I'm facing a problem in calculating surface integration. I defined the function below which calculates surface integration for a discrete dataset using the trapezoidal rule.
< trapz2D.m >
function out = trapz2D(x,y,f)
x = x(:);
y = y(:);
nx = length(x);
ny = length(y);
dx = diff(x);
dy = diff(y);
dS = dy*dx.';
df = (f(1:ny-1,1:nx - 1) + f(2:ny,1:nx-1) + f(1:ny-1,2:nx) + f(2:ny,2:nx))/4;
out = sum(sum(dS.*df));
end
I calculated surface integration using this function and the dataset in the link below, and then compare it to the value obtained from "trapz" which is a MATLAB built-in function.
< surfInteg.m >
clc
close all
clear
format long
load('dataset.mat')
I1 = trapz(y, trapz(x,Pz3,2));
I2 = trapz2D(x,y,Pz3);
disp([I1; I2])
>>>1.0e-12 *
0.307618158054522 - 0.000000000004792i
0.307618158054522 - 0.000000000004792i
I1 = trapz(y, trapz(x,Pz1,2));
I2 = trapz2D(x,y,Pz1);
disp([I1; I2])
>>>1.0e-27 *
-0.135645057561047 + 0.013248760976931i
-0.129284381233370 + 0.013497757971006i
The results are similar for "Pz3", but not for "Pz1". Could someone explain why this is happening?
dataset
Related
I am trying to model the time evolution of a membrane based on the following code in MATLAB.
The basic outline is that the evolution is based on a differential equation
where j=0,1 and x^0 = x, x^1 = y and x^j(s_i) = x^j_i.
My code is the following.
import numpy as np
from matplotlib import pyplot as plt
R0 = 5 #radius
N = 360 #number of intervals
x0 = 2*np.pi*R0/(N/2) #resting membrane lengths
phi = np.linspace(0,2*np.pi, num=360, dtype=float)
R1 = R0 + 0.5*np.sin(20*phi)
X = R1*np.cos(phi)
Y = R1*np.sin(phi)
L = np.linspace(-1,358, num=360, dtype=int)
R = np.linspace(1,360, num=360,dtype=int) #right and left indexing vectors
R[359] = 0
X = R1*np.cos(phi)
Y = R1*np.sin(phi)
plt.plot(X,Y)
plt.axis("equal")
plt.show()
ds = 1/N
ds2 = ds**2
k = 1/10
w = 10**6
for i in range(0,20000):
lengths = np.sqrt( (X[R]-X)**2 + (Y[R]-Y)**2 )
Ex = k/ds2*(X[R] - 2*X + X[L] - x0*( (X[R]-X)/lengths - (X-X[L])/lengths[L]) )
Ey = k/ds2*(Y[R] - 2*Y + Y[L] - x0*( (Y[R]-Y)/lengths - (Y-Y[L])/lengths[L]) )
X = X + 1/w*Ex
Y = Y + 1/w*Ey
plt.plot(X,Y)
plt.axis("equal")
plt.show()
The model is supposed to devolve into a circular membrane, as below
but this is what mine does
Your definition of x0 is wrong.
In the Matlab code, it is equal to
x0 = 2*pi*R/N/2 # which is pi*R/N
while in your Python code it is
x0 = 2*np.pi*R0/(N/2) # which is 4*np.pi*R0/N
Correcting that, the end result is a circular shape, but with a different radius. I'm assuming that this is because of the reduced number of iterations (20000 instead of 1000000).
Edit:
As expected, using the correct number of iterations results in a plot similar to your expected one.
In this MATLAB code, I intend to display a graph, but it shows me an empty graph.
