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- Abstract
- 1. Introduction
- 2. Background: Image formation in phase-contrast imaging
- 3. Signal detection and the ideal observer SNR
- 4. Interpretation of ideal observer SNRs for phase-contrast imaging
- 5. Example: Propagation-based phase-contrast imaging
- 6. Example: Analyzer-based phase-contrast imaging
- 7. Summary
- References

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J Opt Soc Am A Opt Image Sci Vis. Author manuscript; available in PMC 2010 December 6.

Published in final edited form as:

PMCID: PMC2997532

NIHMSID: NIHMS250342

The publisher's final edited version of this article is available at J Opt Soc Am A Opt Image Sci Vis

See other articles in PMC that cite the published article.

Phase-contrast imaging methods exploit variations in an object’s refractive index distribution to permit the visualization of subtle features that may have very similar optical absorption properties. Although phase-contrast is often viewed as being desirable in many biomedical applications, its relative influence on signal detectability when both absorption- and phase-contrast are present remains relatively unexplored. In this work, we investigate the ideal Bayesian observer signal to noise ratio (SNR) in phase-contrast imaging for a signal-known-exactly/background-known exactly detection task involving a weak signal. We demonstrate that this signal detectability measure can be decomposed into three contributions that have distinct interpretations associated with the imaging physics.

Two-dimensional (2D) coherent imaging systems that employ phase-contrast can visualize the projected refractive index of transparent samples and have found widespread use in microscopy and other biomedical imaging applications. When phase-contrast is employed, the measured intensity on the output plane of the imaging system is determined by both the amplitude and phase of the underlying wavefield on the input plane. Well-known examples include in-line holography [1,2], analyzer-crystal-based phase-contrast imaging [3–5], Zernike phase-contrast imaging [6], and differential interference phase-contrast imaging [7]. In recent years, much effort has been devoted to the development of X-ray phase-contrast imaging systems for medical imaging [4,5,8–10] and other applications [2,11–14]. In these methods, the image contrast in the recorded radiographs can arise from a mixture of the absorption and refractive (i.e., phase) properties of the object, resulting in a radiograph with an edge-enhanced appearance that may be diagnostically useful [4,5,8,15].

Although phase-contrast is generally viewed as being desirable in many biomedical applications, its influence on objective measures of image quality when both phase- and absorption-contrast contribute to image formation remains relatively unexplored. Physical measures of image quality such as noise variance, signal contrast, and spatial resolution metrics, have been applied to optimize phase-contrast imaging systems [10,16–20]. While physical measures can provide valuable insights relevant to system optimization, they do not directly reveal how useful the produced images are for facilitating a specified diagnostic task. Task-based, or objective, measures of image quality are fundamentally distinct from physical measures in that they are inherently grounded in statistical decision theory and reveal the average performance of an observer on a specific diagnostic task [21–25]. Our group was among the first to utilize task-based measures of image quality in studies of X-ray phase-contrast imaging [26–30].

The ideal Bayesian observer is a computational algorithm or mathematical model that makes optimal use of all available information in performing a classification task such as signal detection. The performance of the ideal observer provides an upper bound for classification performance that no human or pattern recognition algorithm can surpass for the specified task. When optimizing the data acquisition hardware and protocols of an imaging system, it is advisable to maximize the performance of the ideal observer [31]. Optimizing the imaging system by use of this metric will maximize the amount of information in the raw measurement data that is relevant to performing the specified diagnostic task. Moreover, there may be certain diagnostic tasks for which the performance of the ideal observer is a good predictor of human observer performance [23]. Any numerical observer, including the ideal observer, makes a binary decision by computing a test statistic from knowledge of the observered image data, and comparing it to a decision threshold. The signal-to-noise ratio (SNR) of the test statistic, which is defined explicitly later, represents a convenient summary measure of detection performance. For the ideal observer, we will refer to this quantity as the *ideal observer SNR*.

