In this problem we will take a look at interest point detection

Please complete the code according to the instruction given in above code skeleton. DDONT CHANGE ANYTHING

CODE SKELETON:

import numpy as np

from scipy.ndimage import convolve, maximum$\_$filter

def derivative$\_$filters$()$:

$"""$ Create derivative filters for x and y direction

Returns:

fx: derivative filter in x direction

fy: derivative filter in y direction

$"""$

fx $=$ np$.$array$([[0\mathrm{.}5,0,-0\mathrm{.}5]])$

fy $=$ fx$.$transpose$()$

return fx$,$ fy

def gauss$\_2$d$($sigma$,$ fsize$)$:

$"""$ Create a $2$D Gaussian filter

Args:

sigma: width of the Gaussian filter

fsize: $($h$,$ w$)$ dimensions of the filter

Returns:

$*$normalized$*$ Gaussian filter as $($h$,$ w$)$ np$.$array

$"""$

m$,$ n $=$ fsize

x $=$ np$.$arange$(-$m $/2+0\mathrm{.}5,$ m $/2)$

y $=$ np$.$arange$(-$n $/2+0\mathrm{.}5,$ n $/2)$

xx$,$ yy $=$ np$.$meshgrid$($x$,$ y$,$ sparse$=$True$)$

g $=$ np$.$exp$(-($xx $**2+$ yy $**2)/(2*$ sigma $**2))$

return g $/$ np$.$sum$($g$)$

def compute$\_$hessian$($img$,$ gauss, fx$,$ fy$)$:

$"""$ Compute elements of the Hessian matrix

Args:

img: numpy array with the image

gauss: Gaussian filter

fx: derivative filter in x direction

fy: derivative filter in y direction

Returns:

I$\_$xx: $($h$,$ w$)$ np$.$array of $2$nd derivatives in x direction

I$\_$yy: $($h$,$ w$)$ np$.$array of $2$nd derivatives in y direction

I$\_$xy: $($h$,$ w$)$ np$.$array of $2$nd derivatives in x$-$y direction

$"""$

#

# You code here

#

def compute$\_$criterion$($I$\_$xx$,$ I$\_$yy$,$ I$\_$xy$,$ sigma$)$:

$"""$ Compute criterion function

Args:

I$\_$xx: $($h$,$ w$)$ np$.$array of $2$nd derivatives in x direction

I$\_$yy: $($h$,$ w$)$ np$.$array of $2$nd derivatives in y direction

I$\_$xy: $($h$,$ w$)$ np$.$array of $2$nd derivatives in x$-$y direction

sigma: scaling factor

Returns:

criterion: $($h$,$ w$)$ np$.$array of scaled determinant of Hessian matrix

$"""$

#

# You code here

#

def non$\_$max$\_$suppression$($criterion$,$ threshold$)$:

$"""$ Apply non$-$maximum suppression to criterion values

and return Hessian interest points

Args:

criterion: $($h$,$ w$)$ np$.$array of criterion function values

threshold: criterion threshold

Returns:

rows: $($n$,)$ np$.$array with y$-$positions of interest points

cols: $($n$,)$ np$.$array with x$-$positions of interest points

$"""$

#

# You code here

#

# Set paramters and load the image

sigma $=5$

threshold $=1\mathrm{.}5$e$-3$

color, gray $=$ load$\_$img$("$data$/$a$3$p$2.$png$")$

# Generate filters and compute Hessian

fx$,$ fy $=$ derivative$\_$filters$()$

gauss $=$ gauss$\_2$d$($sigma$,(25,25))$

I$\_$xx$,$ I$\_$yy$,$ I$\_$xy $=$ compute$\_$hessian$($gray$,$ gauss, fx$,$ fy$)$

# Show components of Hessian matrix

plt$.$figure$()$

plt$.$subplot$(1,4,1)$

plot$\_$heatmap$($I$\_$xx$,"$I$\_$xx$")$

plt$.$subplot$(1,4,2)$

plot$\_$heatmap$($I$\_$yy$,"$I$\_$yy$")$

plt$.$subplot$(1,4,3)$

plot$\_$heatmap$($I$\_$xy$,"$I$\_$xy$")$

# Compute and show Hessian criterion

criterion $=$ compute$\_$criterion$($I$\_$xx$,$ I$\_$yy$,$ I$\_$xy$,$ sigma$)$

plt$.$subplot$(1,4,4)$

plot$\_$heatmap$($criterion$,$ "Determinant of Hessian"$)$

# Show all interest points where criterion is greater than threshold

rows, cols $=$ np$.$nonzero$($criterion $>$ threshold$)$

show$\_$points$($color$,$ rows, cols$)$

# Apply non$-$maximum suppression and show remaining interest points

rows, cols $=$ non$\_$max$\_$suppression$($criterion$,$ threshold$)$

show$\_$points$($color$,$ rows, cols$)$

plt$.$show$()$