Laplacian of Gaussian (LOG) Detector

[Lindeberg, 1998]

• Uses the concept of Scale Space.
• Instead of taking zero-crossings (as used for edge detection), consider the point which is a maximum among its 26 neighbors (9 in the same scale + 9 in the scale above + 9 in the scale below = 27, minus the point itself).

$L(x,\sigma )={\sigma }^2(I_{xx}(x,\sigma )+I_{yy}(x,\sigma ))$

• LOG can be used for finding the characteristic scale for a given image location. This is the scale at which the LOG response is maximized.
• LOG can be used for finding scale-invariant regions by searching for 3D (location + scale) extrema of the LOG.
• LOG is also used for edge detection (by finding zero-crossings).

LOG Detector: Flowchart

1. Compute the second derivatives: $I_{xx}(\sigma )+I_{yy}(\sigma )$
2. Apply Gaussian smoothing at different scales ($\sigma$, ${\sigma }^2$, ${\sigma }^3$, ${\sigma }^4$,…).
3. Construct a scale-space representation.
4. Find local maxima in 3D (x, y, scale). This produces a list of (x, y, s) coordinates, where (x, y) is the location and s is the scale.

LOG Detector: Result

The detector identifies blob-like regions at various scales.

Difference of Gaussian (DOG) Detector

[Lowe, 2004]

Approximates the LOG using a Difference of Gaussians (DOG) for computational efficiency.

$D(x,\sigma )=(G(x,k\sigma )-G(x,\sigma ))\ast I(x)$

Where:

• $G(x,\sigma )$ is a Gaussian kernel with standard deviation $\sigma$.
• $k=2^{1/K}$, where K is a constant (determines the number of scales per octave).
• K = 0, 1, 2, …, constant
• $I(x)$ is the input image.

Consider the region where the DOG response is greater than a threshold and the scale lies in a predefined range [$s_{min}$, $s_{max}$].

DOG Detector: Flowchart

5. Apply Gaussian smoothing at different scales to the input image.
6. Compute the difference between successive Gaussian-smoothed images. This creates the Difference of Gaussian (DOG) images.
7. Construct a scale-space pyramid (multiple octaves, where each octave represents a halving of the image resolution).
8. Find local extrema (maxima or minima) in the 3D scale-space (x, y, scale). A point X is selected if it is larger or smaller than all its neighbors in the current scale and adjacent scales.

DOG Detector: Result

Similar to the LOG detector, the DOG detector identifies blob-like features at multiple scales.