In this work, a Fourier optics description of phase-contrast imaging employing linear shift-invariant optical systems is utilized to model phase- and absorption-contrast contributions to the ideal observer SNR for a signal-known-exactly/background-known-exactly (SKE/BKE) detection task. We consider that the observer acts upon a single raw image that contains phase- and absorption-contrast, and no image processing or phase-retrieval operations have been performed. We demonstrate that the squared ideal observer SNR can be decomposed into three contributions that have distinct interpretations associated with the imaging physics and contrast mechanisms. This analysis quantifies the interplay between the imaging physics and spatial resolution response of the imaging system that determines the relative importance of phase-contrast to signal detectability, and may provide insights into the use of the ideal observer SNR as a metric for optimizing phase-contrast imaging systems. It will also permit a quantitative comparison of the ideal observer SNR associated with different phase-contrast imaging systems and operating conditions. Propagation-based and analyzer-based phase-contrast imaging are considered as special cases of our analysis, and numerical studies are conducted to demonstrate the relative magnitudes of the SNR components corresponding to different beam energies and amounts of system blur.

The paper is organized as follows. In Sections 2 and 3 we review some basic principles of phase-contrast imaging and signal detection theory that will be employed in our analysis. The ideal observer SNR corresponding to a SKE/BKE detection task is analyzed in Section 4 for phase-contrast imaging systems employing linear shift-invariant optical systems. The specific cases of propagation-based and analyzer-based X-ray phase-contrast imaging are considered in Sections 5 and 6. We conclude with a summary and discussion of the work in Section 7.

Below we provide a brief review of the relevant imaging physics and a Fourier optics-based description of phase-contrast imaging systems. A comprehensive treatment of phase-contrast image formation can be found in the monograph by Paganin [32].

As depicted in Fig. 1, consider that a compactly supported object is irradiated by a monochromatic scalar plane wave *U _{i}* with wavelength

$$n(\overrightarrow{r})\equiv 1-\mathrm{\Delta}(\overrightarrow{r})+j\beta (\overrightarrow{r}),$$

(1)

where *$\stackrel{\u20d7}{r}$* = (*x, y, z*) and
$j\equiv \sqrt{-1}$. The wavelength dependence of *n*(*$\stackrel{\u20d7}{r}$*) will be suppressed to simplify our notation. In the X-ray wavelength regime, Δ(*$\stackrel{\u20d7}{r}$*) > 0 and the linear X-ray attenuation coefficient *μ*(*$\stackrel{\u20d7}{r}$*) is given by

$$\mu (\overrightarrow{r})=2k\beta (\overrightarrow{r}),$$

(2)

where
$k\equiv {\scriptstyle \frac{2\pi}{\lambda}}$ is the wavenumber. Note that classic radiographic methods are sensitive only to variations in *β*(*$\stackrel{\u20d7}{r}$*), while phase-contrast methods are sensitive to variations in both Δ(*$\stackrel{\u20d7}{r}$*) and *β*(*$\stackrel{\u20d7}{r}$*).

Assuming a forward propagating wavefield and a sufficiently thin object, the transmitted wavefield *U _{o}*(

$${U}_{o}(x,y)=T(x,y){U}_{i},$$

(3)

where the transmission function *T*(*x, y*) can be expressed generally as

$$T(x,y)=M(x,y)exp[j\phi (x,y)].$$

(4)

The amplitude modulus *M*(*x, y*) = exp[−*A*(*x, y*)] and the phase shift (*x, y*) describe how the amplitude and phase of the probing wavefield are perturbed by variations in the object’s complex refractive index distribution *n*(*$\stackrel{\u20d7}{r}$*). Specifically, *A*(*x, y*) and (*x, y*) are related to *n*(*$\stackrel{\u20d7}{r}$*) as

$$A(x,y)=k\int dz\beta (\overrightarrow{r})$$

(5a)

and

$$\phi (x,y)=-k\int dz\mathrm{\Delta}(\overrightarrow{r}),$$

(5b)

where the line-integrals are computed along the beam path that is assumed to be parallel with the *z*-axis in the projection approximation [32].

Many phase-contrast imaging systems utilize a linear shift-invariant (LSI) optical system to render phase-contrast effects visible [33–37]. Consider the generic phase-contrast imaging system shown in Fig. 1. Let the object plane wavefield *U _{o}*(

$${U}_{m}(x,y)={G}_{m}(x,y)\ast {U}_{o}(x,y),$$

(6)

where * denotes a two-dimensional (2D) convolution operation. For example, in propagation-based imaging *G _{m}*(

The intensity of the output wavefield,

$${I}_{m}(x,y)=\phantom{\rule{0.16667em}{0ex}}{\mid {U}_{m}(x,y)\mid}^{2},$$

(7)

represents the measured image that generally contains both mixed absorption- and phase-contrast. As described later, the primary objective of our study is to understand the relative contributions of phase- and absorption-contrast in this image to a summary measure of signal detectability for an object detection task.

