I am currently looking into some image processing project and just wondering how to obtain the low, middle and high frequency components of an image? For example, as this picture showed (I got it from googling without detailed description how to obtained this picture, but presumably using some filtering).
Also, I came across this post of using discrete cosine transform (DCT), and it can help us to get the low and high frequency components of an image. Just wondering how to use DCT to get the middle frequency component?
Link of DCT
I also have very basic knowledge about filtering. I think there are also Gaussian high/low pass filters available to use. And also wavelet based filtering. Just wondering what are the differences between Gaussian, Wavelet and DCT based filtering? Which one should I use?
Typical steps would be:
use a Fourier Transform to bring the image into frequency domain
apply filtering by zero-ing out areas of the fft image
reverse the fourier transform to bring image back to spatial domain
This is a really good example of high/low/mid pass filters in frequency domain: http://paulbourke.net/miscellaneous/imagefilter/
You will want to use MatLab's built in fft our fast fourier transform function. Fourier transforms are an extremely powerful method to filter frequencies. http://www.mathworks.com/help/matlab/ref/fft.html has some great examples on how to use the fft. Once you find the frequencies that make up the image you can take out the undesired frequencies to fit and then reverse fourier transform to obtain the new image.
Related
I have an image where I add a Gaussian noise. I need to use the ideal low pass filter to remove the noise but I cannot really see any examples on the official Matlab documentation. There examples but not with images and I cannot really grasp the concept behind this filter. So could somebody explain how the ideal low pass filter can be used to remove noise?
image = imread('eight.tif');
imshow(image );
noisyImage = imnoise(image,'gaussian',0.02);
imshow(noisyImage);
If you know the standard deviation of the noise, It's good to use a Gaussian filter with that specific standard deviation. Although in most of the cases, it's good to use Bilateral filter (imbilatfilt) which is a gaussian filter with some other features that preserves the edges.
If you don't know what your noise is, It's best to use Wiener filter ([J,noise_out] = wiener2(I,[m n])). This filter observes the frequency behavior of the image and looks for a special pattern which is statistically consistent with noise. In other word, it estimate the noise of image and filter that specific noise for you. noise_out is the estimates of the additive noise power and m,n are the sizes of the filter's kernel (which I suggest something like 5*5 or 7*7).
Of course there are some other filtering methods including handmade ones, but those need more effort and lots of trial and error.
I was reading up on the DWT for the first time and the document stated that it is used to represent time-frequency data of a signal which other transforms do not provide.
But when I look for a usage example of the DWT in MATLAB I see the following code:
X=imread('cameraman.tif');
X=im2double(X);
[F1,F2]= wfilters('db1', 'd');
[LL,LH,HL,HH] = dwt2(X,'db1','d');
I am unable to understand the implementation of dwt2 or rather what is it and when and where we use it. What actually does dwt2 return and what does the above code do?
The first two statements simply read in the image, and convert it so that the dynamic range of each channel is between [0,1] through im2double.
Now, the third statement, wfilters constructs the wavelet filter banks for you. These filter banks are what are used in the DWT. The method of the DWT is the same, but you can use different kinds of filters to achieve specific results.
Basically, with wfilters, you get to choose what kind of filter you want (in your case, you chose db1: Daubechies), and you can optionally specify the type of filter that you want. Different filters provide different results and have different characteristics. There are a lot of different wavelet filter banks you could use and I'm not quite the expert as to the advantages and disadvantages for each filter bank that exists. Traditionally, Daubechies-type filters are used so stick with those if you don't know which ones to use.
Not specifying the type will output both the decomposition and the reconstruction filters. Decomposition is the forward transformation where you are given the original image / 2D data and want to transform it using the DWT. Reconstruction is the reverse transformation where you are given the transform data and want to recreate the original data.
The fourth statement, dwt2, computes the 2D DWT for you, but we will get into that later.
