方式一：
function result = gaborKernel2d( lambda, theta, phi, gamma, bandwidth)
% GABORKERNEL2D
% Version: 2012/8/17 by watkins.song
% Version: 1.0
% Fills a (2N+1)*(2N+1) matrix with the values of a 2D Gabor function.
% N is computed from SIGMA.
%
% LAMBDA - preferred wavelength (period of the cosine factor) [in pixels]
% SIGMA - standard deviation of the Gaussian factor [in pixels]
% THETA - preferred orientation [in radians]
% PHI - phase offset [in radians] of the cosine factor
% GAMMA - spatial aspect ratio (of the x- and y-axis of the Gaussian elipse)
% BANDWIDTH - spatial frequency bandwidth at half response,
% *******************************************************************
%
% BANDWIDTH, SIGMA and LAMBDA are interdependent. To use BANDWIDTH,
% the input value of one of SIGMA or LAMBDA must be 0. Otherwise BANDWIDTH is ignored.
% The actual value of the parameter whose input value is 0 is computed inside the
% function from the input vallues of BANDWIDTH and the other parameter.
%
% pi -1 x'^2+gamma^2*y'^2
% G(x,y,theta,f) = --------------- *exp ([----{-------------------}])*cos(2*pi*f*x'+phi);
% 2*sigma*sigma 2 sigma^2
%
%%% x' = x*cos(theta)+y*sin(theta);
%%% y' = y*cos(theta)-x*sin(theta);
%
% Author: watkins.song
% Email: watkins.song@gmail.com
% calculation of the ratio sigma/lambda from BANDWIDTH
% according to Kruizinga and Petkov, 1999 IEEE Trans on Image Processing 8 (10) p.1396
% note that in Matlab log means ln
slratio = (1/pi) * sqrt( (log(2)/2) ) * ( (2^bandwidth + 1) / (2^bandwidth - 1) );
% calcuate sigma
sigma = slratio * lambda;
% compute the size of the 2n+1 x 2n+1 matrix to be filled with the values of a Gabor function
% this size depends on sigma and gamma
if (gamma <= 1 && gamma > 0)
n = ceil(2.5*sigma/gamma);
else
n = ceil(2.5*sigma);
end
% creation of two (2n+1) x (2n+1) matrices x and y that contain the x- and y-coordinates of
% a square 2D-mesh; the rows of x and the columns of y are copies of the vector -n:n
[x,y] = meshgrid(-n:n);
% change direction of y-axis (In Matlab the vertical axis corresponds to the row index
% of a matrix. If the y-coordinates run from -n to n, the lowest value (-n) comes
% in the top row of the matrix ycoords and the highest value (n) in the
% lowest row. This is oposite to the customary rendering of values on the y-axis: lowest value
% in the bottom, highest on the top. Therefore the y-axis is inverted:
y = -y;
% rotate x and y
% xp and yp are the coordinates of a point in a coordinate system rotated by theta.
% They are the main axes of the elipse of the Gaussian factor of the Gabor function.
% The wave vector of the Gabor function is along the xp axis.
xp = x * cos(theta) + y * sin(theta);
yp = -x * sin(theta) + y * cos(theta);
% precompute coefficients gamma2=gamma*gamma, b=1/(2*sigma*sigma) and spacial frequency
% f = 2*pi/lambda to prevent multiple evaluations
gamma2 = gamma*gamma;
b = 1 / (2*sigma*sigma);
a = b / pi;
f = 2*pi/lambda;
% filling (2n+1) x (2n+1) matrix result with the values of a 2D Gabor function
result = a*exp(-b*(xp.*xp + gamma2*(yp.*yp))) .* cos(f*xp + phi);
%%%%%%%% NORMALIZATION %%%%%%%%%%%%%%%%%%%%
% NORMALIZATION of positive and negative values to ensure that the integral of the kernel is 0.
% This is needed when phi is different from pi/2.
ppos = find(result > 0); %pointer list to indices of elements of result which are positive
pneg = find(result < 0); %pointer list to indices of elements of result which are negative
pos = sum(result(ppos)); % sum of the positive elements of result
neg = abs(sum(result(pneg))); % abs value of sum of the negative elements of result
meansum = (pos+neg)/2;
if (meansum > 0)
pos = pos / meansum; % normalization coefficient for negative values of result
neg = neg / meansum; % normalization coefficient for psoitive values of result
end
result(pneg) = pos*result(pneg);
result(ppos) = neg*result(ppos);
end
方式二：
function [Efilter, Ofilter, gb] = gaborKernel2d_evenodd( lambda, theta, kx, ky)
%GABORKERNEL2D_EVENODD Summary of this function goes here
% Usage:
% gb = spatialgabor(im, wavelength, angle, kx, ky, showfilter)
% Version: 2012/8/17 by watkins.song
% Version: 1.0
%
% Arguments:
% im - Image to be processed.