In many imaging applications where phase-contrast is desirable, the attenuation of the object is weak (relative to a homogeneous background) and the phase perturbation varies slowly [39–41]. Specifically, we will consider that

$$\mid A(x,y)\mid \phantom{\rule{0.16667em}{0ex}}\ll 1$$

(8a)

and

$$\mid \phi (x,y)-\phi (\widehat{x},\widehat{y})\mid \phantom{\rule{0.16667em}{0ex}}\ll 1$$

(8b)

for all locations (*x, y*) and (*$\widehat{x}$, ŷ*) in the object plane that are separated by a distance smaller than the essential support of *G _{m}*(

$${\stackrel{\sim}{I}}_{m}(u,v)=\int {\int}_{-\infty}^{\infty}\mathit{dxdy}\phantom{\rule{0.38889em}{0ex}}{I}_{m}(x,y)exp[-j2\pi (ux+vy)]\equiv {\mathcal{F}}_{2}\{{I}_{m}(x,y)\},$$

(9)

where denotes the 2D Fourier transform operator. Also, let *I _{i}* = |

$$\frac{{\stackrel{\sim}{I}}_{m}(u,v)}{{\mid {\stackrel{\sim}{G}}_{m}(0,0)\mid}^{2}{I}_{i}}=\delta (u,v)-2{\stackrel{\sim}{G}}_{m}^{a}(u,v)\stackrel{\sim}{A}(u,v)-2{\stackrel{\sim}{G}}_{m}^{p}(u,v)\stackrel{\sim}{\phi}(u,v),$$

(10)

where *δ*(*u, v*) is the 2D Dirac delta function,
${\stackrel{\sim}{G}}_{m}^{a}(u,v)$ is the amplitude transfer function (ATF):

$${\stackrel{\sim}{G}}_{m}^{a}(u,v)=\frac{1}{2}\left[\frac{{\stackrel{\sim}{G}}_{m}(u,v)}{{\stackrel{\sim}{G}}_{m}(0,0)}+\frac{{\stackrel{\sim}{G}}_{m}^{\ast}(-u,-v)}{{\stackrel{\sim}{G}}_{m}^{\ast}(0,0)}\right],$$

(11)

and ${\stackrel{\sim}{G}}_{m}^{p}(u,v)$ is the phase transfer function (PTF):

$${\stackrel{\sim}{G}}_{m}^{p}(u,v)=\frac{1}{2j}\left[\frac{{\stackrel{\sim}{G}}_{m}(u,v)}{{\stackrel{\sim}{G}}_{m}(0,0)}-\frac{{\stackrel{\sim}{G}}_{m}^{\ast}(-u,-v)}{{\stackrel{\sim}{G}}_{m}^{\ast}(0,0)}\right].$$

(12)

Here, * _{m}*(

Below we review some features of signal detection theory as applied to objective assessment of image quality. These concepts are applied to analyze ideal observer signal detection performance in phase-contrast imaging in Section 4.

There is growing consensus in the contemporary image science research community that image quality should be measured by the average performance of an observer on a specific diagnostic task [21–23]. The ability to quantify imaging system performance for specific tasks, such as the detection of low-contrast object features, can provide valuable guidance for the design and optimization of imaging systems. We refer the reader to the seminal paper by Wagner and Brown [24] or the book by Barrett and Myers [21] for comprehensive descriptions of the application of statistical decision theory to LSI imaging systems.

The first step in assessing objective measures of image quality is to specify a diagnostic task. In this work, we consider a simple binary detection task where the goal is to detect the presence of a low-contrast signal in a noise-contaminated image. We consider the SKE/BKE task in which a completely specified signal is to be detected against a completely specified background.

