of subjective contour organizations. There are three stages in the computer implementation of subjective contour detection. The first stage is preprocessing of figures where the real contours are segmented according to their curvature discontinuities by Lowe's curve partition method. The next stage is local processing in which each real contour segment selects all the potential subjective contours and their connecting real contour segments. The final stage is global processing to organize the real and subjective contours which can be seen at the same time. Many subjective contour images are tested and good results are produced. ii Contents Abstract ii Contents iii List of Tables v List of Figures vi Acknowledgement viii 1 Introduction 1 1.1 Subjective Contour Chaxacteristics 1 1.2 Subjective Contour Classification 4 1.3 Thesis Motivation 8 1.4 Thesis Organization 12 2 Previous Research 13 2.1 Subjective Contour Theories 13 2.2 Image Segmentation 15 2.3 Shape Completion 18 2.4 Subjective Contour Detection Systems 19 3 Detection of Subjective Contours 22 3.1 Assumptions 22 3.2 Occluding Contours 24 3.3 Shape of Subjective Contour 26 3.3.1 Curvature Continuity of a Curve 26 3.3.2 Supporting Edge Type 27 3.3.3 Subjective Contour Type 30 3.4 Contour Organizations 34 i i i 4 Computer Implementation 39 4.1 Preprocessing 39 4.1.1 Extract Features 40 4.1.2 Lowe's Curve Partitioning 41 4.1.3 Ranking Real Edges 43 4.2 Local Subjective Contour Selection 45 4.2.1 Extend Edge 45 4.2.2 Shape of Subjective Contour 46 4.2.3 Weighing Subjective Contour 47 4.3 Global Contour Organization 48 4.3.1 Data Structure 49 4.3.2 Finding a Contour Chain 50 4.3.3 Grouping of Contour Chains 52 5 Results and Discussion 54 5.1 Subjective Contours with a White Subjective Surface 55 5.2 Subjective Contours with Patches on a Subjective Surface 61 5.3 Subjective Contours Induced by Bars 67 5.4 Subjective Contours with Overlaid Subjective Surfaces 73 6 Conclusions 82 Bibliography 87 A Sources of Figures 92 B Bezier Curve 94 iv List of Tables 3.1 Decision of Current Contour Chain Continuation 37 4.1 Decision of Endpoint Table and Link Table Updates 53 v List of Figures 1.1 Schumann's Figure 2 1.2 Kanizsa Triangle 2 1.3 Edge-based Subjective Contours 5 1.4 Tip-based Subjective Contours 7 1.5 An Example of Test Results 11 3.1 Two Interpretations on a Tip-Based Subjective Contour Image 23 3.2 T-junction on a Subjective Surface and an Inducing Element 25 3.3 Breakpoints and Endpoints on the Blob Outline 27 3.4 Supporting Edge Candidates 28 3.5 Separation Between Supporting Edges 31 3.6 Subjective Contour Type 33 3.7 Organization of Contour Chains 36 4.1 Three Figure Types 40 4.2 Concave and Convex Corner of a Blob 43 4.3 a and /3 Angles 47 5.1 Test Results of Subjective Contours with a White Subjective Surface (1) 56 5.2 Test Results of Subjective Contours with a White Subjective Surface (2) 57 5.3 Test Results of Subjective Contours with a White Subjective Surface (3) 57 5.4 Test Results of Subjective Contours with a White Subjective Surface (4) 58 5.5 Test Results of Subjective Contours with a White Subjective Surface (5) 59 5.6 Test Results of Subjective Contours with a White Subjective Surface (6) 60 5.7 Test Results of Subjective Contours with a White Subjective Surface (7) 60 5.8 Test Results of Subjective Contours with Patches on a Subjective Surface (1) . . 62 5.9 Test Results of Subjective Contours with Patches on a Subjective Surface (2) . . 63 5.10 Test Results of Subjective Contours with Patches on a Subjective Surface (3) . . 64 5.11 Test Results of Subjective Contours with Patches on a Subjective Surface (4) . . 65 5.12 Test Results of Subjective Contours with Patches on a Subjective Surface (5) . . 66 5.13 Test Results of Subjective Contours Induced by Bars (1) 69 5.14 Test Results of Subjective Contours Induced by Bars (2) 70 vi 5.15 Test Results of Subjective Contours Induced by Bars (3) 71 5.16 Test Results of Subjective Contours Induced by Bars (4) 72 5.17 Test Results of Subjective Contours with Overlaid Subjective Surfaces (1) . . . . 74 5.18 Test Results of Subjective Contours with Overlaid Subjective Surfaces (2) . . . . 75 5.19 Test Results of Subjective Contours with Overlaid Subjective Surfaces (3) . . . . 76 5.20 Test Results of Subjective Contours with Overlaid Subjective Surfaces (4) . . . . 77 5.21 Test Results of Subjective Contours with Overlaid Subjective Surfaces (5) . . . . 79 5.22 Test Results of Subjective Contours with Overlaid Subjective Surfaces (6) . . . . 80 vii Acknowledgement I would like to thank my supervisor Dr. Jim Little for his guidance and encouragement throughout my work on this thesis. I would like to acknowledge the graduate students, faculties and staffs of the University of British Columbia Department of Computer Science providing me good working environment. Special thanks to Ms. Xiaomei Han for her friendship, Mr. Xun Li for his help in research of computer vision, and Mr. George Phillips for his expertise on using Latex and converting image file formats. This work was supported by the Natural Sciences and Engineering Research Council of Canada, and the University of British Columbia Department of Computer Science. I would like to thank my mother and my father for their support. viii Chapter 1 Introduction A contour in general is located on a division of two adjacent regions. When two adjacent regions have different brightnesses or colours, we see a real contour. Examples of real contours are the boundary or border of an object, the outline of a figure, and an edge. By comparison, we can also see a contour in the image with a certain arrangement of figures where the two adjacent regions have no real difference but an apparent brightness difference. This type of contour is called a subjective contour [Kanizsa, 1976] because it is physically not present in an image but is provided by human visual perception. The goal of this thesis is to understand the perception of subjective contours and to detect subjective contours by computer. 1.1 Subjective Contour Characteristics The use of subjective contour is an effective technique for art because it can increase brightness without physical gradient [Meyer and Petry, 1987]. When there is a limitation on use of colours on a material such as coins or woodcuts, an artist can either outline or use subjective contours to draw an object. The subjective contour technique on a drawing makes the object looks more intense than the background colour, and thus provides additional colours to the art work. This technique has been recognized for a long time in the art world. Recently, subjective contours have been used widely in designing the logos of organizations. The psychological study of subjective contours began about a century ago started by Fred-1 CHAPTER 1. INTRODUCTION 2 Figure 1.1: Schumann's Figure Figure 1.2: Kanizsa Triangle erich Schumann. Figure 1.1 is one of the example presented in [Schumann, 1904]; Schumann observed a white rectangle in the center of the image with sharply defined contours that are physically not present. In 1955 Gaetano Kanizsa