# Scalar Edge Detectors

### color image operators map

* Definition: Since color images are arrays of three-component color vectors, the use of scalar edge detectors requires pixels’ dimensionality reduction through the conversion of the color image to its luminance-based equivalent* .

Since color images are arrays of three-component color vectors, the use of scalar edge detectors requires pixels’ dimensionality reduction through the conversion of the color image to its luminance-based equivalent. Then traditional (scalar) edge operators, such as those used in gray-scale imaging are applied to the luminance image to obtain the corresponding edge map (Figure 1).

Alternatively, the edge map of the color image can be achieved using component-wise processing. In this way, each of the three color channels are processed separately (Figure 2). The operator then combines the three distinct edge maps to form the output map. The output edge description corresponds to the dominant indicator of the edge activity noticed in the different color bands.

It is well documented in the literature that edge operators are usually sensitive to noise and small variations in intensity, and therefore the achieved edge map contains noise. To improve performance the edge operator’s output in Figures 1 or 2 is often compared with a predefined threshold in order to generate more accurate edge maps. The purpose of the thresholding operation is to increase the accuracy of the edge maps by extracting the structural information which corresponds to the edge discontinuities.

Both scalar edge detection approaches shown in Figures 1-2 can be grouped into two main classes of operators, namely gradient methods (e.g. Canny, Prewitt, Sobel, and isotropic operators), which use the first-order directional derivatives of the image to determine the edge contrast used in edge map formation, and ii) zero-crossing based methods (e.g. Laplacian, LwG, LoG, and DoG operators), which use the second-order directional derivatives to identify locations with zero crossings. The first derivative provides information on the rate of change of the image intensity. Of particular interest is the gradient magnitude denoting the rate of change of the image intensity and the gradient direction denoting the orientation of an edge. Note that when the first derivative achieves a maximum, the second derivative is zero. Thus, operators may localize edges by evaluating the zeros of the second derivatives.

In practice, both gradient and zero-crossing edge operators are approximated through the use of convolution masks. Scalar edge operators do not use the full potential of the spectral image content and thus, they can miss the edges in multichannel images.

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