I used this script to reconstruct an image of the Shepp-Logan phantom.
Basically, it just simply used radon to get sinogram and used iradon to transform it back.
However, I found that a very obvious moire pattern can be seen when I adjust contrast. This is even more obvious if I use my CT image dataset.
Can anyone help me to understand this? Thanks!
img = phantom(512)*1000;
views = 576;
angles = [0:180/576:180-180/576];
sino = radon(img,angles);
img_rec = iradon(sino,angles);
imshow(img_rec,[]);
Full image after being adjusted contrast:
Regions with obvious moire pattern:
This may be happening because of some factors:
From the MATLAB documentation, iradon uses 'Ram-Lak' (known as ramp filter) filtering as default and does not use any windowing to de-emphasizes noise in the high frequencies. You stated "This is even more obvious if I use my CT image dataset", that is because there you have real noise in the images. The documentation itself advises to use some windowing:
"Because this filter is sensitive to noise in the projections, one of the filters listed below might be preferable. These filters multiply the Ram-Lak filter by a window that de-emphasizes higher frequencies."
Other inconvenient is related to the projector itself. The built-in functions radon and iradon from MATLAB does not take into account the detector size and the x-ray length which cross the pixel. These functions are just pixel driving methods, i.e., they basically project geometrically the pixels in the detector and interpolate them.
Possible solutions:
There are more sophisticated projectors today as [1] and [2]. As I stated here, I implemented the distance-driven projector for 2D Computed Tomography (CT) and 3D Digital Breast Tomosynthesis (DBT), so feel free to use it for your experiments.
For example, I generated 3600 equally spaced projections of the phantom with the distance-driven method, and reconstructed it with the iradon function using this line code:
slice = iradon(sinogram',rad2deg(geo.theta));
Related
Currently, I am working at a short project about stereo-vision.
I'm trying to create depth maps of a scenery. For this, I use my phone from to view points and use the following code/workflow provided by Matlab : https://nl.mathworks.com/help/vision/ug/uncalibrated-stereo-image-rectification.html
Following this code I am able to create nice disparity maps, but I want to now the depths (as in meters). For this, I need the baseline, focal length and disparity, as shown here: https://www.researchgate.net/figure/Relationship-between-the-baseline-b-disparity-d-focal-length-f-and-depth-z_fig1_2313285
The focal length and base-line are known, but not the baseline. I determined the estimate of the Fundamental Matrix. Is there a way to get from the Fundamental Matrix to the baseline, or by making some assumptions to get to the Essential Matrix, and from there to the baseline.
I would be thankful for any hint in the right direction!
"The focal length and base-line are known, but not the baseline."
I guess you mean the disparity map is known.
Without a known or estimated calibration matrix, you cannot determine the essential matrix.
(Compare Multi View Geometry of Hartley and Zisserman for details)
With respect to your available data, you cannot compute a metric reconstruction. From the fundamental matrix, you can only extract camera matrices in a canonical form that allow for a projective reconstruction and will not satisfy the true baseline of the setup. A projective reconstruction is a reconstruction that differs from the metric result by an unknown transformation.
Non-trivial techniques could allow to upgrade these reconstructions to an Euclidean reconstruction result. However, the success of these self-calibration techniques strongly depends of the quality of the data. Thus, using images of a calibrated camera is actually the best way to go.
I am trying to solve this problem by using of Monte-Carlo Flooding algorithm. As result I receive set of semicircles (the picture below), but the requested solution is for trapezoid like polygons. Please, can you suggest me an algorithm by which I will be able to transform this semicircles in polygons?
First, you extract your pieces as contours, using Suzuki-Abe algorithm (Suzuki, S. and Abe, K., Topological Structural Analysis of Digitized Binary Images by Border Following. CVGIP 30 1, pp 32-46 (1985)). You'll get all contours out of your image as they are produced.
Then, you approximate contours into polygons using Ramer-Douglas-Peucker algorithm.
THere is well-known library which does it all - OpenCV, see link for details https://docs.opencv.org/2.4.13.2/modules/imgproc/doc/structural_analysis_and_shape_descriptors.html
I found a Matlab implementation of the LKT algorithm here and it is based on the brightness constancy equation.