Consider a generic 2D LSI imaging system described as

$$\mathbf{g}(x,y)=f(x,y)\ast h(x,y)+\mathbf{n}(x,y),$$

(13)

where *f*(*x, y*) represents the input object, *h*(*x, y*) is the point spread function characterizing the system, **n**(*x, y*) is stochastic measurement noise, and **g**(*x, y*) is the measured image. Here and elsewhere, boldface fonts denote stochastic quantities. Note that, in the SKE/BKE detection task, *f*(*x, y*) and *h*(*x, y*) are known and non-random, so statistical fluctuations in **g**(*x, y*) are due only to the measurement noise **n**(*x, y*). In the binary detection task, the decision maker (i.e., observer) must decide whether the object *f*(*x, y*) or *f*(*x, y*) + Δ*f*(*x, y*) produced the observed image data **g**(*x, y*). Here, *f*(*x, y*) can be interpreted as the object background and Δ*f*(*x, y*) represents the signal of interest. This is equivalent to hypothesis testing where the observer must choose between the two hypotheses:

$${H}_{0}:\mathbf{g}(x,y)=f(x,y)\ast h(x,y)+\mathbf{n}(x,y)$$

(14a)

$${H}_{1}:\mathbf{g}(x,y)=(f(x,y)+\mathrm{\Delta}f(x,y))\ast h(x,y)+\mathbf{n}(x,y),$$

(14b)

where hypotheses *H*_{0} and *H*_{1} represent the signal absent and signal present cases, respectively.

The Bayesian ideal observer is a computational observer that makes optimal use of all available statistical information in a signal detection task such as the one described above. The ideal observer operates by computing a scalar likelihood ratio test statistic [43] as

$$\mathbf{t}(\mathbf{g})=\frac{pr(\mathbf{g}\mid {H}_{1})}{pr(\mathbf{g}\mid {H}_{0})},$$

(15)

where *pr*(**g**|*H _{n}*) is the probability density function that describes

$$\mathit{SNR}\equiv \frac{<\mathbf{t}{>}_{1}-<\mathbf{t}{>}_{0}}{\sqrt{{\scriptstyle \frac{1}{2}}{\sigma}_{1}^{2}+{\scriptstyle \frac{1}{2}}{\sigma}_{0}^{2}}},$$

(16)

where < **t** >* _{n}* denotes the conditional mean of

For 2D LSI imaging systems and statistically stationary measurement noise, the ideal observer SNR can be expressed in the 2D Fourier domain [24]. Let
$\stackrel{\sim}{\mathrm{\Delta}f}(u,v)$ denote the 2D FT of the signal Δ*f*(*x, y*). The squared ideal observer SNR can be computed as (*e.g.* see Appendix C in Ref. [31])

$${\mathit{SNR}}^{2}={\iint}_{\infty}\mathit{dudv}\mid \stackrel{\sim}{\mathrm{\Delta}f}{(u,v)}^{2}\mid \frac{{\text{MTF}}^{2}(u,v)}{\underset{\text{NEQ}(u,v)}{\underbrace{N(u,v)}}},$$

(17)

where MTF(*u, v*) = |{*h*(*x, y*)}| is the modulation transfer function of the LSI system and *N*(*u, v*) is the noise power spectrum. Note that, on the right-hand side of Eqn. (17), the quantity

$$\text{NEQ}(u,v)=\frac{{\text{MTF}}^{2}(u,v)}{N(u,v)}$$

(18)

is known as the noise equivalent quanta (NEQ) [44].

Equation (17) provides useful insights into the performance of an LSI imaging system as measured by the squared ideal observer SNR. The quantity NEQ(*u, v*) summarizes the frequency-dependent contribution of the imaging system hardware. The frequency-dependent contribution of the task is described solely by the quantity
$\mid \stackrel{\sim}{\mathrm{\Delta}f}(u,v)\mid $, which weights the corresponding value of NEQ(*u, v*) in determining the ideal observer SNR. In Section 4 the interplay between these contributing factors to detection performance is examined for phase-contrast imaging systems.

Within the Fourier optics framework described above, Hanson [45] investigated several higher-order detection or discrimination tasks that are related to edge detection. Ideal observer performance in these cases can be described by a generalization of Eqn. (17) of the form

$${\mathit{SNR}}_{h}^{2}={\iint}_{\infty}\mathit{dudv}{\mid \stackrel{\sim}{\mathrm{\Delta}f}(u,v)\mid}^{2}W(u,v)\text{NEQ}(u,v),$$

(19)

where *W*(*u, v*) is a real-valued weighting function that generally places greater emphasis on the mid- to high-frequency content of NEQ(*u, v*), and the subscript ‘*h*’ denotes that
${\mathit{SNR}}_{h}^{2}$ corresponds to a higher-order task.

Below the ideal observer SNR corresponding to a SKE/BKE detection task is investigated for phase-contrast imaging systems employing linear shift-invariant optical systems.