published an article on subjective contours [Kanizsa, 1955]. Figure 1.2 is a popular subjective contour image depicted by [Kanizsa, 1955] and is called Kanizsa triangle. Kanizsa noticed the enhancement of brightness, the sharp ap-parent edges, and the depth discontinuities in his figures. There have been many theories of subjective contour perception proposed since 1970 but there is no settled theory today, which indicates that subjective contours are a difficult and complicated subject to study. The following characteristics are observed after examining many subjective contour exam-ples; also see [Kanizsa, 1976] and [Meyer and Petry, 1987] for detail. A subjective contour is normally noticed as an edge or border delimiting a discrete change in apparent brightness. Usually, there is a surface, called a subjective surface [Kanizsa, 1976], bounded by a subjective contour. A subjective surface is a region with the same physical quality as the background, but with a different visual quality that makes it to stand out from its background. This visual quality is caused by brightness enhancement, i.e., the surface adjacent to a darker or brighter coloured surface seems to be more intensive then its physical colour. For example, the sub-jective surface looks brighter than the background when the foreground figures have darker colour than the background; on the other hand, the subjective surface looks darker than the background when the foreground figures have brighter colour than the background. A figure CHAPTER 1. INTRODUCTION 3 is an object in the image, a foreground is part of an image seen closer to the viewer, and a background is an area where the foreground figures are seen against in the image. A subjective surface is usually considered a two dimensional figure and interpreted as a opaque surface parallel or tilted to the image plane. [Brady and Grimson, 1981] have different opinions about the dimensionality of subjective surfaces and they proposed that subjective surfaces are natural three dimensional surfaces. A subjective contour can only be observed on the background coloured area; it cannot be seen by itself. In order to see a subjective contour, there must be a pair of supporting edges [Ullman, 1976] on each end of the subjective contour and the subjective contour continues through the supporting edges. A supporting edge is part of an outline of a figure merged to form the subjective contour or a tip of a figure that touches the subjective contour to support its shape. A figure with supporting edges is called an inducing element [Meyer and Petry, 1987], There are three types of inducing elements: blob, line, and dot [Brady and Grimson, 1981]. A blob is a region with foreground colour, a line is a thin strip with foreground colour, and a dot is one spot with foreground colour. Unlike real contours, subjective contours cannot always be perceived in any images. There must be some evidence of discontinuities in the inducing elements to produce a subjective contour. Discontinuities in the inducing elements or a gap between aligned or continuous edges from two inducing elements suggest that those inducing elements are occluded by an object. Moreover, [Gregory, 1972] explains unlikely gaps are due to eclipsing or occlusion by some near opaque object or surface. A subjective surface is perceived as an occluding surface in front of those inducing elements, and the inducing elements appear to be part of larger figures that continue behind the subjective surface. In this thesis, a subjective contour is considered an occluding contour of an opaque object which has the same colour as the background. CHAPTER 1. INTRODUCTION 4 1.2 Subjective Contour Classification A new classification scheme for subjective contours is presented in the following two subsections. Subjective contours are classified into two categories, edge-based and tip-based, depending on the type of inducing elements that appear in the image. The distinction between the edge-based and tip-based subjective contour is based on whether the supporting edge can suggest the direction of the subjective contour shape or not. Any subjective contour image can be simplified to a black-and-white image separated by the foreground figures and the background colour. Through this thesis black is used as the foreground colour and white is the background colour. Edge-based Subjective Contour The edge-based subjective contour is formed in a blob-based image where parts of the blobs merge to shape the subjective contour. At the end of the blob supporting edge, there is a tangent direction that continues through the subjective contour. The characteristics of the edge-based subjective contour is that the supporting edges suggest the subjective contour shapes. A blob can be regular (Figure 1.3(a), on page 5), or irregular in shape (Figure 1.3(b)). It can have a concave angle corner (Figure 1.3(a) and Figure 1.3(b)) or a curved corner of both concave and convex shape (Figure 1.3(c)). A subjective contour can have the appearance of corner (Figure 1.3(d)) when viewed from far distance; let's call this type of subjective contour a cornered subjective contour. In addition to blobs, line ends and dots can also enhance the shapes of subjective contours if they can help to support the subjective contour shapes (Figure 1.3(e), Figure 1.3(f), and Figure 1.3(g)). A subjective surface does not always appear opaque with a more intensive colour than the background but it can have some patches on it (Figure 1.3(k)). A subjective contour can be closed (Figure 1.3(a)) or opened (Figure 1.3(h)) depending on the subjective surface associated with it. The overlaid subjective contours are seen as one subjective surface occluding another to produce multiple depth levels (Figure 1.3(i) and Figure 1.3(j)). In this kind of image, the CHAPTER 1. INTRODUCTION (a) ci.o (b) (c) c > (e) I I (f) (g) (i) 0) (k) Figure 1.3: Edge-based Subjective Contours (See Appendix A CHAPTER 1. INTRODUCTION 6 surface seen on the top looks brighter than the surfaces below it. The reversible subjective contours are seen on an image with perceptually ambiguous sub-jective contours and their configurations can be perceptually organized in many ways. Each of the subjective contour organization that emerges from the image has a unique set of supporting edges and brightness associated with the subjective contours. Still, only one interpretation of subjective contour organization can be perceived at a time; besides, the perception shifts from one organization to alternative ones. For instance, Figure 1.3(1) shows two simultaneous subjective figures, two triangles, that are depth reversible with respect to each other. Tip-based Subjective Contour The tip-based subjective contour is formed in an image with fines where the ends of the lines touch the subjective contour (Figure 1.4(a) on page 7). There is no tangent at the line end because the dimension of the fine end is just a point. Since the line end is unable to suggest the subjective contour shape, many interpretations of subjective contour shapes are possible when the