Benchmarking the performance of fixed-image receptor digital radiographic systems part 1: a novel method for image quality analysis

Abstract

This is the first part of a two-part study in benchmarking the performance of fixed digital radiographic general X-ray systems. This paper concentrates on reporting findings related to quantitative analysis techniques used to establish comparative image quality metrics. A systematic technical comparison of the evaluated systems is presented in part two of this study. A novel quantitative image quality analysis method is presented with technical considerations addressed for peer review. The novel method was applied to seven general radiographic systems with four different makes of radiographic image receptor (12 image receptors in total). For the System Modulation Transfer Function (sMTF), the use of grid was found to reduce veiling glare and decrease roll-off. The major contributor in sMTF degradation was found to be focal spot blurring. For the System Normalised Noise Power Spectrum (sNNPS), it was found that all systems examined had similar sNNPS responses. A mathematical model is presented to explain how the use of stationary grid may cause a difference between horizontal and vertical sNNPS responses.

Appendix: stationary grid noise

Using the definition of NPS [20], the NNPS is given by

$${\text{NNPS}}\left( {u,v} \right) = \frac{1}{C}\mathop {\lim }\limits_{X,Y \to \infty } \frac{1}{2X \cdot 2Y}\left\langle {\left| {\mathop \int \limits_{ - X}^{X} \mathop \int \limits_{ - Y}^{Y} \left[ {I\left( {x,y} \right) - \bar{I}} \right]e^{ - 2\pi i(ux + vy)} dxdy}\,\, \right|^{2} } \right\rangle$$

(1)

where C is a normalising constant, I(x,y) is the image. Assume

$$I\left( {x,y} \right) = S\left( {x,y} \right) - G\left( {x,y} \right)$$

(2)

where G(x,y) is the radiation perturbation under the grid and S(x,y) is the image without perturbation.

Orthogonal case

If the Fourier line integral, say in the x direction, is orthogonal to the perturbation, then this perturbation can be modelled as: For $- \frac{T}{4} \le x \le \frac{3T}{4}$

$$G(x) = \left\{ {\begin{array}{ll} {{\text{Pcos}}\left( {\frac{2\pi x}{T}} \right),} & { - \frac{T}{4} \le x \le \frac{T}{4}} \\ {0,} & {\rm {Otherwise}} \\ \end{array} } \right.$$

(3)

where P is the perturbation amplitude, T = 1/f₀ and f₀ is the grid frequency. Substituting (2) into (1) and re-arranging gives

$${\text{NNPS}}\left( u \right) = \frac{1}{C}\mathop {\lim }\limits_{X,Y \to \infty } \frac{1}{2X \cdot 2Y}\left\langle {\left| {\mathop \int \limits_{ - X}^{X} \left[ {S\left( {x,y} \right) - \bar{S}} \right]e^{{ - 2\pi i\left( {ux} \right)}} dx - \mathop \int \limits_{ - X}^{X} \left[ {G\left( {x,y} \right) - \bar{G}} \right]e^{{ - 2\pi i\left( {ux} \right)}} dx} \,\,\right|^{2} } \right\rangle$$

(4)

Denoting N′(u) as the 1D noise spectrum and $G_{n}^{{\prime }}$ as the complex Fourier coefficients of G(x), then (4) becomes

$${\text{NNPS}}\left( u \right) = \frac{1}{C}\mathop {\lim }\limits_{X,Y \to \infty } \frac{1}{2X \cdot 2Y}\left\langle {\left| {N^{'} \left( u \right) - \left(\mathop \sum \limits_{ - \infty }^{\infty } G_{n}^{'} \delta \left( {u - nf_{0} }\right) - G^{'} \left( 0 \right) \right)} \,\, \right|^{2} } \right\rangle$$

(5)

where δ(y) is the Dirac delta function and

$$\left| {G_{n}^{{\prime }} } \right| = \left\{ {\begin{array}{lll} {\frac{P}{\pi }} & {n = 0} \\ {\frac{P}{4}} & {n = \pm 1} \\ {\frac{P}{{\pi \left( {1 - n^{2} } \right)}} } & {n = \pm 2, \pm 4, \pm 6, \cdots } \\ \end{array} } \right.$$

(6)

Since G(x,y) is real, only the positive side of the spectrum $G_{n}^{'}$ is needed. The subtraction of G′(0) removes the zero frequency component in the spectrum of $G_{n}^{{\prime }}$. Hence the resultant noise spectrum is the sum of the original noise spectrum and the fundamental and second harmonic frequency components of $G_{n}^{{\prime }}$ as higher frequency components are negligible.

Parallel case

If the Fourier line integral, say in the y direction, is parallel to the perturbation and is over the non-perturbed area, then (1) applies. If the line integral is over the perturbation, then this perturbation can be modelled as

$$G\left( y \right) = P$$

(7)

where P is the perturbation amplitude. Since $G\left( y \right) = \bar{G}(y)$, (4) reduces to (1). Hence when the line integral direction is parallel to the grid lines, there is no grid aliasing peaks in the NNPS.