DCT is similar to the DFT, but it only consider the cosine part with real values. Thus, some features in audio signal processing such as Frequency Domain Linear Prediction (FDLP) uses DCT to convert the time domain signal into frequency domain. The DCT was meant for diagonalizing the resulting co-variance matrices with Toeplitz structure, to give fast estimates of their eigenvectors. Transforming a big co-variance matrix into a thin, close to a diagonal, matrix is an instance of decorrelation. In this way, it compresses the energy of a feature. Is my understanding correct?
I suggest you to follow http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.184.6102&rep=rep1&type=pdf
https://books.google.com.au/books?hl=en&lr=&id=fWviBQAAQBAJ&oi=fnd&pg=PP1&dq=Application+of+Discrete+Cosine+Transform&ots=1sONyl_iAy&sig=CyeWmkhfeY_Me0EfonxAt5NJOCo#v=onepage&q=Application%20of%20Discrete%20Cosine%20Transform&f=false
http://www.cse.iitd.ac.in/~pkalra/siv864/assignment/DCT-Paper.pdf
DCT, when applied to lossy image compression like JPEG, converts the image from spatial to frequency domain. Since the human visual system is not very sensitive to high frequency signals, these terms are removed and the lower frequency terms are converted back to the spatial domain, which results in a compressed image.
DFT vs. DCT : Studies have shown that DCT provides better energy compaction for natural photographic images. This means that DCT has a better capability to pack the energy (information) of the input image into fewer coefficients so that the other coefficients having small amplitude may be discarded easily. In addition, DCT computations are faster and require less computational resources since complex number calculations are avoided. Thus for most real time compression, DCT is preferred to DFT as it provides better approximation at lower complexities.
For the second question, DCT simply converts the image from one form into another, without any changes. It's only after the higher frequency terms are removed do the changes actually occur. It is in the hands of the user to decide how much to remove by specifying the "quality" parameter. If artifacts begin to appear then the quality should be increased.
I think we need to know the way DCT is different from the DFT in term of operation. We can't only say, DCT takes cosine real term of signal instead of sine and cosine term, and it is not discontinuous for the periodic signal.
In term of Energy compaction of DCT, how will you relate the DFT with DCT? I need to know for more clarification.
Thanks Peter Breuer, your answers are really informative. Will you please explain more about " "fewer" FFT coefficients, then why not apply some invertible map in the frequency domain that squishes them up to the left ". I think, if we can apply DFT in a proper way, it may give a better resolution in 2-D spectrogram or image than DCT.
You have answered it, I am also not an expert. The filter I am using that response is already emphasizing on low frequencies and I can omit higher frequencies information based on the response. In this case, the application of DCT is not so useful as I have found in my work. This is telling, DCT may not helpful to get an uncorrelated feature. If we can selectively boost up some frequencies, particularly for high frequencies, it would have more variation in feature. Anyway, I wanted to know an explanation of DFT as an alternative to DCT. Because, due to energy compaction property of DCT, it does not give a better result in recognition task without a nonlinear rectification of data. This is because of losing high frequency information.
The discrete cosine transform (DCT) helps separate the image into parts (or spectral sub-bands) of differing importance (with respect to the image's visual quality). The DCT is similar to the discrete Fourier transform: it transforms a signal or image from the spatial domain to the frequency domain ز
For more information see
https://www.cs.cf.ac.uk/Dave/Multimedia/node231.html
- Badges
- Science topic
- Similar topics
- Mathematics
- Algebra
- Matrix