two ends of lines are connected by a subjective contour. Whenever there is an interrupted black line on a white background, the discontinuity may be caused by an interposed white figure that is whiter than the background. The end of a line can be viewed as sudden line termination and it includes the tip of a blob with sharp angle (Figure 1.4(b)). The bent part of a straight line can be interpreted as the intersection of two straight lines; thus, it contains two line ends at the bend (Figure 1.4(c)). Each of Figures 1.4(a) and 1.4(d) appears as a subjective surface with its subjective contour interrupting a radiating and random line set respectively. A line can be straight (Figure 1.4(a) and Figure 1.4(d)) or curved (Figure 1.4(i) and Figure 1.4(j)) and a subjective contour can be open (Figure 1.4(d) and Figure 1.4(i)) or closed (Figure 1.4(a) and Figure 1.4(j)). Figure 1.4(g) and Figure 1.4(h) show a curved or straight subjective contour between the misaligned line segments terminating the lines on both sides of the subjective contour. Those images give a sense of two adjoining surfaces and a subjective contour is perceived where the two surfaces meet. Dots can help to determine the shape of subjective contours; compare Figure 1.4(e) and CHAPTER 1. INTRODUCTION 7 (a) (b) n u (e) .n. 'U' (f) & & & & & 0) ( j ) (c) (d) (g) (h) WD (d) Six contour chains (/ = 50) with Patches on a Subjective Surface (2) CHAPTER 5. RESULTS AND DISCUSSION 64 JUUl: D C :) c (a) Input image (b) After preprocessing (c) Four contour chains (/ = 22) Figure 5.10: Test Results of Subjective Contours with Patches on a Subjective Surface (3) organization, as in Figure 5.9(d), uses the edges that were not chosen in the dominant contour organization, and as a result six contour chains are found; three closed contour chains that complete the blob outlines, and three contour chains that are connected across the concave corner of each blob satisfying the curvature continuity measure set with the real edges. Figure 5.10(a) has regular figural configuration, and a square subjective surface with regular circle patches in the middle of the image is perceived. The square subjective surface can be seen as located on the top image plane, or alternatively as located behind the image plane seeing through a square hole. The impression of the subjective surface depth can be shifted at will because it is ambiguous. However, the location of a contour separating the square and the rest of the figure will not change. The four corners of the subjective square are defined by the convex corner of the blobs. These blobs are seen as if the circle blobs are cut into the corner shape or the circle blobs are covered by white surfaces with concave corners. Four contour chains are found corresponding to the four sides of a square because each convex corner consists of two supporting edges (see Figure 5.10(c)). No alternative contour organization is found because large areas on the blobs on the border of the subjective contours are occluded that the system couldn't recover any blobs. Figure 5.11(a) has similar figural configuration as Figure 5.10(a) except the circle patchs CHAPTER 5. RESULTS AND DISCUSSION 65 *=* i • • • • < • • -i r. j) 4 (a) Input image (b) After preprocessing 0 o (c) Five contour chains (I = 100) (d) Three contour chains (1 — 100) Figure 5.11: Test Results of Subjective Contours with Patches on a Subjective Surface (4) CHAPTER 5. RESULTS AND DISCUSSION 66 V 9- ^ t a o o ) a i a .4 (a) Input image (b) After preprocessing o o o o o o o (c) Six contour chains (I = 100) (d) Seven contour chains (/ = 100) Figure 5.12: Test Results of Subjective Contours with Patches on a Subjective Surface (5) CHAPTER 5. RESULTS AND DISCUSSION 67 in various sizes. The square subjective surface gives the impression that it is located behind the image plane seeing through a square hole. Five contour chains are found in the dominant contour organization with the given maximum gap size (see Figure 5.11(c)), and some more contour chains are found in the alternative contour organization where the shape of some circle patches at the borders are recovered (see Figure 5.11(d)). Figure 5.12(a) has similar figural configuration as Figure 5.11(a) except the four corners of subjective square are defined by the concave corners on the blobs, and these blobs are seen as if there were the white square corners occluding the round blobs. The square subjective surface has some round patchs, and it is perceived as located on the top of the image plane. Six contour chains are found in the dominant contour organization with the given maximum gap size (see Figure 5.12(c)). The middle square subjective contour is our focuses, and the rest of the contour chains are organized around the image border. The middle square is described by one contour chain because the two segments on each concave corner on the blob are grouped into one supporting edge and the subjective contour continues through the supporting edges. Some more contour chains are found in the alternative contour organization (see Figure 5.12(d)). However, we would not normally perceive the contour chains that each subjective contour connecting to the two adjacent blob outlines. 5.3 Subjective Contours Induced by Bars Subjective contours emerging from bars are tested in this section. The strength of bar sides determine whether the subjective contour organization is dominant or not. The formation of cornered subjective contour is suggested. Also, similarity of bar and line is discussed. Figure 5.13(a) on page 69, the basic subjective contour figure used in this section, has four bars arranged in a cross shape with an opening in the middle. The two shorter sides of each bar are ranked as the SUPPORT type supporting edge candidates because they suggest the sudden termination of a bar. The dominant contour organization is found in the middle of the image as shown in Figure 5.13(c), and the alternative contour organization using the longer sides of CHAPTER 5. RESULTS AND DISCUSSION 68 bars as supporting edges is shown in Figure 5.13(d). Figure 5.14(a) arranged horizontal and vertical bars not aligned with the square shape opening in the middle, and it also produces the reasonable subjective contour organizations (see part (c) and part (d) of Figure 5.14). To demonstrate that the shape and strength of subjective contour depends on inducing element configuration, each bar in Figure 5.13(a) is thickened to make the gaps between adjacent bars close to the center of the image narrower as in Figure 5.15(a). The central white area is seen as a square with round corners, and the corners appear sharper when the gap size is smaller; compare Figure 5.13(c) and Figure 5.15(c). The reason is that the Bezier curve with the bigger curvature will fit into the smaller gap size providing that the subjective contour is curvature continuous. In general, when the object is in a distance, then the smaller a for Gaussian smoothing of the object outline is sufficient compared to the same object, because of the farther object looks smaller and less detailed. In the