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US 6,324,240 B1Granted
Method for beam hardening correction in quantitative computed X-ray tomography
AssigneeThe Board of Trustees of the Leland Stanford Junior University
InventorsChye Hwang Yan, Robert T. Whalen, Sandy Napel
Life of this patent
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Abstract
Each voxel is assumed to contain exactly two distinct materials, with the volume fraction of each material being iteratively calculated. According to the method, the spectrum of the X-ray beam must be known, and the attenuation spectra of the materials in the object must be known, and be monotonically decreasing with increasing X-ray photon energy. Then, a volume fraction is estimated for the voxel, and the spectrum is iteratively calculated.
Claims8
A method for calculating characteristics of an object comprising voxels using an X-ray computer assisted tomography device, comprising the steps of:
- a) projecting through the object an X-ray beam having a known energy spectrum indicated by S(E);
- b) detecting the X-rays with an X-ray detector;
- c) repeating steps (a) and (b) for at least two different orientations of the X-ray beam with respect to the object;
- d) defining a first material and a second material within each voxel, wherein the first and second materials have known distinct attenuation spectra, given by μ1(x,E) and μ2(x,E), respectively, such that a total attenuation coefficient μ(x,E) of the object is given by:
- μ(x,E)=v(x)μ1(x,E)+(1−v(x))μ2(x,E)
- where v(x) is a number between 0 and 1 and indicates a volume fraction of each voxel comprised of the first material;
- e) defining an initial estimated volume fraction vk(x);
- f) calculating an estimated total attenuation coefficient μk(x,E) from the equation;
- μk(x,E)=νk(x)μ1(x,E)+(1−vk(x))μ2(x,E)
- g) calculating a new estimated total attenuation coefficient μk+1(x,E) from the following equation:
- μk+1(x,E)=P−1{Pμk(x,E)−Bμk(x,E)+C(r)}
- where P is an operator defined by Pμ=∫r∫S(E)μ(E)dEdx, B is an operator defined by Bμ=−log∫S(E)exp(∫rμ(E)dx)dE, and C(r) represents X-ray detector output, and r indicates the X-ray beam path;
- h) calculating a new estimated volume fraction vk+1 from the following equation:
- μk+1(x,E)=vk+1(x,E){circumflex over (μ)}1(x,E)+(1−vk+1(x)){circumflex over (μ)}2(x,E)
- where {circumflex over (μ)}1(E)=∫S(E)μ1(E)dE and {circumflex over (μ)}2(E)=∫S(E)μ2(E)dE
- i) repeating steps (f), (g) and (h) using the new estimated volume fraction vk+1(x) instead of the initial estimated volume fraction vk(x).
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A method for calculating characteristics of an object comprising voxels using an X-ray computer assisted tomography device, comprising the steps of:
- a) projecting through the object a first X-ray beam having a known energy spectrum indicated by S1(E);
- b) detecting the first X-ray beam with an X-ray detector;
- c) repeating steps (a) and (b) for at least two different orientations of the first X-ray beam with respect to the object;
- d) projecting through the object a second X-ray beam having a known energy spectrum indicated by S2(E);
- e) detecting the second X-ray beam with an X-ray detector;
- f) repeating steps (d) and (e) for at least two different orientations of the second X-ray beam with respect to the object;
- g) defining a first basis attenuation function and a second basis attenuation function, given by μ1(x,E) and μ2(x,E), respectively, such that a total attenuation coefficient μ(x,E) of the voxel is given by:
- μ(x,E)=A(x)μ1(x,E)+D(x)μ2(x,E)
- where coefficients A and D are real numbers;
- h) defining initial coefficients Ak and Dk;
- i) calculating an estimated total attenuation coefficient μk(x,E) from the equation;
- μk(x)(x,E)=Ak(x)μ1(x,E)+Dk(x)μ2(x,E)
- j) calculating a new estimated total attenuation coefficient μk+1,S1(x,E) associated with spectrum S1(E) from the following equation:
- μk+1,S1(x,E)=PS1−1{PS1μk(x,E)−BS1μk(x,E)+C1(r)}
- where PS1 is an operator defined by PS1μ=∫r∫S1(E)μ(E)dEdx, BS1 is an operator defined by BS1μ=−log∫S1(E)exp(∫r(E)dx)dE, and C1(r) represents X-ray detector output associated with S1(E);
- k) calculating a new estimated total attenuation coefficient μk+,S2(x,E) from the following equation:
- μk+1,S2(x,E)=PS2−1{PS2μk(x,E)−BS2μk(x,E)+CS2(r)}
- where PS2 is an operator defined by PS2μ=∫r∫S2(E)μ(E)dEdx, BS2 is an operator defined by BS2μ=−log ∫S2(E)exp(∫rμ(E)dx)dE, and C2(r) represents X-ray detector output associated with S2(E);
- l) calculate new estimated coefficients Ak+1(x) and Dk+1(x) from the following simultaneous equations:
- μk+1,S1(x,E)=Ak+1(x){circumflex over (μ)}1,S1(x,E)+Dk+1(x){circumflex over (μ)}2,S1(x,E), and
- μk+1,S2(x,E)=Ak+1(x){circumflex over (μ)}1,S2(x,E)+Dk+1(x){circumflex over (μ)}2,S2(x,E),
- where
- {circumflex over (μ)}1,S1(x)=∫S1(E)μ1(x,E)dE, {circumflex over (μ)}1,S2(x)=∫S2(E)μ1(x,E)dE,
- {circumflex over (μ)}2,S1(x)=∫S1(E)μ2(x,E)dE, and {circumflex over (μ)}2,S2(x)=∫S2(E)μ2(x,E)dE;
- m) repeating steps (i), (j), (k) and (1) using the new estimated coefficients Ak+1(x) and Dk+1(x) instead of the initial estimated coefficients Ak(x) and Dk(x).
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