% Proposed Model For Two-Phase Flow Analytical Design Equation for gas
% pipelines
clc
d = 26.75; % diameter in inches
pie = 3.142; %normal pie
A = (pie *d^2)/4; % Compute for the area
qm = 150; %volume flow rate of mixture
pm = 8; %Density of the mixture
Wm = qm* pm; % mass flow rate of mixture i.e oil and gas
Ho = 0.01; %Liquid Hold up
z = 0.85; % compressibility factor
R = 10.73; % Gas constant
f = 0.0254; %friction factor
p0 = 150; % Density of oil
T = 580; % Temperature in R degrees
P0 =1700; % 1700psi
L = 63.068; % Length
P1 = 602.7; % 602.7 psia
%Assume
Mg = 9.8;
formule1 = d*A^2*Mg*(1-Ho);
formule2 = f*Wm^2*z*R*T;
formule = formule1/formule2;
exprs1 = (p0*Ho*z*R*T)/Mg*(1-Ho);
express2 = P2^2-P1^2
drop_p =P1-P2
express3 = 2*p0*Ho*z*R*T*drop_p
better_express = express2 + express3
func1 = d/f
solve( (formule*better_express)- func1*log(P2+ exprs1/P1 + exprs1)^2 + L == 0, P2)
figure
plot(drop_p,L,'r:+')
Please, can anyone help me out? Thanks.
In your code: drop_p = P1-P2 and L = 63.068, so unless P2 is a vector you will get just a single red + in your graph. Like this (for P2 = 1):
Right now, the solve line returns an error, because it should be with '' on the variable to solve to:
solve( (formule*better_express)- func1*log(P2+ exprs1/P1 + exprs1)^2 + L == 0, 'P2')
but this doesn't effect the resulted graph anyway, because the solve output is not assigned to any variable.
this is a part of an assignment for a Fourier-Analysis course.
In this assignment I was asked to implement a matlab function to compute the derivative of a discrete function using the derivative of the Discrete Fourier Transform. The formula I was given was this formula:
The code I wrote is this, using 513 datapoints from -pi to pi:
t = -pi + ((2*pi)/513)*(0:513-1);
for n = 1:513
tmpsum = 0;
for k = 1:N
tmpsum = tmpsum + (1i*k*fft(g(k))*exp(1i*k*n));
end
deriv(n) = real((1/sqrt(N))*tmpsum);
end
It executes fine, but once I plot it against t (e.g. for g = sin(t)) I get a really weird graph, with spikes up to 2000 on the y axis.
What am I doing wrong?
EDIT:
This is the code I currently run:
N = 512;
t = -pi + ((2*pi)/(N))*(0:N-1);
f = sin(t);
deriv = zeros(1,length(t));
ffft = fft(f);
for n = 1:N
tmpsum = 0;
for k = 1:N
tmpsum = tmpsum + ((1i*k*ffft(k))*exp(1i*k*n));
end
deriv(n) = (1/sqrt(N))*tmpsum;
end
plot(t,f,t,deriv);
A filter g is called separable if it can be expressed as the multiplication of two vectors grow and gcol . Employing one dimensional filters decreases the two dimensional filter's computational complexity from O(M^2 N^2) to O(2M N^2) where M and N are the width (and height) of the filter mask and the image respectively.
In this stackoverflow link, I wrote the equation of a Gabor filter in the spatial domain, then I wrote a matlab code which serves to create 64 gabor features.
According to the definition of separable filters, the Gabor filters are parallel to the image axes - theta = k*pi/2 where k=0,1,2,etc.. So if theta=pi/2 ==> the equation in this stackoverflow link can be rewritten as:
The equation above is extracted from this article.
Note: theta can be extented to be equal k*pi/4. By comparing to the equation in this stackoverflow link, we can consider that f= 1 / lambda.
By changing my previous code in this stackoverflow link, I wrote a matlab code to make the Gabor filters separable by using the equation above, but I am sure that my code below is not correct especially when I initialized the gbp and glp equations. That is why I need your help. I will appreciate your help very much.
