Stephan Scholze 
  
2 EDGE DETECTION IN COLOR IMAGES 
Edge detection in color images is not a widely studied subject. In most practical applications, edge extraction is performed 
on the corresponding grey-level intensity image using "standard" edge-detectors such as the Canny-operator (Canny, 1986) 
or adaptions as the Deriche-operator (Deriche, 1987). These operators are based on the norm of the intensity gradient in 
each pixel. Denoting the smoothed grey-level intensity at pixel (x, y) with z(z, y) one would formally compute 
: iN Di” 
IVi(z, y)|| = (2) +5) (I) 
In case of a color image with red, green and blue bands, the image can be represented as a vector valued function 
I c R? 5 R? where I(z, y) — (r(z, y), g(z, y), b(z, y)) represents the RGB-values of the pixel (x, y). We compared 
the following combinations of the derivatives in the individual bands for replacing || Vz(z, y) ||. 
  
2. The Maximum 
We consider the vector valued function I(z, y) — (r(z; y), g(z, y), b(z, y)), representing the multispectral image at pixel 
(z, y). To replace the norm of the intensity gradient || Vi(z, y)|| from the single band case, we choose the maximum of 
the norm of the gradients in the individual color bands at each pixel position: 
I| Vi(z, y)]| — max t] Vr(z, y)ll, lIVoGr, v)]l, || Vb(z, v)]H () 
2.2 The Spatial Gradient 
In case of a scalar field the direction and magnitude of its strongest change are given by the gradient of the field. This idea 
can be extended to vector fields (Lee, 1991) and was recently used for edge detection in color images (Zafiropoulos and 
Schenk, 1999). Again we consider the vector valued image function I(z, y) — (r(z, y), g(z, y), b(x, y)) and an arbitrary 
direction n — (cos à, sin 9) which is defined by the angle ¢ in the image plane. The objective is to find the direction of 
strongest change in I(z, y) at point (z, y). 
First we compute the directional derivative of I(x, y) with respect to n: 
or Or 
ox Oy 
9r = S S n=Jn (3) 
on = 2 
ox oy 
which is equivalent with forming the scalar product of the Jacobian J of the image function I with n. As measure of 
magnitude of change of I(z, y) as a function of n one usually chooses the square of the norm of Jn: 
x In]? = n‘3J" In (4) 
With this choice, note that the n^ J7 Jn is equivalent with the Rayleigh-quotient of D — J7J. Thus the direction oí 
the largest change of I(z, y) at (x, y) is given by max(/?) which is in turn given by the largest eigenvector of D. Since 
D is real, the eigenvalues of D are the squares of the singular values of J. Clearly, computing the spatial gradient is 
computationally expensive, since we have to perform a singular value decomposition at each pixel position. 
2.3 Comparison of both combination schemes 
After edge detection and edge linking, straight line segments are fitted to the edges. A very tight threshold is used for li 
fitting so that curves are not piecewise linearly approximated. Moreover, only line segments above a minimum length (D 
pixels) will be considered for matching. For computational efficiency during further processing steps, the line segments 
are stored in an R-Tree (Guttman, 1981) data-structure. This allows efficient queries for adjacent lines, which will be 
exploited later. Additionally the straight lines are also stored in raster format, i.e. a pixel indexes a line segment if there 
one at that location. 
Since we want to investigate polygonal shapes, only the fitted straight line segments will be used all through the rest of 
the paper. We will not refer to the underlying unfitted data. Thus, in the following the terms ’edge’ and ’line segment 
will be used interchangeably, both referring to the fitted straight line segments. 
In Figure (1) the results of both edge detection schemes are compared with the result obtained by performing the edge 
detection only on the grey-level intensity image. At first glance, the three methods perform similarly well. The number of 
  
816 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 
  
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