The Laplacian operator is a template which implements second-order differencing. (zero-crossing edge detector)

f''(x) = f'(x) - f'(x+1)

= -f(x) + 2f(x+1) - f(x+2)

The 2nd order template horizontal / vertical looks like

-1 2 -1

--------------

0 -1 0

-1 4 -1

0 -1 0

it is important to ensure that the sum of template coefficients is zero, so that edges are not detected in areas of uniform brightness.

Gaussian smoothing first, then Laplacian. Since the convolution operation is associative, we can convolve the Gaussian smoothing filter with the Laplacian filter first of all, and then convolve this hybrid filter with the image to achieve the required result (LoG).

Because of the shape Laplacian of Gaussian filters are often called Mexican hat filters.

So LoG includes smoothing within the differencing action. The Gaussian operator suppresses the influence of points away from the centre of the template, basing differentiation on those points nearer the centre; the standard deviation, σ, is chosen to ensure this action. The transform is circular-symmetric. Since the transform reveals that the LoG operator emits low and high frequencies (those close to the origin, and those far away from the origin) it is equivalent to a band-pass filter. Choice of the value of σ controls the spread of the operator in the spatial domain and the ‘width’ of the band in the frequency domain: setting σ to a high value gives low-pass filtering. If σ is very small, the smoothing has no effect, so LoG becomes the simple Laplacian kernel.
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