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 [2]:
using PyPlot
using NtToolBox
# using Autoreload
# arequire("NtToolBox")

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 [3]:
n = 256*2;

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

In [4]:
f0 = load_image("NtToolBox/src/data/hibiscus.png", n);

Display it.

In [5]:
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 [6]:
cconv = (f, h) -> real(plan_ifft((plan_fft(f)*f).*(plan_fft(h)*h))*((plan_fft(f)*f).*(plan_fft(h)*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 [7]:
# include("NtToolBox/src/ndgrid.jl")
t = [collect(0 : Base.div(n, 2)); collect(-Base.div(n, 2) + 1 : -1)]
(X2, X1) = meshgrid(t, t)
normalize = h -> h./sum(h)
h = sigma -> normalize(exp(-(X1.^2 + X2.^2)./(2*sigma^2)));

Define blurring operator.

In [8]:
blur = (f, sigma) -> cconv(f, h(sigma));

Exercise 1

Test blurring with several blurring size $\si$.

In [9]:
include("NtSolutions/segmentation_1_edge_detection/exo1.jl");
In [10]:
## 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 [11]:
s = [[n]; collect(1:n-1)]
nabla = f -> cat(3, f - f[s, :], f - f[:, s]);

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

In [12]:
v = nabla(f0);

One can display each of its components.

In [14]:
figure(figsize = (10, 10))
imageplot(v[:,:,1], L"\frac{d}{dx}", [1,2,1])
imageplot(v[:,:,2], L"\frac{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 [15]:
sigma = 1
d = sqrt(sum(nabla(blur(f0, sigma)).^2, 3));

Display it.

In [17]:
figure(figsize=(5,5))
imageplot(d[:, :]);