You specified the flag d, so you want only the decomposition filters. You can use wfilters as input into the 2D DWT if you wish, as this will specify the low-pass and high-pass filters that you want to use when decomposing your image. You don't have to do it like this. You can simply specify what filter you want to use, which is how you're calling the function in your code. In other words, you can do this:
[F1,F2]= wfilters('db1', 'd');
[LL,LH,HL,HH] = dwt2(X,F1,F2);
... or you can just do this:
[LL,LH,HL,HH] = dwt2(X,'db1','d');
The above statements are the same thing. Note that there is a 'd' flag on the dwt2 function because you want the forward transform as well.
Now, dwt2 is the 2D DWT (Discrete Wavelet Transform). I won't go into the DWT in detail here because this isn't the place to talk about it, but I would definitely check out this link for better details. They also have fully working MATLAB code and their own implementation of the 2D DWT so you can fully understand what exactly the DWT is and how it's computed.
However, the basics behind the 2D DWT is that it is known as a multi-resolution transform. It analyzes your signal and decomposes your signal into multiple scales / sizes and features. Each scale / size has a bunch of features that describe something about the signal that was not seen in the other scales.
One thing about the DWT is that it naturally subsamples your image by a factor of 2 (i.e. halves each dimension) after the analysis is done - hence the multi-resolution bit I was talking about. For MATLAB, dwt2 outputs four different variables, and these correspond to the variable names of the output of dwt2:
LL - Low-Low. This means that the vertical direction of your 2D image / signal is low-pass filtered as well as the horizontal direction.
LH - Low-High. This means that the vertical direction of your 2D image / signal is low-pass filtered while the horizontal direction is high-pass filtered.
HL - High-Low. This means that the vertical direction of your 2D image / signal is high-pass filtered while the horizontal direction is low-pass filtered.
HH - High-High. This means that the vertical direction of your 2D image / signal is high-pass filtered as well as the horizontal direction.
Roughly speaking, LL corresponds to just the structural / predominant information of your image while HH corresponds to the edges of your image. The LH and HL components I'm not too familiar with, but they're used in feature analysis sometimes. If you want to do a further decomposition, you would apply the DWT again on the LL only. However, depending on your analysis, the other components are used.... it just depends on what you want to use it for! dwt2 only performs a single-level DWT decomposition, so if you want to use this again for the next level, you would call dwt2 on the LL component.
Applications
Now, for your specific question of applications. The DWT for images is mostly used in image compression and image analysis. One application of the 2D DWT is in JPEG 2000. The core of the algorithm is that they break down the image into the DWT components, then construct trees of the coefficients generated by the DWT to determine which components can be omitted before you save the image. This way, you eliminate extraneous information, but there is also a great benefit that the DWT is lossless. I don't know which filter(s) is/are being used in JPEG 2000, but I know for certain that the standard is lossless. This means that you will be able to reconstruct the original data back without any artifacts or quantization errors. JPEG 2000 also has a lossy option, where you can reduce the file size even more by eliminating more of the DWT coefficients in such a way that is imperceptible to the average use.
Another application is in watermarking images. You can embed information in the wavelet coefficients so that it prevents people from trying to steal your images without acknowledgement. The DWT is also heavily used in medical image analysis and compression as the images generated in this domain are quite high resolution and quite large. It would be extremely useful if you could represent the images in the same way but occupying less physical space in comparison to the standard image compression algorithms (that are also lossy if you want high compression ratios) that exist.
One more application I can think of would be the dynamic delivery of video content over networks. Depending on what your connection speed is or the resolution of your screen, you get a lower or higher quality video. If you specifically use the LL component of each frame, you would stream / use a particular version of the LL component depending on what device / connection you have. So if you had a bad connection or if your screen has a low resolution, you would most likely show the video with the smallest size. You would then keep increasing the resolution depending on the connection speed and/or the size of your screen.
This is just a taste as to what the DWT is used for (personally, I don't use it because the DWT is used in domains that I don't personally have any experience in), but there are a lot more applications that are quite useful where the DWT is used.
I am trying to use wavelet transform to represent song in frequency domain using discrete wavelet transform to made decomposition and made the frequency of the singer in place the the song using Matlab
The problem that the dwt and the decomposition mades represent it only in time domain.
How can I represent it in frequency if DWT doesn't represent It what would do?
