Robert E. Alvarez

Dimensionality and noise in energy selective x-ray imaging

PURPOSE To develop and test a method to quantify the effect of dimensionality on the noise in energy selective x-ray imaging. METHODS The Cramèr-Rao lower bound (CRLB), a universal lower limit of the covariance of any unbiased estimator, is used to quantify the noise. It is shown that increasing dimensionality always increases, or at best leaves the same, the variance. An analytic formula for the increase in variance in an energy selective x-ray system is derived. The formula is used to gain insight into the dependence of the increase in variance on the properties of the additional basis functions, the measurement noise covariance, and the source spectrum. The formula is also used with computer simulations to quantify the dependence of the additional variance on these factors. Simulated images of an object with three materials are used to demonstrate the trade-off of increased information with dimensionality and noise. The images are computed from energy selective data with a maximum likelihood estimator. RESULTS The increase in variance depends most importantly on the dimension and on the properties of the additional basis functions. With the attenuation coefficients of cortical bone, soft tissue, and adipose tissue as the basis functions, the increase in variance of the bone component from two to three dimensions is 1.4 × 10(3). With the soft tissue component, it is 2.7 × 10(4). If the attenuation coefficient of a high atomic number contrast agent is used as the third basis function, there is only a slight increase in the variance from two to three basis functions, 1.03 and 7.4 for the bone and soft tissue components, respectively. The changes in spectrum shape with beam hardening also have a substantial effect. They increase the variance by a factor of approximately 200 for the bone component and 220 for the soft tissue component as the soft tissue object thickness increases from 1 to 30 cm. Decreasing the energy resolution of the detectors increases the variance of the bone component markedly with three dimension processing, approximately a factor of 25 as the resolution decreases from 100 to 3 bins. The increase with two dimension processing for adipose tissue is a factor of two and with the contrast agent as the third material for two or three dimensions is also a factor of two for both components. The simulated images show that a maximum likelihood estimator can be used to process energy selective x-ray data to produce images with noise close to the CRLB. CONCLUSIONS The method presented can be used to compute the effects of the object attenuation coefficients and the x-ray system properties on the relationship of dimensionality and noise in energy selective x-ray imaging systems.

Invertibility of the dual energy x-ray data transform

PURPOSE The Alvarez-Macovski method extracts the x-ray energy-dependent information by expanding the attenuation coefficient as a linear combination of functions of energy multiplied by basis set coefficients. Since the basis functions are known a priori, the coefficients represent all the energy-dependent information. The method then computes the line integrals of these coefficients, summarized as a vector A, from measurements with multiple x-ray spectra, summarized as a vector L. The purpose of this paper is to determine the factors that affect the invertibility of the L(A) transformation with a two function basis set and two spectral measurements, the dual energy transformation. METHODS A general invertibility theorem is applied that requires testing for zero values of the Jacobian of the transformation in its input domain. General conditions for invertibility are proved. It is shown that the generalized A vector noise variance is proportional to the generalized measurement noise variance divided by the square of the Jacobian. The relationship between the zero Jacobian values and ambiguous sets of A vector points with the same L values is determined. The effect of zero Jacobian values on an iterative algorithm that inverts L(A) is simulated. RESULTS The choice of a particular valid basis set does not affect invertibility. Nonoverlapping measurement spectra such as those from photon counting detectors with perfect pulse height analysis are invertible. The widely used x-ray tube spectra with different voltages are shown to be invertible. Spectra with the same maximum energy, such as those from layered detectors, approach noninvertibility with small absolute value Jacobian for large object thicknesses. The zero Jacobian values fall on curves in A vector space that, except for a simple artificial case, are close to but not exactly straight lines. With noninvertible spectra, pairs of ambiguous points are located on opposite sides of the zero Jacobian curve. The iterative algorithm has large convergence errors near zero Jacobian curves and converges to the closest ambiguous point to the initial estimate for other points. CONCLUSION The invertibility of dual energy systems is determined by the presence of zero values of the Jacobian of the dual x-ray energy data transformation L(A) in the input domain.