The object/signal model considered for the SKE/BKE detection task is shown in Fig. 2. A weak phase-amplitude signal is embedded in an absorbing semi-infinite background of thickness *L* that possesses no variations in its complex refractive index distribution perpendicular to the optical axis (*z*-axis). The X-ray refractive index distributions of the signal and background will be denoted as

$${n}_{s}(\overrightarrow{r})\equiv 1-{\mathrm{\Delta}}_{s}(\overrightarrow{r})+{j\beta}_{s}(\overrightarrow{r})$$

(20a)

and

$${n}_{b}(z)\equiv 1-{\mathrm{\Delta}}_{b}(z)+{j\beta}_{b}(z),$$

(20b)

respectively. When the signal is present, the object’s transmission function is given by Eqn. (4) with *A*(*x, y*) and (*x, y*) specified as

$$A(x,y)=\underset{{A}_{b}(x,y)}{\underbrace{k\underset{L}{\int}dz{\beta}_{b}(z)}}+\underset{{A}_{d}(\overrightarrow{r})}{\underbrace{k\underset{{t}_{s}(x,y)}{\int}dz({\beta}_{s}(\overrightarrow{r})-{\beta}_{b}(z))}}$$

(21a)

$$\phi (x,y)=-k\underset{L}{\int}dz{\mathrm{\Delta}}_{b}(z)\underset{{\phi}_{d}(\overrightarrow{r})}{\underbrace{-k\underset{{t}_{s}(x,y)}{\int}dz({\mathrm{\Delta}}_{s}(\overrightarrow{r})-{\mathrm{\Delta}}_{b}(z))}},$$

(21b)

where *t _{s}*(

Consider the difference signals

$${A}_{d}(x,y)\equiv k\underset{{t}_{s}(x,y)}{\int}dz({\beta}_{s}(\overrightarrow{r})-{\beta}_{b}(z))$$

(22a)

$$\phi (x,y)\equiv -k\underset{{t}_{s}(x,y)}{\int}dz({\mathrm{\Delta}}_{s}(\overrightarrow{r})-{\mathrm{\Delta}}_{b}(z)),$$

(22b)

which are assumed to be weak in the sense that *A _{d}*(

$${A}_{b}=k\underset{L}{\int}dz{\beta}_{b}(z).$$

(23)

It can be verified that the 2D FT of the ideal phase-contrast wavefield intensity (prior to degradation by the measurement process) in the signal present case satisfies

$${\stackrel{\sim}{I}}_{m,1}(u,v)={I}_{i}{\mid {\stackrel{\sim}{G}}_{m}(0,0)\mid}^{2}exp(-2{A}_{b})\left\{\delta (u,v)-2{\stackrel{\sim}{G}}_{m}^{a}(u,v){\stackrel{\sim}{A}}_{d}(u,v)-2{\stackrel{\sim}{G}}_{m}^{p}(u,v){\stackrel{\sim}{\phi}}_{d}(u,v)\right\},$$

(24)

where the subscript ‘1’ has been added to *I _{m,}*

For the signal absent case, the 2D FT of the ideal wavefield intensity is given by

$${\stackrel{\sim}{I}}_{m,0}(u,v)={I}_{i}{\mid {\stackrel{\sim}{G}}_{m}(0,0)\mid}^{2}exp(-2{A}_{b})\delta (u,v),$$

(25)

where the subscript ‘0’ has been added to *I _{m,}*

We will consider that the measurement process can be described by an incoherent 2D LSI imaging system as

$${\mathbf{g}}_{m}(x,y)={I}_{m}(x,y)\ast h(x,y)+\mathbf{n}(x,y),$$

(26)

where *I _{m}*(

In the binary SKE/BKE detection task under consideration, the decision maker must decide whether *I _{m}*(

$${H}_{0}:{\mathbf{g}}_{m}(x,y)={I}_{m,0}(x,y)\ast h(x,y)+\mathbf{n}(x,y)$$

(27a)

$${H}_{1}:{\mathbf{g}}_{m}(x,y)={I}_{m,1}(x,y)\ast h(x,y)+\mathbf{n}(x,y).$$

(27b)

Below, the ideal observer SNR for phase-contrast imaging is formulated for the detection task described in Section 4.B.

