[CV][Detector] Harris Laplace alogrithm

In [2] there are 2 proposed implementation of Harris Laplace detectors as followings:

I. Standard Harris Laplace detector (without extension):

1. Build the scale-space representation with $\sigma _n = s^n \sigma_0$

2. At each scale level, detecting maximum point in 8 neighbours of that point and more than a threshold:
\[
det(\mu (x, \sigma _n)) - \alpha trace^2(\mu(x, \sigma_n))>threshold_H
\]

here, $\mu$ is second moment matrix (or auto-correlation matrix)

\[
\mu(\textbf{x}, \sigma_I, \sigma_D) = \begin{bmatrix}\mu_{11} & \mu_{12}\\ \mu_{21} & \mu_{22} \end{bmatrix} = \sigma_D^2g(\sigma_I) \star \begin{bmatrix} L_x^2(\boldsymbol{x}, \sigma_D) & L_xL_y(\boldsymbol{x}, \sigma_D)\\ L_xL_y(\boldsymbol{x}, \sigma_D)& L_y^2(\boldsymbol{x}, \sigma_D) \end{bmatrix}
\]

integration scale: $\sigma_I =\sigma_n$ and derivation scale $\sigma_D= k\sigma_I$

In details, given L is image at scale n (I is smoothed with gaussian $\sigma_n$).

- Compute three derivatives of L $(L_x, L_{xy}, L_y)$ using Gaussian-with-$\sigma_D$ derivative kernel.

-Compute $L_x^2 = L_x L_x, L_y^2 = L_y L_y$.

-Convolute $L_x^2, L_{xy}, L_y^2$ with $g(\sigma_I)$ (means that derivation value is weighted by Gaussian with $\sigma_I$) and multiplied by $\sigma_D^2$.

- Take cornerness:
\[
det(\mu (x, \sigma _n)) - \alpha trace^2(\mu(x, \sigma_n))>threshold_H
\]
- Find maximum point in 8-neighbours is greater than $threahold_H$ --> candidate point set
3 For each point in candidate set, compute Laplacian of Gaussian and take point with maximum over scale and greater than Laplacian threshold. Laplacian of Gaussian is given by:
\[
\sigma_n^2 \left| L_{xx}(\boldsymbol{x}, \sigma_n) + L_{yy}(\boldsymbol{x}, \sigma_n) \right| > threshold_L
\]
-Compute Laplacian of Gaussian with $\sigma_n$ at each candidate point and multiply this result by $\sigma_n^2$

Output: $(x, y, \sigma_n)$ - position and scale

Parameters:

Symbol	Meaning	Value
s	scale factor	1.2
$\sigma_0$	initial scale	1 (inference from $\sigma_n = s^n$)
$\sigma_n = s^n$	sigma range	n = (1:17)
$threshold_H$	Harris function threshold	1000
$threshold_L$	Laplacian of Gaussian threshold	10
$\alpha$	in Harris function	0.06
k	$\sigma_D/\sigma_I$	0.6

References:
[1] K. Mikolajczyk, K. and C. Schmid. "Scale and affine invariant interest point detectors". International Journal of Computer Vision. 2004

[2] K. Mikolajczyk's PhD thesis