far distance, the increase in the change of curvature on the subjective contour is compensated for the smaller a, in order for all the points on the curve to have the change of curvature below the maximum change of curvature. Refer to Equation 4.1 on page 42 for the curvature continuity measure. To achieve the maximum change of curvature allowed in the fax distance object, the smaller a on the Gaussian smoothing of the object outline is compensated by the larger change of curvature on both the subjective contours and the supporting edges. The large change of curvature on a subjective contour means that the subjective contour can have larger curvature than the subjective contour in the near distance; therefore, the subjective contour with sharper corner could be seen. Moreover, the subjective contour appears stronger when the supporting edges are longer and each gap in between the two supporting edges is narrower. It is possible to see a cornered subjective contour in Figure 5.15(a) because the effect of extension of the supporting edges is very strong. Figure 5.16(a) arranges bars in the way that some bar supports two chains in one organi-zation. A line can be considered as a special type of a bar when the bar is very narrow. The subjective contour organization of lines will be similar to the dominant contour organization using the shorter sides of the bars as supporting edges (see Figure 5.16(c)). However, the shape CHAPTER 5. RESULTS AND DISCUSSION 69 I I (a) Input image (b) After preprocessing J -^A r (c) One contour chain (/ = 30) (d) Four contour chains (I = 30) Figure 5.13: Test Results of Subjective Contours Induced by Bars (1) CHAPTER 5. RESULTS AND DISCUSSION 70 I I (a) Input image (b) After preprocessing (c) One contour chain (/ = 30) (d) Four contour chains (/ = 30) Figure 5.14: Test Results of Subjective Contours Induced by Bars (2) CHAPTER 5. RESULTS AND DISCUSSION 71 I I (a) Input image (b) After preprocessing J v. -\\ r (c) One contour chain (/ = 30) (d) Four contour chains (I = 30) Figure 5.15: Test Results of Subjective Contours Induced by Bars (3) CHAPTER 5. RESULTS AND DISCUSSION 72 (a) Input image (b) After preprocessing (c) Four contour chains (I = 30) J V_ A r ~~\\ f J \\~ -\\ r (d) Nine contour chains (/ = 30) Figure 5.16: Test Results of Subjective Contours Induced by Bars (4) CHAPTER 5. RESULTS AND DISCUSSION 73 of subjective contour formed by the line ends is not easy to predict because the line end has no length; in fact, it is one pixel wide. Therefore, there is no suggestion about orientation of subjective contour at the line end. Also, unlike a line, the long sides of a bar can become supporting edges for an alternative contour organization (see Figure 5.16(d)). 5.4 Subjective Contours with Overlaid Subjective Surfaces In this section, the more complicated subjective contours are examined. The overlaid subjec-tive contour images (see Figure 5.17(a) to Figure 5.20(a)) show different depth level of the subjective surfaces in one contour organization. The reversible subjective contour image (see Figure 5.21(a)) alternates two or more different depth interpretation of the subjective contour organizations, and the subjective contour perception is unstable. The silhouette image (see Figure 5.22) is the image with reverse of the foreground and the background colour in order to recover the shapes of the original figures by subjective contour organizations. The different maximum gap size, which can decide the shape of subjective contours, are demonstrated in Figure 5.19, Figure 5.20, and Figure 5.21. Figure 5.17 and Figure 5.18 have two dominant square subjective surfaces with the square located at top left corner superimposed on the square located on the bottom right corner and partially occluding circles in their corners. Their alternative organizations are circles. Looking at the result shown in Figure 5.17(c), the bottom subjective square does not touch the border of the top subjective square because there is no supporting edge for the bottom subjective square at the border of the top subjective square, and no effort has been made to extend the subjective contour until it touches another contour chain in this implementation. In contrast, Figure 5.18(c) shows the bottom subjective square continuing until it touches the top subjective square because there are supporting edges for the bottom subjective square which touch the top subjective square. Figure 5.20 is similar to Figure 5.19 except that additional lines shape the subjective con-tours. Part (c) and part (d) in both figure uses the maximum gap size short enough to covert CHAPTER 5. RESULTS AND DISCUSSION 74 (a) Input image (b) After preprocessing (c) Two contour chains (I = 60) (d) Seven contour chains (Z = 60) Figure 5.17: Test Results of Subjective Contours with Overlaid Subjective Surfaces (1) CHAPTER 5. RESULTS AND DISCUSSION 75 (a) Input image (b) After preprocessing (c) Two contour chains (/ = 60) (d) Three contour chains (/ = 60) Figure 5.18: Test Results of Subjective Contours with Overlaid Subjective Surfaces (2) CHAPTER 5. RESULTS AND DISCUSSION 76 (a) Input image (b) After preprocessing (c) One contour chain (I = 40) (d) Six contour chains (/ = 40) (e) Two contour chains (Z = 100) (f) Six contour chains (I = 100) Figure 5.19: Test Results of Subjective Contours with Overlaid Subjective Surfaces (3) CHAPTER 5. RESULTS AND DISCUSSION 77 (a) Input image (b) After preprocessing (c) Six contour chains (I = 28) (d) Six contour chains (I = 28) (e) Two contour chains (I = 100) (f) Six contour chains (7 = 100) Figure 5.20: Test Results of Subjective Contours with Overlaid Subjective Surfaces (4) CHAPTER 5. RESULTS AND DISCUSSION 78 the distance from one blob supporting edge to the adjacent blob supporting edge or line end. When the subjective contour is connected from one blob supporting edge to adjacent blob sup-porting edge as in Figure 5.19(c), the CURVE subjective contour is formed. The contour chain stops when it touches the line end and in Figure 5.20(c) shows corner at a line end as a result. When their maximum gap size is long enough to connect two aligned supporting edges across the gap and produce a LINEAR subjective contour as in part (e) of both figures, two triangle shape contour chains are found. Since the two contour chains intersect each other on subjective contours, the result shows two triangles in the same organization; those contour chains do not share the same supporting edges nor supporting edges from two contour chains are adjacent to each other. The depth order of these two subjective surfaces are ambiguous since either subjective surface can be on top of other subjective surface. The alternative organization (see part (f)) recovers the shape of the occluded blobs. Figure 5.21(a) is an example of reversible subjective contour because the subjective contour image can be organized in many ways changing the arrangement of depth. The central white region is overlap with four bars if long maximum gap size being used, and two crosses one on top of others when shorter maximum gap size is used. Figure 