Let Δ*I _{m}*(

$$\stackrel{\sim}{\mathrm{\Delta}{I}_{m}}(u,v)=-2{I}_{i}{\mid {\stackrel{\sim}{G}}_{m}(0,0)\mid}^{2}exp(-2{A}_{b})\left\{{\stackrel{\sim}{G}}_{m}^{a}(u,v){\stackrel{\sim}{A}}_{d}(u,v)+{\stackrel{\sim}{G}}_{m}^{p}(u,v){\stackrel{\sim}{\phi}}_{d}(u,v)\right\}.$$

(28)

As described by Eqn. (17), the square of the ideal observer SNR for the SKE/BKE detection task under consideration is given by

$${\mathit{SNR}}^{2}={\iint}_{\infty}\mathit{dudv}{\mid \stackrel{\sim}{\mathrm{\Delta}{I}_{m}}(u,v)\mid}^{2}\text{NEQ}(u,v).$$

(29)

On substitution from Eqn. (28) into Eqn. (29), *SNR*^{2} can be expressed as

$${\mathit{SNR}}^{2}={\mathit{SNR}}_{A}^{2}+{\mathit{SNR}}_{\phi}^{2}+{\mathit{SNR}}_{cc}^{2},$$

(30)

where

$${\mathit{SNR}}_{A}^{2}=4{I}_{i}^{2}{\mid {\stackrel{\sim}{G}}_{m}(0,0)\mid}^{4}exp(-4{A}_{b}){\iint}_{\infty}\mathit{dudv}{\mid {\stackrel{\sim}{G}}_{m}^{a}(u,v){\stackrel{\sim}{A}}_{d}(u,v)\mid}^{2}\text{NEQ}(u,v),$$

(31a)

$${\mathit{SNR}}_{\phi}^{2}=4{I}_{i}^{2}{\mid {\stackrel{\sim}{G}}_{m}(0,0)\mid}^{4}exp(-4{A}_{b}){\iint}_{\infty}\mathit{dudv}{\mid {\stackrel{\sim}{G}}_{m}^{p}(u,v){\stackrel{\sim}{\phi}}_{d}(u,v)\mid}^{2}\text{NEQ}(u,v),$$

(31b)

and

$${\mathit{SNR}}_{cc}^{2}=8{I}_{i}^{2}{\mid {\stackrel{\sim}{G}}_{m}(0,0)\mid}^{4}exp(-4{A}_{b})\times {\iint}_{\infty}\mathit{dudv}Re\left\{\left({\stackrel{\sim}{G}}_{m}^{a}(u,v){\stackrel{\sim}{A}}_{d}(u,v)\right){\left({\stackrel{\sim}{G}}_{m}^{p}(u,v){\stackrel{\sim}{\phi}}_{d}(u,v)\right)}^{\ast}\right\}\text{NEQ}(u,v),$$

(31c)

where Re{·} denotes the real-valued component of a complex-valued number. The interpretation of the *SNR*^{2} in terms of these three components represents a key contribution of this work, and will be investigated in the remainder of the article.

The three contributions to *SNR*^{2} have distinct interpretations associated with the imaging physics and contrast mechanisms that provide insight into how phase- and absorption-contrast contribute to signal detection performance. To understand this, consider the intensity image *I _{m,}*

$${I}_{m,1}(x,y)={I}_{i}{\mid {\stackrel{\sim}{G}}_{m}(0,0)\mid}^{2}exp(-2{A}_{b})-{I}_{m}^{A}(u,v)-{I}_{m}^{P}(x,y),$$

(32)

where

$${I}_{m}^{A}(x,y)\equiv 2{I}_{i}{\mid {\stackrel{\sim}{G}}_{m}(0,0)\mid}^{2}exp(-2{A}_{b}){\mathcal{F}}_{2}^{-1}\{{\stackrel{\sim}{G}}_{m}^{a}(u,v){\stackrel{\sim}{A}}_{d}(u,v)\}$$

(33a)

and

$${I}_{m}^{P}(x,y)\equiv 2{I}_{i}{\mid {\stackrel{\sim}{G}}_{m}(0,0)\mid}^{2}exp(-2{A}_{b}){\mathcal{F}}_{2}^{-1}\{{\stackrel{\sim}{G}}_{m}^{p}(u,v){\stackrel{\sim}{\phi}}_{d}(u,v)\}.$$

(33b)

