IntroductionExtracting meaningful contours in a 2D image in an “unsupervised” way is still an unsolved problem. By “meaningful” we mean occluding contours, as well as internal contours representing symmetrical features in the 3D space. The main challenge is the large amount of irrelevant contours as well as the fact that the real contours are never continuous (see Figure 1). How does the visual system perform interpolation of disconnected parts of the contour ignoring irrelevant contours? All previous methods were based on spatially local rules such as co-linearity and co-circularity. Here we present a spatially global interpolation that results in closed and “simple” curves.

The Shortest path as a spatially global interpolation of contours in imagesShweta Gupte, Yunfeng Li & Zygmunt Pizlo

Conclusions:

Acknowledgement

Contact: Shweta Gupte

Email:svaidya@purdue.edu

This research was supported by the NSF.

Reference:

Gray Scale Image Edge detection input Cartesian space Log-polar space

Shortest Path outputCartesian space Log-polar space

Log-polar: The log-polar transformation is a conformal mapping from the points on the Cartesian plane (x,y)  to points in the log-polar plane (ξ,η):

Cartesian Log-polar Mapping

Output in the Cartesian Representation

Log-polar Shortest Path

a) Retina and the area V1 in the cortex b) Idealized log-polar mapping

Figure 2. After Schwartz (1980).

Figure1. (a) A real image with extracted occluding contour of a small chair.

“The circle is a perfectly good figure” (Koffka, 1935, p.151)

Explanation:The shortest path is computed using a modified Dijkstra algorithm in a log-polar representation (Figures 2-4). We begin with Canny edge detection. A pixel on the edge is a node of a fully connected graph. The cost of the path going through an existing edge is lower than the cost of the interpolated path. The fixation point must be inside the region representing the object. The start-end point is selected manually. Alternatively, a number of starting points can be tried. The Dijkstra algorithm has a run time of O(nlogn).

Figures 5-6 illustrate how information about texture can be used to improve contour analysis. A pair of stereo-images is used to compute 3D representation based on texture (not shown). This allows estimation and removal of the ground plane. The resulting 3D points represent individual objects. A 2D convex hull is then computed for the image regions representing individual objects. A number of start and end points are automatically selected around the convex hull and shortest paths in the image are computed between every pair of points. These shortest paths capture important contours of the object. These contours can then be used for precise 3D shape recovery.

ξ = log√x2+y2

η = atan(y/x)

Figure1. (b) Large chair. Figure 1. (c) Rocking horse.

Figure 4. Shortest paths in the log-polar representation

Figure 3. Left: local co-linearity of edges is ignored. Right: interpretation of a junction can change by a spatially remote feature – see Figure 4 for more examples. Red segment marked is the selected segment to move in the scene.

Shortest path in the log-polar and in the image representations is a very effective interpolation tool. It is especially useful for shape analysis because it is a spatially global operator.

Law of Closure: Occluding contour is a closed non-self-intersecting curve.

The shortest path in the log-polar representation (area V1) corresponds to a maximally circular, closed curve in the retinal image. The shortest path is a spatially global interpolation – see examples in Figures 3 and 4.

Shortest path in the retinal image:

a) Gray scale image b) Canny edge detection c) Floor removed d) 2D convex hull e) Multiple shortest pathsFigure 5. Binocular analysis of 3D texture followed by shortest path computation.

a) Gray scale image b) Canny edge detection c) Floor removed d) 2D convex hull e) Multiple shortest pathsFigure 6. Binocular analysis of 3D texture followed by shortest path computation.

Schwartz, E.L. (1980) Computational anatomy and functional architecture of striate cortex: A spatial approach to perceptual coding. Vision Research 20, 645-669.