This numerical tour explores some basic image processing tasks.
Important: Please read the installation page for details about how to install the toolboxes. $\newcommand{\dotp}[2]{\langle #1, #2 \rangle}$ $\newcommand{\enscond}[2]{\lbrace #1, #2 \rbrace}$ $\newcommand{\pd}[2]{ \frac{ \partial #1}{\partial #2} }$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\umax}[1]{\underset{#1}{\max}\;}$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\uargmin}[1]{\underset{#1}{argmin}\;}$ $\newcommand{\norm}[1]{\|#1\|}$ $\newcommand{\abs}[1]{\left|#1\right|}$ $\newcommand{\choice}[1]{ \left\{ \begin{array}{l} #1 \end{array} \right. }$ $\newcommand{\pa}[1]{\left(#1\right)}$ $\newcommand{\diag}[1]{{diag}\left( #1 \right)}$ $\newcommand{\qandq}{\quad\text{and}\quad}$ $\newcommand{\qwhereq}{\quad\text{where}\quad}$ $\newcommand{\qifq}{ \quad \text{if} \quad }$ $\newcommand{\qarrq}{ \quad \Longrightarrow \quad }$ $\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\RR}{\mathbb{R}}$ $\newcommand{\EE}{\mathbb{E}}$ $\newcommand{\Zz}{\mathcal{Z}}$ $\newcommand{\Ww}{\mathcal{W}}$ $\newcommand{\Vv}{\mathcal{V}}$ $\newcommand{\Nn}{\mathcal{N}}$ $\newcommand{\NN}{\mathcal{N}}$ $\newcommand{\Hh}{\mathcal{H}}$ $\newcommand{\Bb}{\mathcal{B}}$ $\newcommand{\Ee}{\mathcal{E}}$ $\newcommand{\Cc}{\mathcal{C}}$ $\newcommand{\Gg}{\mathcal{G}}$ $\newcommand{\Ss}{\mathcal{S}}$ $\newcommand{\Pp}{\mathcal{P}}$ $\newcommand{\Ff}{\mathcal{F}}$ $\newcommand{\Xx}{\mathcal{X}}$ $\newcommand{\Mm}{\mathcal{M}}$ $\newcommand{\Ii}{\mathcal{I}}$ $\newcommand{\Dd}{\mathcal{D}}$ $\newcommand{\Ll}{\mathcal{L}}$ $\newcommand{\Tt}{\mathcal{T}}$ $\newcommand{\si}{\sigma}$ $\newcommand{\al}{\alpha}$ $\newcommand{\la}{\lambda}$ $\newcommand{\ga}{\gamma}$ $\newcommand{\Ga}{\Gamma}$ $\newcommand{\La}{\Lambda}$ $\newcommand{\si}{\sigma}$ $\newcommand{\Si}{\Sigma}$ $\newcommand{\be}{\beta}$ $\newcommand{\de}{\delta}$ $\newcommand{\De}{\Delta}$ $\newcommand{\phi}{\varphi}$ $\newcommand{\th}{\theta}$ $\newcommand{\om}{\omega}$ $\newcommand{\Om}{\Omega}$
using NtToolBox
using PyPlot
WARNING: Method definition ndgrid(AbstractArray{T<:Any, 1}) in module NtToolBox at /Users/gpeyre/.julia/v0.5/NtToolBox/src/ndgrid.jl:3 overwritten at /Users/gpeyre/.julia/v0.5/NtToolBox/src/ndgrid.jl:3. WARNING: Method definition ndgrid(AbstractArray{#T<:Any, 1}, AbstractArray{#T<:Any, 1}) in module NtToolBox at /Users/gpeyre/.julia/v0.5/NtToolBox/src/ndgrid.jl:6 overwritten at /Users/gpeyre/.julia/v0.5/NtToolBox/src/ndgrid.jl:6. WARNING: Method definition ndgrid_fill(Any, Any, Any, Any) in module NtToolBox at /Users/gpeyre/.julia/v0.5/NtToolBox/src/ndgrid.jl:13 overwritten at /Users/gpeyre/.julia/v0.5/NtToolBox/src/ndgrid.jl:13. WARNING: Method definition ndgrid(AbstractArray{#T<:Any, 1}...) in module NtToolBox at /Users/gpeyre/.julia/v0.5/NtToolBox/src/ndgrid.jl:19 overwritten at /Users/gpeyre/.julia/v0.5/NtToolBox/src/ndgrid.jl:19. WARNING: Method definition meshgrid(AbstractArray{T<:Any, 1}) in module NtToolBox at /Users/gpeyre/.julia/v0.5/NtToolBox/src/ndgrid.jl:33 overwritten at /Users/gpeyre/.julia/v0.5/NtToolBox/src/ndgrid.jl:33. WARNING: Method definition meshgrid(AbstractArray{#T<:Any, 1}, AbstractArray{#T<:Any, 1}) in module NtToolBox at /Users/gpeyre/.julia/v0.5/NtToolBox/src/ndgrid.jl:36 overwritten at /Users/gpeyre/.julia/v0.5/NtToolBox/src/ndgrid.jl:36. WARNING: Method definition meshgrid(AbstractArray{#T<:Any, 1}, AbstractArray{#T<:Any, 1}, AbstractArray{#T<:Any, 1}) in module NtToolBox at /Users/gpeyre/.julia/v0.5/NtToolBox/src/ndgrid.jl:44 overwritten at /Users/gpeyre/.julia/v0.5/NtToolBox/src/ndgrid.jl:44.
