Laplacian of Gaussian

The Laplacian of Gaussian

One of the first and also most common blob detectors is based on the Laplacian of the Gaussian (LoG). Given an input image $f(x,y)$, this image is convolved by a Gaussian kernel

$g(x,y,t)={\frac {1}{2\pi t^{2}}}e^{-{\frac {x^{2}+y^{2}}{2t^{2}}}}$

at a certain scale $t$ to give a scale space representation $L(x,y;t)\ =g(x,y,t)*f(x,y)$. Then, the result of applying the Laplacian operator

$\nabla ^{2}L=L_{xx}+L_{yy}$

is computed, which usually results in strong positive responses for dark blobs of extent ${\sqrt {2t}}$ and strong negative responses for bright blobs of similar size. A main problem when applying this operator at a single scale, however, is that the operator response is strongly dependent on the relationship between the size of the blob structures in the image domain and the size of the Gaussian kernel used for pre-smoothing. In order to automatically capture blobs of different (unknown) size in the image domain, a multi-scale approach is therefore necessary.

A straightforward way to obtain a multi-scale blob detector with automatic scale selection is to consider the scale-normalized Laplacian operator

$\nabla _{norm}^{2}L(x,y;t)=t(L_{xx}+L_{yy})$

and to detect scale-space maxima/minima, that are points that are simultaneously local maxima/minima of $\nabla _{norm}^{2}L$ with respect to both space and scale (Lindeberg 1994, 1998). Thus, given a discrete two-dimensional input image $f(x,y)$ a three-dimensional discrete scale-space volume $L(x,y,t)$ is computed and a point is regarded as a bright (dark) blob if the value at this point is greater (smaller) than the value in all its 26 neighbours. Thus, simultaneous selection of interest points $({\hat {x}},{\hat {y}})$ and scales ${\hat {t}}$ is performed according to

$({\hat {x}},{\hat {y}};{\hat {t}})=\operatorname {argmaxminlocal} _{(x,y;t)}(\nabla _{norm}^{2}L(x,y;t))$.

Note that this notion of blob provides a concise and mathematically precise operational definition of the notion of "blob", which directly leads to an efficient and robust algorithm for blob detection. Some basic properties of blobs defined from scale-space maxima of the normalized Laplacian operator are that the responses are covariant with translations, rotations and rescalings in the image domain. Thus, if a scale-space maximum is assumed at a point $(x_{0},y_{0};t_{0})$ then under a rescaling of the image by a scale factor $s$, there will be a scale-space maximum at $(sx_{0},sy_{0};s^{2}t_{0})$ in the rescaled image (Lindeberg 1998). This in practice highly useful property implies that besides the specific topic of Laplacian blob detection, local maxima/minima of the scale-normalized Laplacian are also used for scale selection in other contexts, such as in corner detection, scale-adaptive feature tracking (Bretzner and Lindeberg 1998), in the scale-invariant feature transform (Lowe 2004) as well as other image descriptors for image matching and object recognition.

The scale selection properties of the Laplacian operator and other closely scale-space interest point detectors are analyzed in detail in (Lindeberg 2013a). In (Lindeberg 2013b) it is shown that there exist other scale-space interest point detectors, such as the determinant of the Hessian operator, that perform better than Laplacian operator or its difference-of-Gaussians approximation for image-based matching using local SIFT-like image descriptors.