% wavelength - Wavelength in pixels of Gabor filter to construct
% angle - Angle of filter in degrees. An angle of 0 gives a
% filter that responds to vertical features.
% kx, ky - Scale factors specifying the filter sigma relative
% to the wavelength of the filter. This is done so
% that the shapes of the filters are invariant to the
% scale. kx controls the sigma in the x direction
% which is along the filter, and hence controls the
% bandwidth of the filter. ky controls the sigma
% across the filter and hence controls the
% orientational selectivity of the filter. A value of
% 0.5 for both kx and ky is a good starting point.
% % lambda = 3;
% theta = 90;
% kx = 0.5;
% ky = 0.5;
%
%
% Author: watkins.song
% Email: watkins.song@gmail.com
% Construct even and odd Gabor filters
sigmax = lambda*kx;
sigmay = lambda*ky;
sze = round(3*max(sigmax,sigmay));
[x,y] = meshgrid(-sze:sze);
evenFilter = exp(-(x.^2/sigmax^2 + y.^2/sigmay^2)/2).*cos(2*pi*(1/lambda)*x);
% the imaginary part of the gabor filter
oddFilter = exp(-(x.^2/sigmax^2 + y.^2/sigmay^2)/2).*sin(2*pi*(1/lambda)*x);
evenFilter = imrotate(evenFilter, theta, 'bilinear','crop');
oddFilter = imrotate(oddFilter, theta, 'bilinear','crop');
gb = evenFilter;
Efilter = evenFilter;
Ofilter = oddFilter;
end
方式三：
function gb = gaborKernel2d_gaborfilter( lambda, theta, phi, gamma, bw)
%GABORKERNEL2D_GABORFILTER Summary of this function goes here
% Version: 2012/8/17 by watkins.song
% Version: 1.0
%
% LAMBDA - preferred wavelength (period of the cosine factor) [in pixels]
% SIGMA - standard deviation of the Gaussian factor [in pixels]
% THETA - preferred orientation [in radians]
% PHI - phase offset [in radians] of the cosine factor
% GAMMA - spatial aspect ratio (of the x- and y-axis of the Gaussian elipse)
% BANDWIDTH - spatial frequency bandwidth at half response,
% *******************************************************************
%
% BANDWIDTH, SIGMA and LAMBDA are interdependent. To use BANDWIDTH,
% the input value of one of SIGMA or LAMBDA must be 0. Otherwise BANDWIDTH is ignored.
% The actual value of the parameter whose input value is 0 is computed inside the
% function from the input vallues of BANDWIDTH and the other
% parameter.
% -1 x'^2 + y'^2
%%% G(x,y,theta,f) = exp ([----{-----------------})*cos(2*pi*f*x'+phi);
% 2 sigma*sigma
%%% x' = x*cos(theta)+y*sin(theta);
%%% y' = y*cos(theta)-x*sin(theta);
%
% Author: watkins.song
% Email: watkins.song@gmail.com
% bw = bandwidth, (1)
% gamma = aspect ratio, (0.5)
% psi = phase shift, (0)
% lambda= wave length, (>=2)
% theta = angle in rad, [0 pi)
sigma = lambda/pi*sqrt(log(2)/2)*(2^bw+1)/(2^bw-1);
sigma_x = sigma;
sigma_y = sigma/gamma;
sz=fix(8*max(sigma_y,sigma_x));
if mod(sz,2)==0
sz=sz+1;
end
% alternatively, use a fixed size
% sz = 60;
[x y]=meshgrid(-fix(sz/2):fix(sz/2),fix(sz/2):-1:fix(-sz/2));
% x (right +)
% y (up +)
% Rotation
x_theta = x*cos(theta)+y*sin(theta);
y_theta = -x*sin(theta)+y*cos(theta);
gb=exp(-0.5*(x_theta.^2/sigma_x^2+y_theta.^2/sigma_y^2)).*cos(2*pi/lambda*x_theta+phi);
end
方式四：
function gb = gaborKernel2d_wiki( lambda, theta, phi, gamma, bandwidth)
% GABORKERNEL2D_WIKI 改写的来自wiki的gabor函数
% Version: 2012/8/17 by watkins.song
% Version: 1.0
%
% LAMBDA - preferred wavelength (period of the cosine factor) [in pixels]
% SIGMA - standard deviation of the Gaussian factor [in pixels]
% THETA - preferred orientation [in radians]
% PHI - phase offset [in radians] of the cosine factor
% GAMMA - spatial aspect ratio (of the x- and y-axis of the Gaussian elipse)
% BANDWIDTH - spatial frequency bandwidth at half response,
% *******************************************************************
%
% BANDWIDTH, SIGMA and LAMBDA are interdependent. To use BANDWIDTH,
% the input value of one of SIGMA or LAMBDA must be 0. Otherwise BANDWIDTH is ignored.
