ASCI Imaging Workshop 1995 25-27 October 1995 Venray, The Netherlands
-5-Segmentation of overlapping objects
Lucas J. van Vliet1 and Piet W. Verbeek
Pattern Recognition Group of the Faculty of Applied Physics Delft University of Technology
Lorentzweg 1, 2628 CJ Delft, The Netherlands
Traditional segmentation schemes are not capable of segmentation and subsequent labeling of overlapping objects. The reason is that overlap areas need multiple labels. As a first step one can reduce the overlap areas by representing objects by their edges. Using a priori knowledge about the shape of objects one can fit parametric models to the data. Examples are fitting procedures by least-squares models and the generalized Hough transform. Our method assumes no a priori knowledge of the objects to be detected. The only restriction we apply is a maximum curvature constraint to avoid discontinuities at corners. Using this constraint all lines are approximately straight at the scale of the selected window size. This is guaranteed for all highly oversampled images. The point-spread-function rounds all corners to a minimum contour radius and the oversampling factor scales this minimum contour radius according our requirements. In this abstract we focus our attention on contours. These contours can be either the original objects or the result of an edge detection scheme.
From single-valuedness to multi-valuedness
Based upon the grey level of a pixel it can be classified as an object or a background pixel. In the case of overlapping contours the points of overlap are members of more than one object. Such points ask for a multiple label. A local feature that fills the need of multi-valuedness is orientation. Points where contours overlap support several distinct orientations. In order to detect which orientations pass through a certain point several techniques can be applied. For now we choose a local orientation histogram [1] method based on a modified local Hough transform. This orientation transform works as follows. A window scans over the image. Straight lines passing through the origin (the center of the window) can be characterized by xcosφ+ysinφ= 0, with x and y the window coordinates with respect to the window center. Thus a pixel at position (x,y) inside the window lies on a line through its center with orientation φ(x,y) = –tan–1(x/y). Like in Hough transforms the orientation parameter is discretized in a fixed number of bins at orientations {φk} which accumulate the intensity of pixels for which
φk < φ(x,y) < φk+1. If we slide the window over the entire image, the 2D image is transformed into a 3D image in which the first two coordinates are the original image coordinates and the third coordinate is the local orientation. Note that this 3D image is intrinsically periodic in the orientation dimension. In principle we now could analyze the histograms of all 2D pixels separately. For a pixel with n peaks in the histogram, the
n peak positions could be attributed to the pixel as n orientation values. This is a
multi-valued 2D image. We prefer the single-multi-valued 3D image and shall use it henceforth.
1 This work was partially supported by the Rolling Grants Program of the Foundation for Fundamental
ASCI Imaging Workshop 1995 25-27 October 1995 Venray, The Netherlands
-6-Object separation in the (X,Y,Φ) space
The (X,Y,Φ)-space can be interpreted as follows. High values at location (x,y,φ) indicate that the current orientation φ is likely to be present at the current position (x,y). Thresholding can be applied to binarize the (X,Y,Φ)-space. It produces an image in which all likely (x,y,φ) are set to 1 and everything else is set to zero. An important property of objects in (X,Y,Φ)-space is due to the maximum curvature constraint that was posed upon all objects. The maximum curvature constraint for individual objects guarantees that these objects are also connected in (X,Y,Φ)-space. While overlapping objects are connected in (X,Y)-space, they are separated in (X,Y,Φ)-space.
Figure 1 demonstrates the separation of two overlapping contours using labeling in (X,Y,Φ)-space. The original image is transformed into orientation space. The resulting image was binarized using the isodata threshold level. A 3-D labeling routine with periodic boundary conditions in the Φ-direction yields the desired result: separate labels for overlapping objects.
a) b)
c) d)
Figure 1: a) original image of overlapping circles; b) slices with constant orientation; c) orientation transform: (X,Y,Φ)-space; d) labeling in orientation space, periodic in Φ; [1] J. Birk, R. Kelly, N. Chen and L. Wilson, Image feature extraction using diameter-limited gradient