5.21(c) is an example of using shorter maximum gap size when adjusting the weight to select the horizontal and vertical subjective contours first. The horizontal and vertical parts of a contour chain axe connected by curved subjective contours near the center of the image. The cross in the bottom, oriented at 45°, resulted in four chains because of occlusion from the cross shape subjective surface on its top, and the supporting edges for the top cross are adjacent to the supporting edges for the bottom cross. The alternative contour organization is shown in Figure 5.21(d). There is a white subjective circle in the middle of the image touching the blob tips. This type of subjective contour is organized by the tip-based subjective contour, and it could not be found by the system because the system detects the edge-based subjective contour only. The priority to select horizontal and vertical subjective contours is just one way of finding the subjective contour organizations. The other reversible subjective contour arrangement can reverse the depth of the two crosses in Figure 5.21(c) that a diagonal cross on the top of the CHAPTER 5. RESULTS AND DISCUSSION 79 * • (a) Input image (c) Five contour chains (I = 40) (b) After preprocessing (d) Two contour chains (I = 40) (e) Six contour chains (/ = 100) (f) Two contour chains (/ = 100) Figure 5.21: Test Results of Subjective Contours with Overlaid Subjective Surfaces (5) CHAPTER 5. RESULTS AND DISCUSSION 80 (a) Input image (b) After preprocessing —I (c) One contour chain (I = 150) (d) One contour chain (I = 150) Figure 5.22: Test Results of Subjective Contours with Overlaid Subjective Surfaces (6) CHAPTER 5. RESULTS AND DISCUSSION 81 horizontal and vertical cross. Each reversible subjective contour organization uses a some set of supporting edges but the subjective contour connections are arranged in the different way. Figure 5.21(e) is an example of using longer maximum gap size when selecting the horizontal and vertical subjective contours first. Two contour chains in the bar shape are found first and its shows that they are crossing each other because the model does not try to label the depth when a subjective contour crosses another subjective contour. The other two contour chains that shape diagonal bars are discontinued under the horizontal and vertical subjective bars on the top. The alternative organization is shown in Figure 5.21(f). When using the long maximum gap size, the four bars can be on any depth order because this image produces reversible subjective contour figures. Figure 5.22(a) shows the silhouette of two figures—a triangle and a rectangle—overlaid. There is no depth information specifying whether the triangle is in front of or behind the rectangle. This kind of image is an extreme case of overlaid objects where all objects have the same colour; consequently, we cannot distinguish each object from the colour information. The only clue to distinguish each object is the outline continuity. The subjective contours cannot be seen on the figure; however, we can see subjective contours on the background coloured area. Therefore, we reverse the figure and the background of such image to recover the overlaid objects. A silhouette is the shadow of such objects; hence, it simply put the objects on the background and make the background of the objects as foreground colour because the shadow area is suppose to be the foreground in the silhouette image. The white part of the image is ambiguous in depth whether the white triangle in front or the white rectangle in front. The system first selects a rectangle subjective contour organization because this contour chain has the shortest LINEAR type subjective contour. Note that each of the two adjacent corners on the rectangle are grouped as one supporting edge because the two segments on each concave corner is grouped together and one segment in the middle is shared by both corners. The alternative organization gives a triangle shaped subjective contour chain. In the silhouette image, there is no dominant contour organization and the subjective contour organization shifts from one to the other because the depth of the subjective surfaces are ambiguous. Chapter 6 Conclusions Until today, not many computer vision systems are capable of detecting subjective contours. Each system limits its input and the subjective contours it deals with, and some parameters must be entered to adjust the system. In this thesis, we present a model of subjective contour detection system based on the approach that finds boundary of subjective surfaces and performs subjective contour organization with less limitations on the input images and the subjective contours that can be detected. In particular, we use figural cues to find supporting edges and apply the perceptual organization to find subjective contours. This thesis presents steps involved in detecting subjective contours. Moreover, a new classification scheme of subjective contours is presented in Chapter 1. Based on the classification, the edge-based subjective contour is investigated. A subjective detection model is presented based on four criteria: no prior knowledge is nec-essary to detect subjective contour; a subjective contour is a special type of occluding contours; the shape of a subjective contour is determined by the viewing condition; and it is possible to have multiple subjective contour organizations from one image. The model emphasizes contours rather than surface because perception of subjective contour is local phenomenon of surface perception. The main concern in this thesis is the overall organization of the subjective contours and our focuses is on which supporting edges to connect rather than the exact shape of a subjective contour. The rules for subjective contour organization are described and the model explains different types of subjective contour organizations. 82 CHAPTER 6. CONCLUSIONS 83 The algorithms for local subjective contour selection and global contour organizations have been developed. The observer's viewing distance is translated into the maximum gap size allowed between the two supporting edges that are connected by a subjective contour. The consistent curvature continuity measure is used to find the contour discontinuities on the curve to segment the real and subjective contours. The factors affecting the perception of subjective contours are identified and incorporated in the algorithm. The computer implementation of subjective contour detection is performed in three stages: the preprocessing, local subjective contour selection, and global contour organization. The preprocessing identifies figures, and the blob outHnes are segmented according to their curvature discontinuity by Lowe's curve partition method. The local processing processes each supporting edge which selects potential subjective contours depending on the maximum gap size. The global processing chooses the subjective contours among the potential subjective contours and groups the supporting edges and the subjective contours into contour organizations. The implementation of