The algorithm calculates the Image gradients in x and y direction by convolving the image with appropriate 2x2 horizontal and vertical edge gradient operators.
The brightness constancy equation in the classic literature has on its right hand side the difference between two successive frames.
However, in the implementation referred to by the aforementioned link, the right hand side is the difference of convolution.
It_m = conv2(im1,[1,1;1,1]) + conv2(im2,[-1,-1;-1,-1]);
Why couldn't It_m be simply calculated as:
it_m = im1 - im2;
As you mentioned, in theory only pixel by pixel difference is stated for optical flow computation.
However, in practice, all natural (not synthetic) images contain some degree of noise. On the other hand, differentiating is some kind of high pass filter and would stress (high pass) noise ratio to the signal.
Therefore, to avoid artifact caused by noise, usually an image smoothing (or low pass filtering) is carried out prior to any image differentiating (we have such process in edge detection too). The code does exactly this, i.e. apply and moving average filter on the image to reduce noise effect.
It_m = conv2(im1,[1,1;1,1]) + conv2(im2,[-1,-1;-1,-1]);
(Comments converted to an answer.)
In theory, there is nothing wrong with taking a pixel-wise difference:
Im_t = im1-im2;
to compute the time derivative. Using a spatial smoother when computing the time derivative mitigates the effect of noise.
Moreover, looking at the way that code computes spatial (x and y) derivatives:
Ix_m = conv2(im1,[-1 1; -1 1], 'valid');
computing the time derivate with a similar kernel and the valid option ensures the matrices It_x, It_y and Im_t have compatible sizes.
The temporal partial derivative(along t), is connected to the spatial partial derivatives (along x and y).
Think of the video sequence you are analyzing as a volume, spatio-temporal volume. At any given point (x,y,t), if you want to estimate partial derivatives, i.e. estimate the 3D gradient at that point, then you will benefit from having 3 filters that have the same kernel support.
For more theory on why this should be so, look up the topic steerable filters, or better yet look up the fundamental concept of what partial derivative is supposed to be, and how it connects to directional derivatives.
Often, the 2D gradient is estimated first, and then people tend to think of the temporal derivative secondly as independent of the x and y component. This can, and very often do, lead to numerical errors in the final optical flow calculations. The common way to deal with those errors is to do a forward and backward flow estimation, and combine the results in the end.
One way to think of the gradient that you are estimating is that it has a support region that is 3D. The smallest size of such a region should be 2x2x2.
if you do 2D gradients in the first and second image both using only 2x2 filters, then the corresponding FIR filter for the 3D volume is collected by averaging the results of the two filters.
The fact that you should have the same filter support region in 2D is clear to most: thats why the Sobel and Scharr operators look the way they do.
You can see the sort of results you get from having sanely designed differential operators for optical flow in this Matlab toolbox that I made, in part to show this particular point.
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.
Things are like this:
I have some graphs like the pictures above and I am trying to classify them to different kinds so the shape of a character can be recognized, and here is what I've done:
I apply a 2-D FFT to the graphs, so I can get the spectral analysis of these graphs. And here are some result:
S after 2-D FFT
T after 2-D FFT
I have found that the same letter share the same pattern of magnitude graph after FFT, and I want to use this feature to cluster these letters. But there is a problem: I want the features of interested can be presented in a 2-D plane, i.e in the form of (x,y), but the features here is actually a graph, with about 600*400 element, and I know the only thing I am interested is the shape of the graph(S is a dot in the middle, and T is like a cross). So what can I do to reduce the dimension of the magnitude graph?
I am not sure I am clear about my question here, but thanks in advance.
You can use dimensionality reduction methods such as
k-means clustering
SVM
PCA
MDS
Each of these methods can take 2-dimensional arrays, and work out the best coordinate frame to distinguish / represent etc your letters.
One way to start would be reducing your 240000 dimensional space to a 26-dimensional space using any of these methods.
This would give you an 'amplitude' for each of the possible letters.
But as #jucestain says, a network classifiers are great for letter recognition.