Several functions are implemented to load and display images.
First we load an image.
path to the images
name = "NtToolBox/src/data/lena.png"
n = 256
M = load_image(name, n);
We can display it. It is possible to zoom on it, extract pixels, etc.
imageplot(M[Int(n/2 - 25) : Int(n/2 + 25), Int(n/2 - 25) : Int(n/2 + 25)], "Zoom", [1, 2, 2])
PyObject <matplotlib.text.Text object at 0x32dc25ad0>
An image is a 2D array, that can be modified as a matrix.
imageplot(-M, "-M", [1,2,1])
imageplot(M[end:-1:1,1:size(M, 2)], "Flipped", [1,2,2])
PyObject <matplotlib.text.Text object at 0x332945e50>
Blurring is achieved by computing a convolution with a kernel.
Compute the low pass Gaussian kernel. Warning, the indexes need to be modulo $n$ in order to use FFTs.
sigma = 5
X = [0:n/2; -n/2:-2]'
Y = [0:n/2; -n/2:-2]
h = exp(-(X.^2 .+ Y.^2)/(2*(sigma)^2))
h = h/sum(h)
imageplot(fftshift(h))
Compute the periodic convolution ussing FFTs
Mh = conv2(Array{Float64, 2}(M), h)
Mh = Mh[1:255, 1:255] + Mh[257:511, 1:255] + Mh[1:255, 257:511] + Mh[257:511, 257:511];
Display
imageplot(M, "Image", [1, 2, 1])
imageplot(Mh, "Blurred", [1, 2, 2])
PyObject <matplotlib.text.Text object at 0x32ec90890>
Several differential and convolution operators are implemented.
(G_x, G_y) = Images.imgradients(M)
imageplot(G_x, "d/ dx", [1, 2, 1])
imageplot(G_y, "d/ dy", [1, 2, 2])
WARNING: the order of outputs has switched (`grad1, grad2 = imgradients(img)` rather than `gradx, grady = imgradients`). Silence this warning by providing a kernelfun, e.g., imgradients(img, KernelFactors.ando3).
in depwarn(::String, ::Symbol) at ./deprecated.jl:64
in imgradients(::Array{Float32,2}) at /Users/gpeyre/.julia/v0.5/ImageFiltering/src/specialty.jl:50
in include_string(::String, ::String) at ./loading.jl:441
in execute_request(::
The 2D Fourier transform can be used to perform low pass approximation and interpolation (by zero padding).
Compute and display the Fourier transform (display over a log scale). The function fftshift is useful to put the 0 low frequency in the middle. After fftshift, the zero frequency is located at position $(n/2+1,n/2+1)$.
Mf = plan_fft(M)
Mf*M
Lf = fftshift(log(abs(Mf*M) + 1e-1))
imageplot(M, "Image", [1, 2, 1])
imageplot(Lf, "Fourier transform", [1, 2, 2])
PyObject <matplotlib.text.Text object at 0x336290890>
Exercise 1: To avoid boundary artifacts and estimate really the frequency content of the image (and not of the artifacts!), one needs to multiply M by a smooth windowing function h and compute fft2(M*h). Use a sine windowing function. Can you interpret the resulting filter ?
include("NtSolutions/introduction_3_image/exo1.jl")
PyObject <matplotlib.text.Text object at 0x332bd2350>
Exercise 2: Perform low pass filtering by removing the high frequencies of the spectrum. What do you oberve ?
include("NtSolutions/introduction_3_image/exo2.jl")
PyObject <matplotlib.text.Text object at 0x32e7330d0>