subjective contour detection system is limited to detecting the sub-jective contour on the black-and-white image which we can perceive subjective contours without preset knowledge of the object shapes, i.e., we use the subjective contour image with strong subjective contour effect. The straight subjective contours as well as the curved subjective con-tours can be found by the system. The shape of a curved subjective contour is approximated by the Bezier curve. The gap size between the two supporting edge endpoints and the orientation of the supporting edges with respect to the subjective contour are considered into the weight of the subjective contour. The smaller the gap size and the less curvature between the supporting edges, the stronger the subjective contour is. Many subjective contour images have been tested on the subjective contour detection sys-tem. The straight as well as the curved subjective contours have been detected, and the dominant and alternative contour organizations are found in each image. In addition, we have demonstrated the effects of additional line ends that help to shape the subjective contours, and the results of different maximum gap size that could find different subjective contour organi-zations. The system sometimes gives the different interpretation of the contour organizations, CHAPTER 6. CONCLUSIONS 84 especially the alternative contour organizations, then the perception of human observer due to the assumptions used in the system. In general, the system produces good results. The subjective contour detection system is capable of selecting and grouping subjective contours based on the four subjective contour detection criteria. The system can find the sub-jective contour regardless of the inducing element orientations along the subjective contour; therefore, it gives the flexibility to find the subjective contour with patches on the subjective surface. Moreover, the inducing element outline, the supporting contour, and the subjective contour together form a T-junction indicating occlusion that is independent of the subjective surface orientation. The similarity of lines and bars are discussed, and the formation of curved subjective contour is suggested. The subjective contours might cross each other in one contour organization because the model does not label the depth of the surfaces. The system is robust because the slight change in the maximum gap size would not change the contour organiza-tions. It requires the large change in the maximum gap size to result in the different contour organization if it is possible to have some different contour organizations. The immediate application of the subjective contour detection system is to uncover cam-ouflaged figures in the image. Camouflaged figures have textures and patterns similar to the surroundings and they appear to be part of the surroundings. However, often the pattern on the boundary of a camouflaged figure and its background are misaligned, and we see the occlu-sion of the background by a surface. In fact, the subjective figure in general is a camouflaged figure. Therefore, the same clues to detect subjective contours also apply to finding camou-flaged figures. The other application of the subjective contour detection system is to separate two or more occluded objects with similar textures and colours, or to separate the overlapping shadows. The former case is similar to detecting the reversible subjective contours, and the latter case is similar to detecting the subjective contours on the overlapping silhouettes of many objects. The subjective contour detection system can be extended in the following three areas: the strength of subjective contours, the tip-based subjective contours, and the depth labeling of subjective contours. The strength of subjective contour is dependent on the length of supporting CHAPTER 6. CONCLUSIONS 85 edges, the figure thickness perpendicular to the supporting edge, and orientation of the sup-porting edge pair that connect to the subjective contour. The longer the supporting edge, the thicker the inducing element along the supporting edge, and the smaller the curvature between the supporting edge pair, the stronger subjective contour it produces. We haven't considered the blob thickness perpendicular to the supporting edge in the strength of subjective contour, so this is one future work possibility. In addition, we can explore more about the strength of subjective contour due to its orientation, i.e., the subjective contour is stronger in the horizontal and vertical direction. The brightness associated with the subjective surface also determines the subjective contour strength but the brightness measure is rather subjective. In summary, the strength of subjective contour adds more clues to the subjective contour detection. In this implementation, line ends and dots are treated as tips because the image is based on the blobs and and distinguishing between tips is not important. In contrast, we have to distinguish between line ends and dots in the image based on line ends because line ends can produce subjective contours but more than one dot cannot produce a subjective contour. To modify the system to deal with a tip-base subjective contour image, we make two endpoints at the line end to consider the line end as if it was a very thin bar. There is no direction at the line end, so either a T, Y, or arrow junction is formed at a line end when there is a subjective contour connecting to the line end. In a certain image, the implementation result gives the subjective contours crossing each other in one contour organization. This happens because there is no depth labeling of subjective contours. In most cases, the depth of subjective contours are ambiguous when they are crossing each other, and the depth of each subjective surface shifts as the contour organization changes. One possible improvement to the subjective contour detection system is to apply the depth labeling to the subjective surfaces, and we can present the combination of different depth of the subjective surfaces in many contour organizations. The subjective contour uses the same depth information as its subjective surface. Depth labeling of the subjective contours will improve subjective contour organizations. The subjective contour detection system can interact with the perceptual organization to CHAPTER 6. CONCLUSIONS 86 notice grouping and structures in the image; in particular, the figure-ground separation in which the system gives the contours separating the two regions. Moreover, the subjective contour detection is located in the intermediate-level of computer vision because it uses the processing results from low-level vision system that gives the local features and processes the data into global description about the object boundaries. The subjective contour detection system can be connected to high-level vision systems to produce the description of the input image. In conclusion, perception of a subjective contour is understood in this thesis as a percep-tion of an invisible occluding contour. The subjective contour detection system is implemented and many subjective contours are detected. Although the system is not capable of detecting every types of subjective contour, we found some clues and constraints for contour perception by modeling the system. Human can recover the partially occluded object shape by com-pleting the missing outline, and subjective contour is an extreme case of occluding contour perception. Therefore, understanding subjective contour is important for understanding hu-man visual perception. We should continue research in the topic of subjective contours to have better understanding of contour perception. Bibliography [Asada and Brady, 1986] Haruo Asada and Michael Brady. The curvature primal sketch. