DifferenceofGaussians

Difference of Gaussians (DoG) is an image processing operator that computes the difference between two Gaussian-blurred versions of the same image. By subtracting a smoother version from a less smooth version, DoG acts as a band-pass filter and emphasizes structures that lie between the two smoothing scales. It is commonly used as an efficient approximation to the Laplacian of Gaussian (LoG), which is more expensive to compute directly.

Mathematically, for an image I, the DoG at scales sigma1 and sigma2 is DoG_sigma1,sigma2(I) = I * G_sigma2 − I * G_sigma1, where G_sigma denotes a Gaussian kernel with standard deviation sigma. In practice sigma2 is chosen as a constant multiple of sigma1, often sigma2 = k * sigma1 with k > 1 (for example around 1.6). The result changes sign depending on local structure and can be used to detect both bright-on-dark and dark-on-bright features.

In scale-space applications, a DoG pyramid is built by creating blurred images at a sequence of increasing sigmas and subtracting adjacent levels. Extrema of the resulting DoG images across space and scale identify potential features, a principle exploited in blob detectors and feature descriptors such as SIFT.

Advantages include computational efficiency, since Gaussian blurs are separable and DoG avoids second derivatives. It provides a robust, broadband response to features of a characteristic size while being relatively simple to implement. Limitations include its approximate nature relative to LoG, dependence on scale parameters, and susceptibility to noise and edge responses if not paired with appropriate normalization and thresholding.