I have a little bit confused on the difference between stationary wavelet transform and un-decimated wavelet transform.
So, can anyone tell me, if there is a difference between them?
They are the same (see wikipedia). The a trous algorithm is the most efficient way to do this with typical short wavelet filters. See this link for its implementation in time and frequency domains.
Related
At the link there is the documentation on how opencv matches images with cv::matchTemplate. I am interested in equation 5.
In literature I have found that this equation is usually computed via a truncated Fourier transform or with other truncated expansion on a basis set to better the performances. Obviously, this is an approximation.
I have implemented the same formula reported in the link without any discrepancies. The results are slightly different. I would like to know how opencv implements this metrics.
I have read the source code, but they use a foggy calls to kernel functions. Do they uses Fourier transform or other basis sets like rectangular basis functions?
I recently learned about the definition of shift invariant, but I still don’t understand why does the dual-tree complex wavelet transform have this property. Can someone explain it simply?
The short explanation is that the DTCWT is not exactly shift-invariant. A shift-invariant invertible wavelet transform is the à trous algorithm, also called cycle spinning (see an overview here). The principle behind the DTCWT is explained here in a very accessible way.
Compared to the FFT, the DWT is not shift invariant because at each scale, the odd samples are treated very differently from the even samples. In the frequency domain, the even samples are covered by the real part, the odd samples by the imaginary part.
The DTCWT paper explains that following this principle, a shift invariant transform can be built, but not with compact basis functions (localised in time). The DTCWT uses basis functions that are compact and as close to shift-invariant as possible.
I am trying to rewrite a matlab code to cpp and I still blocked with this line :
[c, l]=wavedec(S,4,'Dmey');
Is there something like that in opencv ?
if someone have an idea about it try to share it with and thanks in advance.
Maybe if you would integrate your codes with Python, then PyWavelet might be an option.
I'm seeing the function that you're looking for is in there. That function is called Discrete Meyer (Dmey). Not sure what you're planning to do with that though, maybe you're processing some images or something, but Dmey is OK, not very widely used. You might want to just find some GitHub codes and integrate to whatever you're doing to see if it would work first, and based on those you can also change the details of your currently posted function (might find something more efficient).
1-D wavelet decomposition (wavedec)
In your code, c and l stand for coefficients and level. You're passing level four with a Dmey function. If you'd have one dimensional data, the following map is how your decomposition would look like, roughly I guess:
There are usually two types of decomposition models that are being used in Wavelets, one is called packet which is similar to a Full Binary Tree, from architecture standpoint:
The other one, which is the one you're most likely using, is less computationally expensive, because it does not decompose both branches of a tree, if you will. It'd just do the mathematical decomposition in one branch of the tree. Maybe, these images would shed some lights:
1 D
2 D
Notes:
If you have a working model in MatLab, you might want to see the C/C++ Code Generation in MatLab, will automatically convert MatLab codes to C++.
References:
Images are from Wikipedia or mathworks.com
Wiki
mathworks
Wavelet 2D
I need to solve some large (N~1e6) Laplacian matrices that arise in the study of resistor networks. The rest of the network analysis is being handled with boost graph and I would like to stay in C++ if possible. I know there are lots and lots of C++ matrix libraries but no one seems to be a clear leader in speed or usability. Also, the many questions on the subject, here and elsewhere seem to rapidly devolve into laundry lists which are of limited utility. In an attempt to help myself and others, I will try to keep the question concise and answerable:
What is the best library that can effectively handle the following requirements?
Matrix type: Symmetric Diagonal Dominant/Laplacian
Size: Very large (N~1e6), no dynamic resizing needed
Sparsity: Extreme (maximum 5 nonzero terms per row/column)
Operations needed: Solve for x in A*x=b and mat/vec multiply
Language: C++ (C ok)
Priority: Speed and simplicity to code. I would really rather avoid having to learn a whole new framework for this one problem or have to manually write too much helper code.