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8(1):2-14, January 1986. [Barrow and Tenenbaum, 1978] H. G. Barrow and J. M. Tenenbaum. Recovering intrinsic scene characteristics from images. In A. R. Hanson and E. M. Riseman, editors, Computer Vision Systems, pages 3-26. Academic Press, 1978. [Bradley, 1987] Drake R. Bradley. Cognitive contours and perceptual organization. In Susan Petry and Glenn E. Meyer, editors, The perception of illusory contours, pages 201-212. Springer-Verlag, 1987. [Bradley and Dumais, 1975] Drake R. Bradley and Susan T. Dumais. Ambiguous cognitive contours. Nature, 257:582-584, October 16 1975. [Brady and Grimson, 1980] Mike Brady and W. E. L. Grimson. Shape encoding and subjective contours. First Annual National Conference on Artificial Intelligence, pages 15-17, 1980. [Brady and Grimson, 1981] Michael Brady and W. Eric L. Grimson. The perception of subjec-tive surfaces. Technical Report 666, MIT Artificial Intelligence Laboratory, Novem-ber 1981. [Canny, 1983] J. F. Canny. Finding edges and lines in images. Technical Report 720, MIT Artificial Intelligence Laboratory, 1983. [Coren, 1972] Stanley Coren. Subjective contours and apparent depth. Psychological Review, 79(4):359-367, 1972. [Darrell and Pentland, 1991] Trevor Darrell and Alex Pentland. On the representation of oc-cluded shapes. In IEEE Conference on Computer Vision and Pattern Recognition, pages 728-729, 1991. [Day and Jory, 1978] R. H. Day and M. K. Jory. Subjective contours, visual acuity, and line contrast. In John C. Armington, John Krauskopf, and B. R. Wooten, editors, Visual Psychophysics and Physiology, pages 331-340. Academic Press, 1978. 87 BIBLIOGRAPHY [Day and Jory, 1980] R. H. Day and M. K. Jory. A note on a second stage in the formation of illusory contours. Perception and Psychophysics, 27(1):89-91, 1980. [Day and Kasperczyk, 1983a] R. H. Day and R. T. Kasperczyk. Amodal completion as a basis for illusory contours. Perception and Psychophysics, 33(4):355-364, 1983. [Day and Kasperczyk, 1983b] Ross H. Day and Richard T. Kasperczyk. Illusory contours in line patterns with apparent depth due to either perspective or overlay. Perception, 12:485-490, 1983. [Freeman and Davis, 1977] Herbert Freeman and Larry S. Davis. A corner-finding algorithm for chain-coded curves. IEEE Transactions on Computers, pages 297-303, March 1977. [Frisby and Clatworthy, 1975] John P. Frisby and Jeremy L. Clatworthy. Illusory contours: curious cases of simultaneous brightness contrast? Perception, 4:349-357, 1975. [Gillam, 1987] Barbara Gillam. Perceptual grouping and subjective contours. In Susan Petry and Glenn E. Meyer, editors, The perception of illusory contours, pages 268-273. Springer-Verlag, 1987. [Gregory, 1972] R. L. Gregory. Cognitive contours. Nature, 238:51-52, July 7 1972. [Grossberg and Mingolla, 1985] Stephen Grossberg and Ennio Mingolla. Neural dynamics of perceptual grouping: textures, boundaries, and emergent segmentations. Perception and Psychophysics, 38(2):141-171, 1985. [Halpern, 1981] Diane F. Halpern. The determinants of illusory-contour perception. Perception, 10:199-213, 1981. [Halpern and Salzman, 1983] Diane F. Halpern and Billie Salzman. The multiple determination of illusory contours: 1. a review. Perception, 12:281-291, 1983. [Halpern, Salzman, Harrison, and Widaman, 1983] Diane F. Halpern, Billie Salzman, Wayne Harrison, and Keith Widaman. The multiple determination of illusory contours: 2. an empirical investigation. Perception, 12:293-303, 1983. [Haralick, Mackworth, and Tanimoto, 1989] R. M. Haralick, A. K. Mackworth, and S. L. Tan-imoto. Computer vision update. Technical Report TR89-12, Department of Com-puter Science, U.B.C, June 1989. [Heitger and von der Heydt, 1993] Friedrich Heitger and Riidiger von der Heydt. A computa-tional model of neural contour processing: figure-ground segregation and illusory contours. In IEEE 4th International Conference on Computer Vision, pages 32-40, 1993. BIBLIOGRAPHY 89 [von der Heydt, Peterhans, and Baumgartner, 1984] R. von der Heydt, E. Peterhans, and G. Baumgartner. Illusory contours and cortical neuron responses. Science, 224:1260-1262, June 1984. [Horn, 1983] B. K. P. Horn. The curve of least energy. ACM Transactions on Mathematical Software, 9:441-460, December 1983. [Huttenlocher and Wayner, 1991] Daniel P. Huttenlocher and Peter C. Wayner. Finding convex edge groupings in an image. In IEEE Conference on Computer Vision and Pattern Recognition, pages 406-412, 1991. [Jory and Day, 1979] Max K. Jory and Ross H. Day. The relationship between brightness contrast and illusory contours. Perception, 8:3-9, 1979. [Kanizsa, 1955] Gaetano Kanizsa, Translated by Walter Gerbino. Quasi-perceptual margins in homogeneously stimulated fields. In Susan Petry and Glenn E. Meyer, editors, The perception of illusory contours, pages 40-49. Springer-Verlag, 1987. [Kanizsa, 1976] Gaetano Kanizsa. Subjective contours. Scientific American, 234:48-52, April 1976. [Kanizsa, 1979] Gaetano Kanizsa. Organization in vision: essays on gestalt perception. Praeger, 1979. [Kawabata, 1984] N. Kawabata. Perception at the blind spot and similarity grouping. Percep-tion and Psychophysics, 36(2):151—158, 1984. [Kellman and Loukides, 1987] Philip J. Kellman and Martha G. Loukides. Object perception and subjective contours. In Susan Petry and Glenn E. Meyer, editors, The percep-tion of illusory contours, pages 151-164. Springer-Verlag, 1987. [Kennedy, 1978a] John M. Kennedy. Illusory contours and the ends of lines. Perception, 7:605-607, 1978. [Kennedy, 1978b] John M. Kennedy. Illusory contours not due to completion. Perception, 7:187-189, 1978. [Kennedy, 1979] John M. Kennedy. Subjective contours, contrast, and assimilation. In Calvin F. Nodine and Dennis F. Fisher, editors, Perception and Pictorial Rep-resentation, pages 167-195. Praeger Publishers, 1979. [Kennedy, 1987] John M. Kennedy. Lo, perception abhors not a contradiction. In Susan Petry and Glenn E. Meyer, editors, The perception of illusory contours, pages 253-261. Springer-Verlag, 1987. [Lowe, 1989] David G. Lowe. Organization of smooth image curves at multiple scales. Inter-national Journal of Computer Vision, 3:119-130, 1989. BIBLIOGRAPHY 90 [McCafferty, 1990] James D. McCafferty. Human and machine vision: computing perceptual organisation. Ellis Horwood, 1990. [Meyer and Petry, 1987] Glenn E. Meyer and Susan Petry. Top-down