The first term on the right-hand side of Eqn. (32) represents the intensity of the incident wavefield that has been attenuated by the background, while the second two terms represent contributions that arise from absorption-contrast and phase-contrast associated with the signal. In terms of these quantities, each of the components in Eqn. (30) can be assigned a simple interpretation. Namely, ${\mathit{SNR}}_{A}^{2}$ and ${\mathit{SNR}}_{\phi}^{2}$ describe the ideal observer performance for the SKE/BKE tasks of detecting the single-contrast signals ${I}_{m}^{A}(x,y)$ and ${I}_{m}^{P}(x,y)$, respectively. Alternatively, the component ${\mathit{SNR}}_{cc}^{2}$ describes detection performance for a detection task involving a cross-correlation of the absorption- and phase-contrast signal components. Specifically, this task involves the use of ${I}_{m}^{P}(x,y)$ as a template that is cross-correlated with ${I}_{m}^{A}(x,y)$.

The quantities
${\mathit{SNR}}_{A}^{2}$ and
${\mathit{SNR}}_{\phi}^{2}$ can be alternatively interpreted in terms of the higher- order detection tasks. For example,
${\mathit{SNR}}_{A}^{2}$ corresponds to Eqn. (19) with
$\stackrel{\sim}{\mathrm{\Delta}f}(u,v)={\stackrel{\sim}{A}}_{d}(u,v)$ and
$W(u,v)=4{I}_{i}^{2}{\mid {\stackrel{\sim}{G}}_{m}(0,0)\mid}^{4}exp(-4{A}_{b}){\mid {\stackrel{\sim}{G}}_{m}^{a}(u,v)\mid}^{2}$. This reveals that
${\mathit{SNR}}_{A}^{2}$ can be interpreted as the ideal observer performance metric for a higher-order binary detection task involving the absorption signal *A _{d}*(

In the following two Sections, the components of *SNR*^{2} are examined for the cases of propagation-based and analyzer-based phase-contrast imaging.

Below, we apply the findings described in Section 4 to examine the squared ideal observer *SNR* for propagation-based phase-contrast imaging in the contrast transfer function formulation. For additional details regarding this method, we refer the reader to Refs. [42] and [48].

Assuming the object is irradiated by a monochromatic plane-wave, the impulse response *G _{m}*(

$${\stackrel{\sim}{G}}_{m}^{a}(u,v)=cos(\pi \lambda {z}_{m}({u}^{2}+{v}^{2}))$$

(34a)

and

$${\stackrel{\sim}{G}}_{m}^{p}(u,v)=-sin(\pi \lambda {z}_{m}({u}^{2}+{v}^{2})).$$

(34b)

In this case, the components of *SNR*^{2} in Eqn. (31) are given by

$${\mathit{SNR}}_{A}^{2}=4{I}_{i}^{2}exp(-4{A}_{b}){\iint}_{\infty}\mathit{dudv}{\mid {\stackrel{\sim}{A}}_{d}(u,v)\mid}^{2}{\text{cos}}^{\text{2}}(\pi \lambda {z}_{m}({u}^{2}+{v}^{2}))\text{NEQ}(u,v),$$

(35)

$${\mathit{SNR}}_{\phi}^{2}=4{I}_{i}^{2}exp(-4{A}_{b}){\iint}_{\infty}\mathit{dudv}{\mid {\stackrel{\sim}{\phi}}_{d}(u,v)\mid}^{2}{\text{sin}}^{\text{2}}(\pi \lambda {z}_{m}({u}^{2}+{v}^{2}))\text{NEQ}(u,v),$$

(36)

and

$${\mathit{SNR}}_{cc}^{2}=-8{I}_{i}^{2}exp(-4{A}_{b}){\iint}_{\infty}\mathit{dudv}Re\left\{{\stackrel{\sim}{A}}_{d}(u,v){\stackrel{\sim}{\phi}}_{d}^{\ast}(u,v)\right\}\times cos(\pi \lambda {z}_{m}({u}^{2}+{v}^{2}))sin(\pi \lambda {z}_{m}({u}^{2}+{v}^{2}))\text{NEQ}(u,v).$$

(37)

In the above expressions we have employed the fact that, for the Fresnel propagator, |* _{m}*(0, 0)| = |e

Simplified expressions that yield additional insights can be obtained when the near-field condition
$\pi \lambda ({u}_{c}^{2}+{v}_{c}^{2}){z}_{m}\ll 1$ is satisifed. Here, *u _{c}* and

$${\stackrel{\sim}{G}}_{m}^{a}(u,v)\approx 1$$

(38a)

and

$${\stackrel{\sim}{G}}_{m}^{p}(u,v)\approx -\pi \lambda {z}_{m}({u}^{2}+{v}^{2}),$$