% The actual value of the parameter whose input value is 0 is computed inside the
% function from the input vallues of BANDWIDTH and the other
% parameter.
% -1 x'^2 + y'^2
%%% G(x,y,theta,f) = exp ([----{-----------------})*cos(2*pi*f*x'+phi);
% 2 sigma*sigma
%%% x' = x*cos(theta)+y*sin(theta);
%%% y' = y*cos(theta)-x*sin(theta);
%
% Author: watkins.song
% Email: watkins.song@gmail.com
% calculation of the ratio sigma/lambda from BANDWIDTH
% according to Kruizinga and Petkov, 1999 IEEE Trans on Image Processing 8 (10) p.1396
% note that in Matlab log means ln
slratio = (1/pi) * sqrt( (log(2)/2) ) * ( (2^bandwidth + 1) / (2^bandwidth - 1) );
% calcuate sigma
sigma = slratio * lambda;
sigma_x = sigma;
sigma_y = sigma/gamma;
% Bounding box
nstds = 4;
xmax = max(abs(nstds*sigma_x*cos(theta)),abs(nstds*sigma_y*sin(theta)));
xmax = ceil(max(1,xmax));
ymax = max(abs(nstds*sigma_x*sin(theta)),abs(nstds*sigma_y*cos(theta)));
ymax = ceil(max(1,ymax));
xmin = -xmax; ymin = -ymax;
[x,y] = meshgrid(xmin:xmax,ymin:ymax);
% Rotation
x_theta = x*cos(theta) + y*sin(theta);
y_theta = -x*sin(theta) + y*cos(theta);
% Gabor Function
gb= exp(-.5*(x_theta.^2/sigma_x^2+y_theta.^2/sigma_y^2)).*cos(2*pi/lambda*x_theta+phi);
end
方式五：
function [GaborReal, GaborImg] = gaborKernel_matlab( GaborH, GaborW, U, V, sigma)
%GABORKERNEL_MATLAB generate very beautiful gabor filter
% Version: 2012/8/17 by watkins.song
% Version: 1.0
% 用以生成 Gabor
% GaborReal: 核实部 GaborImg: 虚部
% GaborH,GaborW: Gabor窗口 高宽.
% U,V: 方向 大小
% ||Ku,v||^2
% G(Z) = ---------------- exp(-||Ku,v||^2 * Z^2)/(2*sigma*sigma)(exp(i*Ku,v*Z)-exp(-sigma*sigma/2))
% sigma*sigma
%
% 利用另外一个gabor函数来生成gabor filter, 通过u,v表示方向和尺度.
% 这里的滤波器模板的大小是不变的，变化的只有滤波器的波长和方向
% v: 代表波长
% u: 代表方向
% 缺省输入前2个参数,后面参数 Kmax=2.5*pi/2, f=sqrt(2), sigma=1.5*pi;
% GaborH, GaborW, Gabor模板大小
% U,方向因子{0,1,2,3,4,5,6,7}
% V,大小因子{0,1,2,3,4}
% Author: watkins.song
% Email: watkins.song@gmail.com
HarfH = fix(GaborH/2);
HarfW = fix(GaborW/2);
Qu = pi*U/8;
sqsigma = sigma*sigma;
Kv = 2.5*pi*(2^(-(V+2)/2));
%Kv = Kmax/(f^V);
postmean = exp(-sqsigma/2);
for j = -HarfH : HarfH
for i = -HarfW : HarfW
tmp1 = exp(-(Kv*Kv*(j*j+i*i)/(2*sqsigma)));
tmp2 = cos(Kv*cos(Qu)*i+Kv*sin(Qu)*j) - postmean;
%tmp3 = sin(Kv*cos(Qu)*i+Kv*sin(Qu)*j) - exp(-sqsigma/2);
tmp3 = sin(Kv*cos(Qu)*i+Kv*sin(Qu)*j);
GaborReal(j+HarfH+1, i+HarfW+1) = Kv*Kv*tmp1*tmp2/sqsigma;
GaborImg(j+HarfH+1, i+HarfW+1) = Kv*Kv*tmp1*tmp3/sqsigma;
end
end
end
最后调用方式都一样：
% 测试用程序
theta = [0 pi/8 2*pi/8 3*pi/8 4*pi/8 5*pi/8 6*pi/8 7*pi/8];
lambda = [4 6 8 10 12];
phi = 0;
gamma = 1;
bw = 0.5;
% 计算每个滤波器
figure;
for i = 1:5
for j = 1:8
gaborFilter=gaborKernel2d(lambda(i), theta(j), phi, gamma, bw);
% 查看每一个滤波器
%figure;
%imshow(real(gaborFilter),[]);
% 将所有的滤波器放到一张图像中查看，查看滤波器组
subplot(5,8,(i-1)*8+j);
imshow(real(gaborFilter),[]);
end
end