如何将 Gabor 小波应用于图像? [英] How to apply Gabor wavelets to an image?

问题描述

如何在图像上应用这些 Gabor 滤波器小波?

How do i apply these Gabor filter wavelets on an image ?

close all;
clear all;
clc;

% Parameter Setting
R = 128;
C = 128;
Kmax = pi / 2;
f = sqrt( 2 );
Delt = 2 * pi;
Delt2 = Delt * Delt;

% Show the Gabor Wavelets
for v = 0 : 4
    for u = 1 : 8
        GW = GaborWavelet ( R, C, Kmax, f, u, v, Delt2 ); % Create the Gabor wavelets
          figure( 2 );
     subplot( 5, 8, v * 8 + u ),imshow ( real( GW ) ,[]); % Show the real part of Gabor wavelets

    end

    figure ( 3 );
     subplot( 1, 5, v + 1 ),imshow ( abs( GW ),[]); % Show the magnitude of Gabor wavelets

end



function GW = GaborWavelet (R, C, Kmax, f, u, v, Delt2)


k = ( Kmax / ( f ^ v ) ) * exp( 1i * u * pi / 8 );% Wave Vector

kn2 = ( abs( k ) ) ^ 2;

GW = zeros ( R , C );

for m = -R/2 + 1 : R/2

    for n = -C/2 + 1 : C/2

        GW(m+R/2,n+C/2) = ( kn2 / Delt2 ) * exp( -0.5 * kn2 * ( m ^ 2 + n ^ 2 ) / Delt2) * ( exp( 1i * ( real( k ) * m + imag ( k ) * n ) ) - exp ( -0.5 * Delt2 ) );

    end

end

这是我的图像尺寸

推荐答案

Gabor 滤波器的典型用途是计算多个方向的滤波器响应，例如用于边缘检测.

A typical use of Gabor filters is to calculate filter responses at each of several orientations, e.g. for edge detection.

您可以使用卷积定理，通过取逆图像和滤波器的傅立叶变换的元素乘积的傅立叶变换.这是基本公式:

You can convolve a filter with an image using the Convolution Theorem, by taking the inverse Fourier transform of the element-wise product of the Fourier transforms of the image and the filter. Here is the basic formula:

%# Our image needs to be 2D (grayscale)
if ndims(img) > 2;
    img = rgb2gray(img);
end
%# It is also best if the image has double precision
img = im2double(img);

[m,n] = size(img);
[mf,nf] = size(GW);
GW = padarray(GW,[n-nf m-mf]/2);
GW = ifftshift(GW);
imgf = ifft2( fft2(img) .* GW );

通常情况下，FFT 卷积对于大小 > 20 的内核更胜一筹.有关详细信息，我推荐 C 中的数值食谱，它对该方法及其警告进行了与语言无关的良好描述.

Typically, the FFT convolution is superior for kernels of size > 20. For details, I recommend Numerical Recipes in C, which has a good, language agnostic description of the method and its caveats.

您的内核已经很大了，但是使用 FFT 方法它们可以和图像一样大，因为无论如何它们都被填充到那个大小.由于 FFT 的周期性，该方法执行循环卷积.这意味着过滤器将环绕图像边界，因此我们还必须填充图像本身以消除这种边缘效应.最后，由于我们想要所有过滤器的总响应(至少在典型实现中)，我们需要依次将每个应用到图像，并对响应求和.通常只使用 3 到 6 个方向，但在多个尺度(不同的内核大小)上进行过滤也很常见，因此在这种情况下会使用更多的过滤器.

Your kernels are already large, but with the FFT method they can be as large as the image, since they are padded to that size regardless. Due to the periodicity of the FFT, the method performs circular convolution. That means that the filter will wrap around the image borders, so we have to pad the image itself as well to eliminate this edge effect. Finally, since we want the total response to all the filters (at least in a typical implementation), we need to apply each to the image in turn, and sum the responses. Usually one uses just 3 to 6 orientations, but it is also common to do the filtering at several scales (different kernel sizes) as well, so in that context a larger number of filters is used.

你可以用这样的代码完成整个事情:

You can do the whole thing with code like this:

img = im2double(rgb2gray(img)); %# 
[m,n] = size(img); %# Store the original size.

%# It is best if the filter size is odd, so it has a discrete center.
R = 127; C = 127;

%# The minimum amount of padding is just "one side" of the filter.
%# We add 1 if the image size is odd.
%# This assumes the filter size is odd.
pR = (R-1)/2;
pC = (C-1)/2;
if rem(m,2) ~= 0; pR = pR + 1; end;
if rem(n,2) ~= 0; pC = pC + 1; end;
img = padarray(img,[pR pC],'pre'); %# Pad image to handle circular convolution.

GW = {}; %# First, construct the filter bank.
for v = 0 : 4
    for u = 1 : 8
        GW =  [GW {GaborWavelet(R, C, Kmax, f, u, v, Delt2)}];
    end
end

%# Pad all the filters to size of padded image.
%# We made sure padsize will only be even, so we can divide by 2.
padsize = size(img) - [R C];
GW = cellfun( ...
        @(x) padarray(x,padsize/2), ...
        GW, ...
        'UniformOutput',false);

imgFFT = fft2(img); %# Pre-calculate image FFT.

for i=1:length(GW)
    filter = fft2( ifftshift( GW{i} ) ); %# See Numerical Recipes.
    imgfilt{i} = ifft2( imgFFT .* filter ); %# Apply Convolution Theorem.
end

%# Sum the responses to each filter. Do it in the above loop to save some space.
imgS = zeros(m,n);
for i=1:length(imgfilt)
    imgS = imgS + imgfilt{i}(pR+1:end,pC+1:end); %# Just use the valid part.
end

%# Look at the result.
imagesc(abs(imgS));

请记住，这基本上是最低限度的实现.您可以选择使用边界的副本而不是零来填充，对图像应用窗口函数或使填充尺寸更大以获得频率分辨率.每一个都是我上面概述的技术的标准增强，通过 Google 和 维基百科.另请注意，我没有添加任何基本的 MATLAB 优化，例如预分配等.

Keep in mind that this is essentially the bare minimum implementation. You can choose to pad with replicates of the border instead of zeros, apply a windowing function to the image or make the pad size larger in order to gain frequency resolution. Each of these is a standard augmentation to the technique I've outlined above and should be trivial to research via Google and Wikipedia. Also note that I haven't added any basic MATLAB optimizations such as pre-allocation, etc.

最后一点，如果您的过滤器总是比图像小得多，您可能希望跳过图像填充(即使用第一个代码示例).这是因为，向图像添加零会在填充开始处创建人工边缘特征.如果过滤器很小，循环卷积的环绕不会引起问题，因为只会涉及过滤器填充中的零.但是一旦过滤器足够大，环绕效应就会变得严重.如果必须使用大型过滤器，则可能需要使用更复杂的填充方案，或者裁剪图像的边缘.

As a final note, you may want to skip the image padding (i.e. use first code example) if your filters are always much smaller than the image. This is because, adding zeros to an image creates an artificial edge feature where the padding begins. If the filter is small, the wraparound from circular convolution doesn't cause problems, because only the zeros in the filter's padding will be involved. But as soon as the filter is large enough, the wraparound effect will become severe. If you have to use large filters, you may have to use a more complicated padding scheme, or crop the edge of the image.

这篇关于如何将 Gabor 小波应用于图像?的文章就介绍到这了，希望我们推荐的答案对大家有所帮助，也希望大家多多支持IT屋！