Let's show now my code:
function [fSiz,filters1,filters2,c1OL,numSimpleFilters] = init_gabor(rot, RF_siz)
image=imread('xxx.jpg');
image_gray=rgb2gray(image);
image_gray=imresize(image_gray, [100 100]);
image_double=double(image_gray);
rot = [0 45 90 135]; % we have four orientations
RF_siz = [7:2:37]; %we get 16 scales (7x7 to 37x37 in steps of two pixels)
minFS = 7; % the minimum receptive field
maxFS = 37; % the maximum receptive field
sigma = 0.0036*RF_siz.^2 + 0.35*RF_siz + 0.18; %define the equation of effective width
lambda = sigma/0.8; % it the equation of wavelength (lambda)
G = 0.3; % spatial aspect ratio: 0.23 < gamma < 0.92
numFilterSizes = length(RF_siz); % we get 16
numSimpleFilters = length(rot); % we get 4
numFilters = numFilterSizes*numSimpleFilters; % we get 16x4 = 64 filters
fSiz = zeros(numFilters,1); % It is a vector of size numFilters where each cell contains the size of the filter (7,7,7,7,9,9,9,9,11,11,11,11,......,37,37,37,37)
filters1 = zeros(max(RF_siz),numFilters);
filters2 = zeros(numFilters,max(RF_siz));
for k = 1:numFilterSizes
for r = 1:numSimpleFilters
theta = rot(r)*pi/180;
filtSize = RF_siz(k);
center = ceil(filtSize/2);
filtSizeL = center-1;
filtSizeR = filtSize-filtSizeL-1;
sigmaq = sigma(k)^2;
for x = -filtSizeL:filtSizeR
fx = exp(-(x^2)/(2*sigmaq))*cos(2*pi*x/lambda(k));
f1(x+center,1) = fx;
end
for y = -filtSizeL:filtSizeR
gy = exp(-(y^2)/(2*sigmaq));
f2(1,y+center) = gy;
end
f1 = f1 - mean(mean(f1));
f1 = f1 ./ sqrt(sum(sum(f1.^2)));
f2 = f2 - mean(mean(f2));
f2 = f2 ./ sqrt(sum(sum(f2.^2)));
p = numSimpleFilters*(k-1) + r;
filters1(1:filtSize,p)=f1;
filters2(p,1:filtSize)=f2;
convv1=imfilter(image_double, filters1(1:filtSize,p),'conv');
convv2=imfilter(double(convv1), filters2(p,1:filtSize),'conv');
figure
imagesc(convv2);
colormap(gray);
end
end
I think the code is correct provided your previous version of Gabor filter code is correct too. The only thing is that if theta = k * pi/4;, your formula here should be separated to:
fx = exp(-(x^2)/(2*sigmaq))*cos(2*pi*x/lambda(k));
gy = exp(-(G^2 * y^2)/(2*sigmaq));
To be consistent, you may use
f1(1,x+center) = fx;
f2(y+center,1) = gy;
or keep f1 and f2 as it is but transpose your filters1 and filters2 thereafter.
Everything else looks good to me.
EDIT
My answer above works for theta = k * pi/4;, with other angles, based on your paper,
x = i*cos(theta) - j*sin(theta);
y = i*sin(theta) + j*cos(theta);
fx = exp(-(x^2)/(2*sigmaq))*exp(sqrt(-1)*x*cos(theta));
gy = exp(-(G^2 * y^2)/(2*sigmaq))*exp(sqrt(-1)*y*sin(theta));
The final code will be:
function [fSiz,filters1,filters2,c1OL,numSimpleFilters] = init_gabor(rot, RF_siz)
image=imread('xxx.jpg');
image_gray=rgb2gray(image);
image_gray=imresize(image_gray, [100 100]);
image_double=double(image_gray);
rot = [0 45 90 135];
RF_siz = [7:2:37];
minFS = 7;
maxFS = 37;
sigma = 0.0036*RF_siz.^2 + 0.35*RF_siz + 0.18;
lambda = sigma/0.8;
G = 0.3;
numFilterSizes = length(RF_siz);
numSimpleFilters = length(rot);
numFilters = numFilterSizes*numSimpleFilters;
fSiz = zeros(numFilters,1);
filters1 = zeros(max(RF_siz),numFilters);
filters2 = zeros(numFilters,max(RF_siz));
for k = 1:numFilterSizes
for r = 1:numSimpleFilters
theta = rot(r)*pi/180;
filtSize = RF_siz(k);
center = ceil(filtSize/2);
filtSizeL = center-1;
filtSizeR = filtSize-filtSizeL-1;
sigmaq = sigma(k)^2;
for x = -filtSizeL:filtSizeR
fx = exp(-(x^2)/(2*sigmaq))*exp(sqrt(-1)*x*cos(theta));
f1(1, x+center) = fx;
end
for y = -filtSizeL:filtSizeR
gy=exp(-(y^2)/(2*sigmaq))*exp(sqrt(-1)*y*sin(theta));
f2(y+center,1) = gy;
end
f1 = f1 - mean(mean(f1));
f1 = f1 ./ sqrt(sum(sum(f1.^2)));
f2 = f2 - mean(mean(f2));
f2 = f2 ./ sqrt(sum(sum(f2.^2)));
p = numSimpleFilters*(k-1) + r;
filters1(1:filtSize,p)=f1;
filters2(p,1:filtSize)=f2;
convv1=imfilter(image_double, filters1(1:filtSize,p),'conv');
convv2=imfilter(double(convv1), filters2(p,1:filtSize),'conv');
figure
imagesc(imag(convv2));
colormap(gray);
end
end
Got one more problem with matrix multiplication in Matlab. I have to plot Taylor polynomials for the given function. This question is similar to my previous one (but this time, the function is f: R^2 -> R^3) and I can't figure out how to make the matrices in order to make it work...