and bottom-up: The illusory contour as a microcosm of issues in perception. In Susan Petry and Glenn E. Meyer, editors, The perception of illusory contours, pages 3-20. Springer-Verlag, 1987. [Minguzzi, 1987] Gian Franco Minguzzi. Anomalous figures and the tendency to continuation. In Susan Petry and Glenn E. Meyer, editors, The perception of illusory contours, pages 71-75. Springer-Verlag, 1987. [Nitzberg and Mumford, 1990] Mark Nitzberg and David Mumford. The 2.1-d sketch. In IEEE 3rd International Conference on Computer Vision, pages 138-144, 1990. [Parks, 1980] Theodore E. Parks. Subjective figures: some unusual concomitant brightness effects. Perception, 9:239-241, 1980. [Parks, 1984] Theodore E. Parks. Illusory figures: A (mostly) atheoretical review. Psychological Bulletin, 95(2):282-300, 1984. [Redies, Crook, and Creutzfeldt, 1986] C. Redies, J. M. Crook, and O. D. Creutzfeldt. Neu-ronal responses to borders with and without luminance gradients in cat visual cor-tex and dorsal lateral geniculate nucleus. Experimental Brain Research, 61:469-481, 1986. [Richardson, 1979] Barry L. Richardson. The nonequivalence of abrupt and diffuse illusory contours. Perception, 8:589-593, 1979. [Rock, 1983] Irvin Rock. The logic of perception. The MIT Press, 1983. [Rock and Anson, 1979] Irvin Rock and Richard Anson. Elusory contours as the solution to a problem. Perception, 8:665-681, 1979. [Rosenfeld and Johnston, 1973] Azriel Rosenfeld and Emily Johnston. Angle detection on dig-ital curves. IEEE Transaction on Computers, pages 875-878, September 1973. [Rosenfeld and Kak, 1981] Azriel Rosenfeld and Avinash C. Kak. Digital picture processing, volume 1, 2. Academic Press, 1981. [Rutkowski, 1979] Wallace S. Rutkowski. Shape completion. Computer Graphics and Image Processing, 9:89-101, 1979. [Sambin, 1987] Marco Sambin. A dynamic model of anomalous figures. In Susan Petry and Glenn E. Meyer, editors, The perception of illusory contours, pages 131-142. Springer-Verlag, 1987. BIBLIOGRAPHY 91 [Schumann, 1904] F. Schumann, Translated by Anne Hogg. Contributions to the analysis of visual perception-first paper: some observations on the combination of visual im-pressions into units. In Susan Petry and Glenn E. Meyer, editors, The perception of illusory contours, pages 21-34. Springer-Verlag, 1987. [Shapley and Gordon, 1985] Robert Shapley and James Gordon. Nonlinearity in the perception of form. Perception and Psychophysics, 37(l):84-88, 1985. [Skrzypek and Ringer, 1992] Josef Skrzypek and Brian Ringer. Neural network models for illusory contour perception. IEEE, pages 681-683, 1992. [Smith and Over, 1975] Andrew Smith and Ray Over. Tilt aftereffects with subjective contours. Nature, 257:581-582, October 16 1975. [Trytten and Tuceryan, 1991] Deborah A. Trytten and Mihran Tuceryan. Segmentation and grouping of object boundaries using energy minimization. In IEEE Conference on Computer Vision and Pattern Recognition, pages 730-731, 1991. [Ullman, 1976] Shimon Ullman. Filling-in the gaps: the shape of subjective contours and a model for their generation. Biological Cybernetics, 25:1-6, 1976. [Ullman and Sha'ashua, 1988] Shimon Ullman and Amnon Sha'ashua. Structural saliency: the detection of globally salient structures using a locally connected network. Technical Report 1061, MIT Artificial Intelligence Laboratory, July 1988. [Webb and Pervin, 1984] Jon A. Webb and Edward Pervin. The shape of subjective contours. Proceedings of the National Conference on Artificial Intelligence, pages 340-343, August 1984. [Williams, 1990] Lance R. Williams. Perceptual organization of occluding contours. Interna-tional Conference on Computer Vision, pages 133-137, December 1990. [Zucker and Cavanagh, 1985] Steven W. Zucker and Patrick Cavanagh. Subjective figures and texture perception. Spatial Vision, 1(2):131-139, 1985. Appendix A Sources of Figures Figure Reference Figure 1.1 [Schumann, 1904], Figure 2.7, page 26. Figure 1.2 [Kanizsa, 1955], Figure 4.11, page 44. Figure 1.3a [Kanizsa, 1976], figure at middle in middle illustration on page 51. Figure 1.3b [Kanizsa, 1976], figure at right in bottom illustration on page 49. Figure 1.3c [Kanizsa, 1976], figure at left in bottom illustration on page 49. Figure 1.3d [Sambin, 1987], Figure 14.19a, page 141. Figure 1.3e [Kanizsa, 1976], figure at left in middle illustration on page 49. Figure 1.3f [Minguzzi, 1987], Figure 7.3a, page 72. Figure 1.3g [Minguzzi, 1987], Figure 7.3b, page 72. Figure 1.3h [Kanizsa, 1976], figure at right in top illustration on page 50. Figure 1.3i [Kanizsa, 1955], Figure 4.16, page 45. Figure 1.3j [Parks, 1980], Figure 3.(a), page 240. Figure 1.3k [Brady and Grimson, 1981], Figure 26, page 29. Figure 1.31 [Bradley and Dumais, 1975], Figure 3, page 583. Figure 1.4a [Kennedy, 1978a], Figure l.(a), page 606. Figure 1.4b [Kennedy, 1987], Figure 28.2, page 255. Figure 1.4c [Kennedy, 1978b], Figure l.(a), page 188. Figure 1.4d [Gillam, 1987], Figure 30.8, page 272. Figure 1.4e [Day and Jory, 1980], Figure 2A, page 89. Figure 1.4f [Day and Jory, 1980], Figure 1, page 89. Figure 1.4g [Brady and Grimson, 1981], Figure 18, page 18. Figure 1.4h [Kanizsa, 1979], Figure 12.12, page 204. Figure 1.4i [Gillam, 1987], part of Figure 30.10 turned side way, page 272. Figure 1.4j [Parks, 1984], Figure 5.(b), page 291. Figure 1.4k [Minguzzi, 1987], Figure 7.5, page 73. Figure 1.41 [Zucker and Cavanagh, 1985], Figure 1. left, page 132. 92 Figure Reference Figure 1.5a [Kanizsa, 1955], Figure 4.11, page 44. Figure 5.1 [Kanizsa, 1976], figure at right in bottom illustration on page 49. Figure 5.2 [Richardson, 1979], Figure 4a, page 593. Figure 5.3 [Kanizsa, 1976], figure at left in bottom illustration on page 49. Figure 5.4 [Kanizsa, 1976], figure at left in middle illustration on page 49. Figure 5.5 [Kanizsa, 1976], figure at right in middle illustration on page 49. Figure 5.6 [Brady and Grimson, 1981] Figure 23 left, page 27. Figure 5.7 [Brady and Grimson, 1981] Figure 23 right, page 27. Figure 5.8 [Brady and Grimson, 1981], Figure 26, page 29. Figure 5.9 Hand drawn subjective contour image. Figure 5.10 Hand drawn subjective contour image. Figure 5.11 [Kanizsa, 1979], Figure 10.11, page 179. Figure 5.12 [Kanizsa, 1979], Figure 10.9, page 177. Figure 5.13 [Sambin, 1987], Figure 14.19a, page 141. Figure 5.14 [Sambin, 1987], Figure 14.19b, page 141. Figure 5.15 Based on Figure 5.14, thicken the bars. Figure 5.16 Based on Figure 5.14, multiply the bars. Figure 5.17 [Parks, 1980], Figure 3.(a), page 240. Figure 5.18 Hand drawn subjective contour image. Figure 5.19 [Bradley and Dumais, 1975], Figure 3 without lines, page 583. Figure 5.20 [Bradley and Dumais, 1975], Figure 3, page 583. Figure 5.21 [Kellman and Loukides, 1987], Figure 3c colour inverted, page 160. Figure 5.22 [Bradley, 1987], Figure 22.6a, page 206. 93 Appendix B Bezier Curve A Bezier curve described in this section takes three control points, PQ, PI, and Pi- The curve originates from point PQ, and does not always passes through point Pi, and terminates at point P2. Furthermore, the curve is tangent to PQP\\ at point PQ and P\\P