Edge Detection¶

Important: Please read the installation page for details about how to install the toolboxes. $\newcommand{\dotp}[2]{\langle #1, #2 \rangle}$ $\newcommand{\enscond}[2]{\lbrace #1, #2 \rbrace}$ $\newcommand{\pd}[2]{ \frac{ \partial #1}{\partial #2} }$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\umax}[1]{\underset{#1}{\max}\;}$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\uargmin}[1]{\underset{#1}{argmin}\;}$ $\newcommand{\norm}[1]{\|#1\|}$ $\newcommand{\abs}[1]{\left|#1\right|}$ $\newcommand{\choice}[1]{ \left\{ \begin{array}{l} #1 \end{array} \right. }$ $\newcommand{\pa}[1]{\left(#1\right)}$ $\newcommand{\diag}[1]{{diag}\left( #1 \right)}$ $\newcommand{\qandq}{\quad\text{and}\quad}$ $\newcommand{\qwhereq}{\quad\text{where}\quad}$ $\newcommand{\qifq}{ \quad \text{if} \quad }$ $\newcommand{\qarrq}{ \quad \Longrightarrow \quad }$ $\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\RR}{\mathbb{R}}$ $\newcommand{\EE}{\mathbb{E}}$ $\newcommand{\Zz}{\mathcal{Z}}$ $\newcommand{\Ww}{\mathcal{W}}$ $\newcommand{\Vv}{\mathcal{V}}$ $\newcommand{\Nn}{\mathcal{N}}$ $\newcommand{\NN}{\mathcal{N}}$ $\newcommand{\Hh}{\mathcal{H}}$ $\newcommand{\Bb}{\mathcal{B}}$ $\newcommand{\Ee}{\mathcal{E}}$ $\newcommand{\Cc}{\mathcal{C}}$ $\newcommand{\Gg}{\mathcal{G}}$ $\newcommand{\Ss}{\mathcal{S}}$ $\newcommand{\Pp}{\mathcal{P}}$ $\newcommand{\Ff}{\mathcal{F}}$ $\newcommand{\Xx}{\mathcal{X}}$ $\newcommand{\Mm}{\mathcal{M}}$ $\newcommand{\Ii}{\mathcal{I}}$ $\newcommand{\Dd}{\mathcal{D}}$ $\newcommand{\Ll}{\mathcal{L}}$ $\newcommand{\Tt}{\mathcal{T}}$ $\newcommand{\si}{\sigma}$ $\newcommand{\al}{\alpha}$ $\newcommand{\la}{\lambda}$ $\newcommand{\ga}{\gamma}$ $\newcommand{\Ga}{\Gamma}$ $\newcommand{\La}{\Lambda}$ $\newcommand{\si}{\sigma}$ $\newcommand{\Si}{\Sigma}$ $\newcommand{\be}{\beta}$ $\newcommand{\de}{\delta}$ $\newcommand{\De}{\Delta}$ $\newcommand{\phi}{\varphi}$ $\newcommand{\th}{\theta}$ $\newcommand{\om}{\omega}$ $\newcommand{\Om}{\Omega}$

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
warnings.filterwarnings('ignore')

%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]:

plt.figure(figsize=(5,5))
imageplot(f0)

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]:

plt.figure(figsize=(10,10))
imageplot(v[:,:,0], "d/dx", [1,2,1])
imageplot(v[:,:,1], "d/dy", [1,2,2])