For a project in the University I am working with several "Quality Assessement" metrics on Finger-Vein images.
Now I try to implement a metric that uses the Radon Transform and I got stuck at some point doing this in Matlab.
My problem is as follows:
I got the following formula for the Radon Transform. In the first steps I used the built in one in Matlab, but for further implementing the metric I need the derivation of the thing for the Curvature of the curve.
the delta is the dirac-delta function.
Derivation:
So my intention is to calculate the Radon Transform on my own with the formula but my problem is that F(x,y) is the gray value of the pixel located at (x,y). And so I need a Function F(x,y) that gives me the gray value of the pixel that I can put in to calculate the derivates and the double integral.
How can I get such a function? Or got I do some kind of "Curve Fitting" with my values of the pixels that I get a function?
Thanks in advance.
As I understand your question, there are two things that you could do:
Compute the derivatives of the Radon transform numerically (as suggested by Ander Biguri in a comment above). If you compute the Radon transform carefully, it will be a band-limited function, making the computation of derivatives possible. See this paper for some ideas on how to enforce a band-limited transform:
"The generalized Radon transform: sampling, accuracy and memory considerations" (PDF).
Compute the derivatives of the image numerically, then sample those derivatives to compute your C function. That is, you compute dF/dx, dF/dy, d^2F/dx^2, and whichever derivatives you need as images. You can interpolate into these derivatives if you need more precision.
IMO the best way to compute derivatives of a discrete image is through Gaussian derivatives. Note that this applies to both solutions above. For example dF/dx (Fx) can be computed by (see here for more details):
h = fspecial('gaussian',[1,2*cutoff+1],sigma);
dh = h .* (-cutoff:cutoff) / (-sigma^2);
Fx = conv2(dh,h,F,'same');
PS: sorry for all the self-references, but I have worked on these topics quite a bit in the past. :)
Related
Im im trying to validate my engineering work using Matlab. I have a series of x and y data that I have plotted on a Log-Log Graph.
The result is a curve.
What I need to do is to apply a curve fit to this graph, and show what the equation of the fit is?
I have tried other answers on here and tried using polyfit and polyval but they aren't really doing what I need but what I lack is the forthwith understanding.
Kind regards
Apply polyfit to logx and logy instead of x and y, and then, to use the fitted result apply polyval to log(x) and use exp() on the result to get the actual fitted y:
logx = log(x);
logy = log(y);
fitp = polyfit(logx, logy, n);
newy = exp(polyval(fitp, log(newx)));
Fitting in the log-space may be undesirable. Most likely you want to show the equation that best fits the data, not a transformation of the data. As a result, I would fit the linear data, then transform it for visualization as necessary. If that's acceptable, polyfit and polyval should work.
If you believe fitting in the log-space is important, I've used lsqcurvefit before, but this requires both the optimization toolbox and some idea of which function you'd like to fit (i.e. is your data best represented by 10^x or x^2?). There's also the curve-fitting toolbox, which might be worth looking into if there are issues you could identify interactively with a GUI but not easily put into words. This provides a 'fit' function that could be useful too.
I want to take FFT of following forms:
Fourier in spherical coordinates
So in fact, my f(theta,phi) is a 3D image voxel, ie it has f(radius,theta,phi). If I had to take integral along dx, that would amount to just selecting my IMG(ylocation,:,zlocation)=currentvoxel and running fft(currentvoxel) on it. However, now I have to only take fft along phi. Which is hard because it requires for me to find all the locations with theta=currenttheta and for each r=currentr, I have to find all the phi's to take fft.
I don't even know how to unroll the phi to take fft. When it's Cartesian coordinates, it's easy, just start from x=0,..x=end and run fft. Is phi case the same? Get all the phi's on a certain r,theta and sort them 0 to 2pi and take fft?
Or maybe sine/cosine transforms like this:
cosine/sine transform
If this was a f(x,y), I think the way to do this in MATLAB would be to take dct(f*(x.^2+y.^2)). In other words running a discrete cosine transform. Am I right?
Extra clarification on second part:
The integral I'm interested in is: bessel
where j_l(kr): expanded j_l
and
expanded further
finally the F_(l)(t) is given by previous: cosine/sine transform
To create a simulated phantom sinogram, it is common to firstly create the 2-d digital image:
I = phantom(xxx);
and then apply the rand() transform
R = radon(I, theta);
However, since the phantom image has its analytical expression, it will be more accurate to simulate the parallel projection if the radon transform is performed on the analytical expression, i.e., on the analog image directly.
Is it possible in Matlab?
How would it be more accurate? Much of the artifacts are either caused or amplified by the discrete CT-datasets. The point of the phantom is usually to study these artifacts. All of the algorithms that I have seen have been discrete, all of the results everywhere are from discrete datasets with discrete algorithms. I'd keep with the crowd.
