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Sep 15 2014

Correlated noise reduction

Tag: Math,Noise,softwareadmin @ 10:28 am
The noise of the components of the A-vector is highly correlated and the previous post showed a way to produce low noise images analogous to conventional, non-energy selective images by “whitening” the A-vector data. That is good but is there a way to use the correlation to produce lower noise material selective images such as bone or soft tissue canceled? It turns out there are many methods that are seemingly different but are all based on the correlation. Al Macovski introduced the idea and his group at Stanford published several papers on it. It has been used in commercial systems. For example, the Fuji Corporation used an elaborate iterative method to reduce the noise in their “sandwich” photostimulable screen detector system[4]. Other companies like GE are more secretive but I think that they used a similar method with their voltage switching flat panel system.


In this post, I will describe a linear least mean squares method, which is a simplified version of the approach introduced by Cao et. al.[2], who also did her work at Stanford. This approach has straight-forward theory, is easy to implement and is effective at reducing noise. One problem with the approach is that it may change the quantitative values of the data in CT and Kalendar et al.[5] published an enhancement that may retain the quantitative information. However, if quantitative data are important, then the software can extract data from the underlying images guided by an operator using the noise-reduced image.

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Aug 22 2014

Dimensionality and noise in energy selective x-ray imaging-Part 2

Tag: Math,Noise,Physics,softwareadmin @ 11:34 am

The previous post in this series discussed the mathematics behind the increase in noise with the dimensionality, the number of basis functions used to approximate the attenuation coefficient. The series of posts is based on my recently published paper, Dimensionality and noise in energy selective x-ray imaging, available for free download here. This post describes simulations of the increase in noise with an object composed of body materials and an x-ray tube spectrum. The next post will show how to make low-noise images with the same properties as conventional x-ray images from the energy spectrum data. The main purpose of these last two posts is providing and explaining the code to reproduce the images in the paper.

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Jul 23 2014

Dimensionality and noise in energy selective x-ray imaging-Part 1

Tag: Math,Noise,Physicsadmin @ 2:18 pm
In the next few posts I will discuss my paper, Dimensionality and noise in energy selective x-ray imaging, available for free download here. I will elaborate on the physical and mathematical background and explain how to reproduce the figures.

With my approach to energy selective imaging, the x-ray attenuation coefficient is approximated as a linear combination of functions of energy multiplied by constants that are independent of energy. The number of functions required is the dimensionality. The basic premise of the paper is that the dimensionality is really a pragmatic tradeoff between more information, which requires larger dimensionality, and the increase in noise, which requires higher dose and more expensive equipment to reduce it to a level where the resultant images are clinically useful. The bottom line of the paper is that with biological materials such as soft tissue, bone, and fat, only two dimensions are practical but if an externally administered contrast agent with a high atomic number element such as iodine is included then three and maybe more dimensions are possible.

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May 07 2014

The singular value decomposition

Tag: Math,softwareadmin @ 1:59 pm
Not only is the singular value decomposition (SVD) fundamental to matrix theory but it is also widely used in data analysis. I have used it several times in my posts. For example, here and here, I used the singular values to quantify the intrinsic dimensionality of attenuation coefficients. In this post, I applied the SVD to give the optimal basis functions to approximate the attenuation coefficient and compared them to the material attenuation coefficient basis set[1]. All of these posts were based on the SVD approximation theorem, which allows us to find the nearest matrix of a given rank to our original matrix. This is an extremely powerful result because it allows us to reduce the dimensionality of a problem while still retaining most of the information.
In this post, I will discuss the SVD approximation theorem from an intuitive basis. The math here will be even less rigorous than my usual low standard since my purpose is to get an understanding of how the theorem works and what are its limitations. If you want a mathematical proof, you can find it in many places like Theorems 5.8 and 5.9 of the book Numerical Linear Algebra[2] by Trefethen and Bau. These proofs do not provide much insight into the approximation so I will provide two ways of looking at the theorem: a geometric interpretation and an algebraic interpretation.

more → Continue reading “The singular value decomposition”

Oct 18 2013

Why is polynomial estimator variance so large?

Tag: Math,Noiseadmin @ 9:11 am
Anyone with experience in energy selective imaging is struck by the terrible performance of polynomial estimators discussed in my last post. This is most likely due to the fact that in the past the number of spectra was almost always equal to the dimension of the A-vector. In this case, as I showed in the last post, any estimator that solves the deterministic, noise free equations is the maximum likelihood estimator (MLE). With equal number of spectra and dimension, the polynomial estimator is accurate for low-noise data so it provides an ’efficient’ estimator. That is its covariance is equal to the Cramèr-Rao lower bound (CRLB). In this post, I examine the reason for the poor performance with more measurements than the A-vector dimension.

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Oct 01 2013

Estimators for Energy-selective imaging—Part 1

Tag: Implementation,Math,Noise,Physicsadmin @ 5:46 pm

In a previous post I described the application of statistical estimator theory to energy selective x-ray imaging. I introduced a linearized model for the signal and noise and in a subsequent post I described a linear maximum likelihood estimator (MLE) that achieved the Cramèr-Rao lower bound (CRLB). In many applications, such as CT, the linear model is not sufficiently accurate. In this post, I will start the discussion of my paper[3] “Estimator for photon counting energy selective x-ray imaging with multi-bin pulse height analysis.” The paper describes an estimator that is accurate for a wide dynamic range that also achieves the CRLB and has other desirable properties such as fast and predictable computation time and being implementable in a clinical institution as opposed to a physics lab. This post frames the discussion by describing general aspects of computing the A-vector from energy selective measurements and several estimators that are widely used and their properties.

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Aug 02 2012

NQ detector statistics

Tag: Math,Noiseadmin @ 10:32 am

Another way to get energy selective data is to measure the total number of counts and their total energy. This may not seem to provide energy-dependent information but if you look at it from the point of view of energy weighting, there is information. We can look at the total counts as the integral of the spectrum multiplied by a constant function of energy, where the constant is one. On the other hand, the total energy is the integral of the spectrum multiplied by the function f(E)=E. This weights the higher energies more than the lower and therefore has different information than the counts. This is actually enough difference to give SNR comparable to two-bin PHA.

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Nov 09 2011

Monte Carlo simulation of SNR with energy information

Tag: Implementation,Math,Noise,Physicsadmin @ 11:41 am

In this post, I continue the discussion of my paper “Near optimal energy selective x-ray imaging system performance with simple detectors”, which is available for free download here. I will describe a Monte Carlo simulation of the SNR with energy information discussed in my last post. The simulation traces the paths of individual photons. This is, perhaps, more fundamental than my previous simulations that relied on statistical models for the detector data. I develop models for the random path lengths and use them to simulate the imaging task for SNR described in my last post. The model is validated by comparing the energy spectrum of photons transmitted through an object to the theoretical formula. I also provide an estimate of the errors in the Monte Carlo results.

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Nov 07 2011

SNR with energy information

Tag: Implementation,Math,Noise,Physicsadmin @ 10:40 am

In the next posts I will discuss some of the results in my recent paper, which is available for free download here. The paper discusses fundamental limits on the signal to noise ratio of x-ray detectors with energy spectrum information. It also describes how we can design practical systems with low energy resolution detectors whose performance gets close to the optimal limit.

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Oct 29 2011

Logarithm of PHA with deadtime

Tag: Math,Noiseadmin @ 10:23 am

The logarithm of the data is often used in x-ray imaging systems because it is (very) approximately proportional to the line integral. The statistics of the log of the counts in photon counting detectors are are different than those of the counts as summarized in my last post and I derive them in this post. I then test the formulas using a Monte Carlo simulation.

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