Thank you
When we say "frequency transform" or talk about "representing frequency" we are usually talking about the Fourier Transform, implemented as the DFT, or discrete Fourier transform. Andre is correct in the comments below when he says that the DWT is also a type of frequency transform; however, wen we say "represent song in frequency domain" it usually means DFT, not DWT.
That being said, I don't recommend the DWT for music and sound analysis because the analysis bands are fixed at one-octave, which is simply too wide to do anything meaningful with. There are other techniques related to wavelets that are more effective for audio, but I don't gather from your question that you are using one of them.
In addition to the DFT, which is usually implemented as the FFT, or fast Fourier transform, you may also want to read about the STFT (short-time Fourier transform).
I have some dynamic light scattering data. The machine pumps out the autocorrelation function, and a count-rate.
I can do a simple fit to the ACF
ACF = exp(-D*q^2*t)
and obtain the diffusion coefficient.
I want to obtain the same D from the power spectrum. I have been able to create a power spectrum in two ways -- from the Fourier transform of the ACF, and from the count rate. Both agree, but the power spectrum does not look like in the one in the books, so I'm not sure how to use it to work out the line width.
Attached is an image from a PDF that shows what you should get, and what I get from MATLAB. Can anyone make sense of whats going on?
I have used the code of answer #3 on this question. The resulting autocorrelation comes out exactly the same as
the machine gives me and
using MATLAB's autocorr command on the photoncount data.
Thank you for your time.
When you compute the Fourier transform from short sequences of data it often looks very noisy. There are a number of reasons for this. One reason is that the statistics of individual Fourier components are not Gaussian, and so averaging the spectra across multiple samples of data will only slowly improve the quality of the estimate.
Another causes of "noisiness" in empirical spectra behavior is that you are applying (to a finite data sample) a transform which involves a pathological sinc function and which assumes an infinite length signal. To diminish this problem, it helps to apply a "windowing-function" to your data before computing the Fourier transform. One of the more complicated but also more powerful windowing approaches is the use of so-called 'Slepian tapers'.
MATLAB conveniently implements well-known windows in functions such as hamming and hann.
I am working with the same image and I also need to remove the texture from the image posted in this link
How can I remove the texture from an image using matlab?
Discussions were made on this and I'm quite confused which filter(gaussian LPF or ideal lowpass) is really needed and what is the reason behind this.Which frequencies contribute for this texture????please can someone explain me!
An ideal low pass filter will keep all spatial frequencies below a nominal spatial frequency, and remove all spatial frequencies above it. Unfortunately, a true ideal low pass filter has infinite support (i.e., has an infinitely large non-zero spatial extend). Even a practical approximation to an ideal low pass filter has large spatial support.
A Gaussian, on the other hand, isn't ideal in terms of which frequencies it filters out. A Gaussian in the spatial domain turns out to be a Gaussian in the spatial frequency domain. That is, it doesn't produce very sharp spatial frequency selectivity. The advantage though is that the spatial support of the filter is small. People use Gaussian filters for this because they are convenient mostly. Filtering with a Gaussian tends to look "natural" compared to ideal low pass filters, which can generate ringing artifacts.
A Lanczos filter (windowed sinc filter) is also another choice as it will have a small spatial support and will approximate an ideal filter better than a Gaussian.
However, which is better for your image largely depends on what you want to do. While there's significant theory behind it, qualitative choices like this in image processing are largely an art.
The type of filter you are looking for is ideally nonlinear:
smoothing in areas without large-scale gradients (edges), and
little smoothing close to edges to be preserved.
Here are two alternatives:
The Kuwahara filter:
http://homepage.tudelft.nl/e3q6n/publications/1999/PAA99DRBDPVLV/PAA99DRBDPVLV.pdf
Enhanced shortening flow (Figure 8) in:
http://www.cs.jhu.edu/~misha/Fall07/Papers/intro-to-scalespace.pdf
In the second filter (Enhanced shortening flow), you can
vary the scale parameter and the nonlinear function,
h(Lw) on page 17. Thus, more trimming possibilities.
Ideally, the filter is completely isotropic
(same frequency effect on each possible angle).
Michael