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This numerical tour explores some basic image processing tasks.
from __future__ import division
import nt_toolbox as nt
from nt_solutions import introduction_3_image as solutions
%matplotlib inline
%load_ext autoreload
%autoreload 2
Several functions are implemented to load and display images.
First we load an image.
path to the images
name = 'lena'
n = 256
M = load_image(name, [])
M = rescale(crop(M, n))
We can display it. It is possible to zoom on it, extract pixels, etc.
imageplot(M, 'Original', 1, 2, 1)
imageplot(crop(M, 50), 'Zoom', 1, 2, 2)
An image is a 2D array, that can be modified as a matrix.
imageplot(-M, '-M', 1, 2, 1)
imageplot(M(n: -1: 1, : ), 'Flipped', 1, 2, 2)
Blurring is achieved by computing a convolution with a kernel.
compute the low pass kernel
k = 9
h = ones(k, k)
h = h/ sum(h(: ))
compute the convolution
Mh = perform_convolution(M, h)
display
imageplot(M, 'Image', 1, 2, 1)
imageplot(Mh, 'Blurred', 1, 2, 2)
Several differential and convolution operators are implemented.
G = grad(M)
imageplot(G(: , : , 1), 'd/ dx', 1, 2, 1)
imageplot(G(: , : , 2), 'd/ dy', 1, 2, 2)
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 = fft2(M)
Lf = fftshift(log(abs(Mf) + 1e-1))
imageplot(M, 'Image', 1, 2, 1)
imageplot(Lf, 'Fourier transform', 1, 2, 2)
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 ? ompute kernel h ompute FFT isplay
solutions.exo1()
## Insert your code here.
Exercise 2
Perform low pass filtering by removing the high frequencies of the spectrum. What do you oberve ? isplay
solutions.exo2()
## Insert your code here.
It is possible to do image interpolating by adding high frequencies
p = 64
n = p*4
M = load_image('boat', 2*p); M = crop(M, p)
Mf = fftshift(fft2(M))
MF = zeros(n, n)
sel = n/ 2-p/ 2 + 1: n/ 2 + p/ 2
sel = sel
MF(sel, sel) = Mf
MF = fftshift(MF)
Mpad = real(ifft2(MF))
imageplot(crop(M), 'Image', 1, 2, 1)
imageplot(crop(Mpad), 'Interpolated', 1, 2, 2)
A better way to do interpolation is to use cubic-splines. It avoid ringing artifact because the spline kernel has a smaller support with less oscillations.
Mspline = image_resize(M, n, n)
imageplot(crop(Mpad), 'Fourier (sinc)', 1, 2, 1)
imageplot(crop(Mspline), 'Spline', 1, 2, 2)