The Laplacian of Gaussian (Marr-Hildreth Operator)

It is common for a single image to contain edges having widely different sharpnesses and scales, from blurry and gradual to crisp and abrupt. Edge scale information is often useful as an aid toward image understanding. For instances, edges at low resolution tend to indicate gross shapes, whereas texture tends to become important at higher resolutions. An edge detected over a wide range of scale is more likely to be physically significant in the scene than an edge found only within a narrow range of scale. Furthermore, the effects of noise are usually most deleterious at the finer scales.

Marr and Hildreth advocated the need for an operator that can be tuned to detect edges at a particular scale. Their method is based filtering the image with a Gaussian kernel selected for a particular edge scale. The Gaussian smoothing operation serves to band-limit the image to a small range of frequencies, reducing the noise sensitivity problem when detecting zero crossings. The image is filtered over a variety of scales and the Laplacian zero crossings are computed at each. This produces a set of edge maps as a function of edge scale. Each edge point can be considered to reside in a region of scale space, for which edge point location is a function of x, y and s . Scale space has been successfully used to refine and analyze edge maps.

The Gaussian has some very desirable properties that facilitate this edge detection procedure. First, the Gaussian function is smooth and localized in both the spatial and frequency domains, providing a good compromise between the need for avoiding false edges and for minimizing errors in edge position. . In fact, the Gaussian is the only real-valued function that minimizes the product of spatial and frequency-domain spreads. The Laplacian of Gaussuan essentially acts as a bandpass filter because of its differential and smoothing behavior. Second, the Gaussian is separable, which helps make computation very efficient.

Omitting the scaling factor, the Gaussian filter can be written

	(5.8.6)

Its frequency response, is also Gaussian:

The parameter is inversely related to the cutoff frequency.

Because the convolution and Laplacian operations are both linear and shift invariant, their computation order can be interchanged.

	(5.8.7)

Here we take advantage of the fact that the derivative is linear operator. Therefore, Gaussian filtering following by differentiation is the same as filtering with the derivative of a Gaussian. The right-hand side of Eq.(5.8.7) usually provides for more efficient computation since can be prepared in advance as a result of its image independence. The Laplacian of Gaussian(LoG) filter, therefore has the following impulse response.

	(5.8.8)