You can find the analytical expression for the radon transform for the Shepp-Logan phantom from Peter Toft's PhD (http://orbit.dtu.dk/files/5529668/Binder1.pdf), pages 199-201
... Due to the constant excitation on the circle the Radon transform
is merely the length of the line crossing the circle ...
If Q ellipses are given with a set of parameters then due to the
linearity the Radon transform is a sum ...
The analytical expression for the phantom itself can be found from Wikipedia, or the original article if you wish to cite it:
L. A. Shepp and B. F. Logan,
Bell Laboratories,
THE FOURIER RECONSTRUCTION OF A HEAD SECTION,
IEEE TRANSACTION ON NUCLEAR SCIENCE,VOL.NS-21, JUNE 1974
I have a matlab script that performs curve fitting on a set of curves using polynomials of third, second and first order (using polyfit with the desired order) and also using DCT of 4,3 and 2 coefficients (invoking dct for the whole array and then truncating just the first 4,3 or 2 coeffs).
I'm able to get a graphical view of the accuracy of each curve fitting using polyval and idct for the 2 types of curve fitting, but I was wondering if there is any way of getting a numeric value of the accuracy that makes sense for both approaches (dct and polyfit).
I'm sure this is more a maths question rather than a Matlab question, but maybe there is some way to obtain a simple and elegant way in a array-based algorithm that I haven't thought of yet.
Thanks in advance for your comments!
EDIT: What about correlation? :D
In the cuve fitting tool there should be a residual that uses standard deviation. If you are interested in another way to do it maybe you should use rmse for the entire curve, scripting a function that does something like:
input args : y1 = (curve going to be fitted), y2 = (fitted curve)
For each value in y, sum up the difference y1-y2 squared
Divide with the number of entries
Provided you are now left with a number, return its square root
See http://en.wikipedia.org/wiki/Root-mean-square_deviation#Formula for more.
This question has already confused me several days. While I referred to senior students, they also cannot give a reply.
We have ten ODEs, into which each a noise term should be added. The noise is defined as follows. since I always find that I cannot upload a picture, the formula below maybe not very clear. In order to understand, you can either read my explanation or go the this address: Plos one. You could find the description of the equations directly above the Support Information in this address
The white noise term epislon_i(t) is assumed with Gaussian distribution. epislon_i(t) means that for equation i, and at t timepoint, the value of the noise.
the auto-correlation of noise are given:
(EQ.1)
where delta(t) is the Dirac delta function and the diffusion matrix D is defined by
(EQ.2)
Our problem focuses on how to explain the Dirac delta function in the diffusion matrix. Since the property of Dirac delta function is delta(0) = Inf and delta(t) = 0 if t neq 0, we don't know how to calculate the epislonif we try to sqrt of 2D(x, t)delta(t-t'). So we simply assume that delta(0) = 1 and delta(t) = 0 if t neq 0; But we don't know whether or not this is right. Could you please tell me how to use Delta function of diffusion equation in MATLAB?
This question associates with the stochastic process in MATLAB. So we review different stochastic process to inspire our ideas. In MATLAB, the Wienner process is often defined as a = sqrt(dt) * rand(1, N). N is the number of steps, dt is the length of the steps. Correspondingly, the Brownian motion can be defined as: b = cumsum(a); All of these associate with stochastic process. However, they doesn't related to the white noise process which has a constraints on the matrix of auto-correlation, noted by D.
Then we consider that, we may simply use randn(1, 10) to generate a vector representing the noise. However, since the definition of the noise must satisfy the equation (2), this cannot enable noise term in different equation have the predefined partial correlation (D_ij). Then we try to use mvnrnd to generate a multiple variable normal distribution at each time step. Unfortunately, the function mvnrnd in MATLAB return a matrix. But we need to return a vector of length 10.
We are rather confused, so could you please give me just a light? Thanks so much!
NOTE: I see two hazy questions in here: 1) how to deal with a stochastic term in a DE and 2) how to deal with a delta function in a DE. Both of these are math related questions and http://www.math.stackexchange.com will be a better place for this. If you had a question pertaining to MATLAB, I haven't been able to pin it down, and you should perhaps add code examples to better illustrate your point. That said, I'll answer the two questions briefly, just to put you on the right track.
What you have here are not ODEs, but Stochastic differential equations (SDE). I'm not sure how you're using MATLAB to work with this, but routines like ode45 or ode23 will not be of any help. For SDEs, your usual mathematical tools of separation of variables/method of characteristics etc don't work and you'll need to use Itô calculus and Itô integrals to work with them. The solutions, as you might have guessed, will be stochastic. To learn more about SDEs and working with them, you can consider Stochastic Differential Equations: An Introduction with Applications by Bernt Øksendal and for numerical solutions, Numerical Solution of Stochastic Differential Equations by Peter E. Kloeden and Eckhard Platen.
Coming to the delta function part, you can easily deal with it by taking the Fourier transform of the ODE. Recall that the Fourier transform of a delta function is 1. This greatly simplifies the DE and you can take an inverse transform in the very end to return to the original domain.