function example
clf;
M = 40;
N = 20;
% domain of f(x)
x1 = linspace(0,2*pi,M).'*ones(1,N);
x2 = ones(M,1)*linspace(0,2*pi,N);
[y1,y2,y3] = F(x1,x2);
mesh(y1,y2,y3,...
'facecolor','w',...
'edgecolor','k');
axis equal;
axis vis3d;
axis manual;
hold on
% point for our Taylor polynom
xx1 = 3;
xx2 = 0.5;
[yy1,yy2,yy3] = F(xx1,xx2);
% plots one discrete point
plot3(yy1,yy2,yy3,'ro');
[y1,y2,y3] = T1(xx1,xx2,x1,x2);
mesh(y1,y2,y3,...
'facecolor','w',...
'edgecolor','g');
% given function
function [y1,y2,y3] = F(x1,x2)
% constants
R=2; r=1;
y1 = (R+r*cos(x2)).*cos(x1);
y2 = (R+r*cos(x2)).*sin(x1);
y3 = r*sin(x2);
function [y1,y2,y3] = T1(xx1,xx2,x1,x2)
dy = [
-(R + r*cos(xx2))*sin(xx1) -r*cos(xx1)*sin(xx2)
(R + r*cos(xx2))*cos(xx1) -r*sin(xx1)*sin(xx2)
0 r*cos(xx2) ];
y = F(xx1, xx2) + dy.*[x1-xx1; x2-xx2];
function [y1,y2,y3] = T2(xx1,xx2,x1,x2)
% ?
I know that my code is full of mistakes (I just need to fix my T1 function). dy represents Jacobian matrix (total derivation of f(x) - I hope I got it right...). I am not sure how would the Hessian matrix in T2 look, by I hope I will figure it out, I'm just lost in Matlab...
edit: I tried to improve my formatting - here's my Jacobian matrix
[-(R + r*cos(xx2))*sin(xx1), -r*cos(xx1)*sin(xx2)...
(R + r*cos(xx2))*cos(xx1), -r*sin(xx1)*sin(xx2)...
0, r*cos(xx2)];
function [y1,y2,y3]=T1(xx1,xx2,x1,x2)
R=2; r=1;
%derivatives
y1dx1 = -(R + r * cos(xx2)) * sin(xx1);
y1dx2 = -r * cos(xx1) * sin(xx2);
y2dx1 = (R + r * cos(xx2)) * cos(xx1);
y2dx2 = -r * sin(xx1) * sin(xx2);
y3dx1 = 0;
y3dx2 = r * cos(xx2);
%T1
[f1, f2, f3] = F(xx1, xx2);
y1 = f1 + y1dx1*(x1-xx1) + y1dx2*(x2-xx2);
y2 = f2 + y2dx1*(x1-xx1) + y2dx2*(x2-xx2);
y3 = f3 + y3dx1*(x1-xx1) + y3dx2*(x2-xx2);