Month: June 2017

Signal
Filter
Normalized cross-correlation

Matlab cross-correlation doc
onion = rgb2gray(imread(‘onion.png’));
peppers = rgb2gray(imread(‘peppers.png’));
imshowpair(peppers, onion,’montage’)

Matlab cross-correlation doc
c = normxcorr2(onion,peppers);
figure, surf(c), shading flat;

pkg load image;
function index = find_template_1D(t, s)

endfunction

s = [-1 0 0 1 1 1 0 -1 -1 0 1 0 0 -1];
t = [1 1 0];

disp('Signal:'), disp([1:size(s, 2); s]);
disp('Template:'), disp([1:size(t, 2); t]);

index = find_template_1D(t, s);
disp('Index:'), disp(index);

10 15 20
23 90 27
33 31 30
sor-> 10 15 20 23 27 30 31 33 90
replace
10 15 20
23 27 27
33 31 30
Median filter is smooth nicer

pkg load image;

img = imread('moon.png');
imshow(img);

%% Add salt & pepper noise
noisy_img = imnoise(img, 'salt & pepper', 0.02);
imshow(noisy_img);

median_filtered = medfilt2(noisy_img);
imshow(median_filtered);

division is a linear operation?
true becase (x + y)/z = x/z + y/z

Boundary Issues
full, same, valid

Boundary issues
methods:
clip filter(black):imfilter(f,g,0)
wrap around(f,g,’circular’)
copy edge(f,g,’replicate’)
reflect across edge(f,g,’symmetric’)

pkg load image;

img = imread('fall-leaves.png');
imshow(img);

filter_size = 21;
filter_sigma = 3;
filter = fspecial('gaussian', filter_size, filter_sigma);

smoothed = imfilter(img, filter, 0);

The reflection method of handling boundary conditions in filtering is good because
the created imagery has the same statistic as the original image

An operator H is linear if two properties hold(f1 and f2 are some functions, a is a constant):

Additivity(things sum):
-H(f1 + f2) = H(f1) + H(f2)(looks like distributive law)

Multiplicative scaling(Homogeneity of degree 1)
(constant scales):
-H(a*f1) = a*H(f1)

Because it is sums ans multiplies, the “filtering” operation we were doing are linear:

An impulse function…
it’s just a value of 1 at a single location

An impulse response
-if I have unknown system and I put in an impulse, the response is called the impulse response
-so if the black box is linear you can describe H by h(x)

Assuming center coordinate is “reference point”.

kernel is size MxM, and our image was NxN.
filter whole image is M*M*N*N

Correlation vs Convolution(flip in both dimensions, bottom to top, right to left)

Shift invariant:
linear & shift invariant

Associative
(f*g)*h = f*(g*h)
Identity:
unit impulse e = […,0,0,1,0,0,…] f*e = f
Differentiation: α/αx(f*g) = αf/αx*g

Nearest neighboring pixels have the most influence
F(x, y)
H(u, v)
1 2 1
2 4 2
1 2 1
this kernel is an approximation of a Gaussian function
h(u, v) = 1/2πσ^2 e ^(- u^2+v^2/σ^2)

Gaussian filters
Variance(σ^2), standard deviation(σ): determines extent of smoothing

hsize = 31;
sigma = 5;
h = fspecial('gaussian', hsize, sigma);

surf(h);
imagesc(h);

outim = imfilter(im, h);
imshow(outim);

for sigma = 1:3:10
	h = fspecial('gaussian', fsize, sigma);
	out = imfilter(im, h);
	imshow(out);
	pause;
end

img = imread('saturn.png');
imshow(img);

noise = randn(size(img)) .* 25;
noisy_img = img + noise;
imshow(noisy_img);

filter_size = 11;
filter_sigma = 2;
pkg load image;
filter = fspecial('gaussian', filter_size, filter_sigma);

smoothed = imfilter(noisy_img, filter);

When filtering with a Gaussian, the sigma is most important- it defines the blur kernel’s scale with respect to the image.
Altering the normalization coefficient does not effect the blur, only the brightness.

F(x, y)
G(x, y)

Correlation filtering – uniform weights
2k + 1 * 2k + 1;
G[i, j] = 1/(2k + 1)^2 ΣΣF[i + u, j + v]
Loop over all pixels in neighborhood around image pixel F[i,j]

nonuniform weights
Now generalize to allow different wieghts depending on neighboring pixel’s relative position:
G[i, j] = ΣΣH[u, v]F[i+u, j+v]
This is called cross-correlation, denoted G = HF

averaging filter -> box filter
Blurry spot as a function

To blur a single pixel into a “blurry” spot, we would need to filter the spot with something that looks like a blurry spot – higher value in the middle, falling off to the edges.

Noise is just a function added to an images, we could remove the noise by subtracting the noise again
-> true but we don’t know the noise function so we can’t actually do the subtraction.

-can add weights to our moving average
-weight [1, 1, 1, 1, 1] / 5
Non-uniform weights
Σ　[1, 4, 6, 4, 1]/16

To do the moving average computation the number of weight should be
-odd: makes it easier to have a middle pixel

noise = randn(size(im)).*sigma;
Sigma = 2, 8, 32, 64

noise = randn(size(img)).* 2;
output = im + noise;

First attempt at a solution
Replace each pixel with an average of all the values in its neighborhood – a moving average;
original -> smoothed

Average assumptions
1. the true value of pixels are similar to the true value of pixels nearby.
2. the noise added to each pixel is done independently.

some_number = randn();
disp(some_number);

some_numbers = randn([1 5]);
disp(some_numbers);

some_matrix = randn([2 3]);
disp(some_matrix);

noise = randn([2 3]);
disp(noise);

noise = randn([1 100]);
[n, x] = hist(noise, [-3 -2 -1 0 1 2 3]);
disp([x; n]);
plot(x, n);