Edge Detection

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This numerical tour explores local differential operators (grad, div, laplacian) and their use to perform edge detection.

In [1]:
from __future__ import division

import numpy as np
import scipy as scp
import pylab as pyl
import matplotlib.pyplot as plt

from nt_toolbox.general import *
from nt_toolbox.signal import *

import warnings

%matplotlib inline
%load_ext autoreload
%autoreload 2

Diffusion and Convolution

To obtain robust edge detection method, it is required to first remove the noise and small scale features in the image. This can be achieved using a linear blurring kernel.

Size of the image.

In [2]:
n = 256*2

Load an image $f_0$ of $N=n \times n$ pixels.

In [3]:
f0 = load_image("nt_toolbox/data/hibiscus.bmp",n)

Display it.

In [4]:

Blurring is achieved using convolution: $$ f \star h(x) = \sum_y f(y-x) h(x) $$ where we assume periodic boundary condition.

This can be computed in $O(N\log(N))$ operations using the FFT, since $$ g = f \star h \qarrq \forall \om, \quad \hat g(\om) = \hat f(\om) \hat h(\om). $$

In [5]:
cconv = lambda f, h: np.real(pyl.ifft2(pyl.fft2(f)*pyl.fft2(h)))

Define a Gaussian blurring kernel of width $\si$: $$ h_\si(x) = \frac{1}{Z} e^{ -\frac{x_1^2+x_2^2}{2\si^2} }$$ where $Z$ ensure that $\hat h(0)=1$.

In [6]:
t = np.hstack((np.arange(0,n//2+1),np.arange(-n//2+1,0)))
[X2, X1] = np.meshgrid(t, t)
normalize = lambda h: h/np.sum(h)
h = lambda sigma: normalize(np.exp(-(X1**2 + X2**2)/(2*sigma**2)))

Define blurring operator.

In [7]:
blur = lambda f, sigma: cconv(f, h(sigma))

Exercise 1

Test blurring with several blurring size $\si$.

In [8]:
run -i nt_solutions/segmentation_1_edge_detection/exo1
In [9]:
## Insert your code here.

Gradient Based Edge Detectiors

The simplest edge detectors only make use of the first order derivatives.

For continuous functions, the gradient reads $$ \nabla f(x) = \pa{ \pd{f(x)}{x_1}, \pd{f(x)}{x_2} } \in \RR^2. $$

We discretize this differential operator using first order finite differences. $$ (\nabla f)_i = ( f_{i_1,i_2}-f_{i_1-1,i_2}, f_{i_1,i_2}-f_{i_1,i_2-1} ) \in \RR^2. $$ Note that for simplity we use periodic boundary conditions.

Compute its gradient, using (here decentered) finite differences.

In [10]:
s = np.hstack(([n-1],np.arange(0,n-1)))
nabla = lambda f: np.concatenate(((f - f[s,:])[:,:,np.newaxis], (f - f[:,s])[:,:,np.newaxis]), axis=2)

One thus has $ \nabla : \RR^N \mapsto \RR^{N \times 2}. $

In [11]:
v = nabla(f0)

One can display each of its components.

In [12]:
imageplot(v[:,:,0], "d/dx", [1,2,1])
imageplot(v[:,:,1], "d/dy", [1,2,2])

A simple edge detector is simply obtained by obtained the gradient magnitude of a smoothed image.

A very simple edge detector is obtained by simply thresholding the gradient magnitude above some $t>0$. The set $\Ee$ of edges is then $$ \Ee = \enscond{x}{ d_\si(x) \geq t } $$ where we have defined $$ d_\si(x) = \norm{\nabla f_\si(x)}, \qwhereq f_\si = f_0 \star h_\si. $$

Compute $d_\si$ for $\si=1$.

In [13]:
sigma = 1
d = np.sqrt(np.sum(nabla(blur(f0, sigma))**2, 2))

Display it.

In [14]:

Exercise 2

For $\si=1$, study the influence of the threshold value $t$.

In [15]:
run -i nt_solutions/segmentation_1_edge_detection/exo2
In [16]:
## Insert your code here.

Exercise 3

Study the influence of $\si$.

In [17]:
run -i nt_solutions/segmentation_1_edge_detection/exo3
In [18]:
## Insert your code here.

Zero-crossing of the Laplacian

Defining a Laplacian requires to define a divergence operator. The divergence operator maps vector field to images. For continuous vector fields $v(x) \in \RR^2$, it is defined as $$ \text{div}(v)(x) = \pd{v_1(x)}{x_1} + \pd{v_2(x)}{x_2} \in \RR. $$ It is minus the adjoint of the gadient, i.e. $\text{div} = - \nabla^*$.

It is discretized, for $v=(v^1,v^2)$ as $$ \text{div}(v)_i = v^1_{i_1+1,i_2} + v^2_{i_1,i_2+1}. $$

In [19]:
def div(v):
    v0 = v[:,:,0]
    v1 = v[:,:,1]
    t = np.hstack((np.arange(1,n),[0]))
    return v0[t,:] - v0 + v1[:,t] - v1

The Laplacian operatore is defined as $\Delta=\text{div} \circ \nabla = -\nabla^* \circ \nabla$. It is thus a negative symmetric operator.

In [20]:
delta = lambda f: div(nabla(f))

Display $\Delta f_0$.

In [21]:

Check that the relation $ \norm{\nabla f} = - \dotp{\Delta f}{f}. $

In [22]:
dotp = lambda a, b: np.sum(a*b)
print("Should be 0: %.10f " %(dotp(nabla(f0), nabla(f0)) + dotp(delta(f0), f0)))
Should be 0: -0.0000000000 

The zero crossing of the Laplacian is a well known edge detector. This requires first blurring the image (which is equivalent to blurring the laplacian). The set $\Ee$ of edges is defined as: $$ \Ee = \enscond{x}{ \Delta f_\si(x) = 0 } \qwhereq f_\si = f_0 \star h_\si . $$

It was proposed by Marr and Hildreth:

Marr, D. and Hildreth, E., Theory of edge detection, In Proc. of the Royal Society London B, 207:187-217, 1980.

Display the zero crossing.

In [23]:
from nt_toolbox.plot_levelset import *
sigma = 4
plot_levelset(delta(blur(f0, sigma)), 0, f0)