Extra love to answers with a minimal working example...
If you want to write your own solver, in terms of simplicity, it's hard to beat Gauss-Seidel iteration. The update step is one line, and it can be parallelized easily. Successive over-relaxation (SOR) is only slightly more complicated and converges much faster.
Conjugate gradient is also straightforward to code, and should converge much faster than the other iterative methods. The important thing to note is that you don't need to form the full matrix A, just compute matrix-vector products A*b. Once that's working, you can improve the convergance rate again by adding a preconditioner like SSOR (Symmetric SOR).
Probably the fastest solution method that's reasonable to write yourself is a Fourier-based solver. It essentially involves taking an FFT of the right-hand side, multiplying each value by a function of its coordinate, and taking the inverse FFT. You can use an FFT library like FFTW, or roll your own.
A good reference for all of these is A First Course in the Numerical Analysis of Differential Equations by Arieh Iserles.
Eigen is quite nice to use and one of the fastest libraries I know:
http://eigen.tuxfamily.org/dox/group__TutorialSparse.html
There is a lot of related post, you could have look.
I would recommend C++ and Boost::ublas as used in UMFPACK and BOOST's uBLAS Sparse Matrix
After some studying, I created a small app that calculates DFTs (Discrete Fourier Transformations) from some input. It works well enough, but it is quite slow.
I read that FFTs (Fast Fourier Transformations) allow quicker calculations, but how are they different? And more importantly, how would I go about implementing them in C++?
If you don't need to manually implement the algorithm, you could take a look at the Fastest Fourier Transform in the West
Even thought it's developed in C, it officially works in C++ (from the FAQ)
Question 2.9. Can I call FFTW from
C++?
Most definitely. FFTW should compile
and/or link under any C++ compiler.
Moreover, it is likely that the C++
template class is
bit-compatible with FFTW's
complex-number format (see the FFTW
manual for more details).
FFT has n*log(n) compexity compared to DFT which has n^2.
There are lot of literature about that, and I strongly advise that you check that first, because such wide topic can not be full explaned here.
http://en.wikipedia.org/wiki/Fast_Fourier_transform (check external links )
If you need library I advise you to use existing one, for instance.
http://www.fftw.org/
This library has efficiently implementation of FFT and is also used in propariaretery software (MATLAB for instance)
Steven Smith's book The Scientist and Engineer's Guide to Digital Signal Processing , specifically Chapter 8 on the DFT and Chapter 12 on the FFT, does a much better job of explaining the two transforms that I ever could.
By the way, the whole book is available for free (link above) and it's a very good introduction to signal processing.
Regarding the C++ code request, I've only used the Fastest Fourier Transform in the West (already cited by superexsl) or DSP libraries such as those from TI or Analog Devices.
The results of a correctly implemented DFT are essentially identical to the results of a correctly implemented FFT (they differ only by rounding errors). As others have pointed out here, the major difference is that of performance. DFT has O(n^2) operations while the FFT has O(nlogn) operations.
The best, most readable publication I have ever found (the one I still refer to) is The Fast Fourier Transform and its Applications by E Oran Brigham. The first few chapters provide a very thorough overview of the continuous and discrete forms of the Fourier Transform. He then uses that to develop the fast version of the DFT based on the Cooley-Tukey Algorithm for the radix-2 (n is a power of 2) and mixed-radix cases (though the latter being somewhat more shallow treatise than the former).
The basic approach in the radix-2 algorithm to perform a linear time operation on the input X and to recursively split the result in half and perform a similar linear time operation on the two halves. The mixed radix case is similar, though you need to divide X into equal portions each time, so it helps if n doesn't have any large prime factors.
I've found this nice explanation with some algorithms described.
FastFourierTransform
About implementation,
first i'd make sure your implementation returns correct results (compare the output from matlab or octave - which have built in fourier transformates)
optimize when necessary, use profilers
don't use unnecesary for loops