(38b)

and therefore the difference signal in Eqn. (28) reduces to

$$\stackrel{\sim}{\mathrm{\Delta}{I}_{m}}(u,v)\approx -2{I}_{i}exp(-2{A}_{b})\left\{{\stackrel{\sim}{A}}_{d}(u,v)-\pi \lambda {z}_{m}({u}^{2}+{v}^{2}){\stackrel{\sim}{\phi}}_{d}(u,v)\right\}.$$

(39)

Accordingly, Eqns. (35)–(37) can be expressed as

$${\mathit{SNR}}_{A}^{2}=4{I}_{i}^{2}exp(-4{A}_{b}){\iint}_{\infty}\mathit{dudv}{\mid {\stackrel{\sim}{A}}_{d}(u,v)\mid}^{2}\text{NEQ}(u,v),$$

(40)

$${\mathit{SNR}}_{\phi}^{2}=4{I}_{i}^{2}exp(-4{A}_{b}){\pi}^{2}{\lambda}^{2}{z}_{m}^{2}{\iint}_{\infty}\mathit{dudv}{\mid {\stackrel{\sim}{\phi}}_{d}(u,v)\mid}^{2}\underset{{W}_{\phi}(u,v)}{\underbrace{{({u}^{2}+{v}^{2})}^{2}}}\text{NEQ}(u,v),$$

(41)

and

$${\mathit{SNR}}_{cc}^{2}=8{I}_{i}^{2}exp(-4{A}_{b})\pi \lambda {z}_{m}{\iint}_{\infty}\mathit{dudv}Re\left\{{\stackrel{\sim}{A}}_{d}(u,v){\stackrel{\sim}{\phi}}_{d}^{\ast}(u,v)\right\}\underset{{W}_{cc}(u,v)}{\underbrace{({u}^{2}+{v}^{2})}}\text{NEQ}(u,v).$$

(42)

The functions *W _{}*(

In the spatial domain, the difference signal in Eqn. (39) is given by

$$\mathrm{\Delta}{I}_{m}(x,y)\approx -2{I}_{i}exp(-2{A}_{b})\left\{{A}_{d}(x,y)+\frac{\lambda {z}_{m}}{4\pi}{\nabla}^{2}{\phi}_{d}(x,y)\right\},$$

(43)

where ^{2} denotes the 3D Laplacian operator. If we define

$${\mathit{neq}}^{{\scriptstyle \frac{1}{2}}}(x,y)\equiv {\mathcal{F}}_{2}^{-1}\left\{\frac{\text{MTF}(u,v)}{\sqrt{N(u,v)}}\right\},$$

(44)

Eqns. (40)–(42) can be expressed as

$${\mathit{SNR}}_{A}^{2}=4{I}_{i}^{2}exp(-4{A}_{b}){\iint}_{\infty}\mathit{dxdy}{\left|{A}_{d}(x,y)\ast {\mathit{neq}}^{{\scriptstyle \frac{1}{2}}}(x,y)\right|}^{2},$$

(45)

$${\mathit{SNR}}_{\phi}^{2}=\frac{{I}_{i}^{2}exp(-4{A}_{b}){\lambda}^{2}{z}_{m}^{2}}{4{\pi}^{2}}{\iint}_{\infty}\mathit{dxdy}{\left|{\nabla}^{2}{\phi}_{d}(x,y)\ast {\mathit{neq}}^{{\scriptstyle \frac{1}{2}}}(x,y)\right|}^{2},$$

(46)

and

$${\mathit{SNR}}_{cc}^{2}=\frac{2}{\pi}{I}_{i}^{2}exp(-4{A}_{b})\lambda {z}_{m}\times {\iint}_{\infty}\mathit{dudy}Re\left\{\left({A}_{d}(x,y)\ast {\mathit{neq}}^{{\scriptstyle \frac{1}{2}}}(x,y)\right){\left({\nabla}^{2}{\phi}_{d}(x,y)\ast {\mathit{neq}}^{{\scriptstyle \frac{1}{2}}}(x,y)\right)}^{\ast}\right\}.$$

(47)

Equation (40) shows that, for low-contrast signals and near-field imaging geometries,
${\mathit{SNR}}_{A}^{2}$ corresponds to the ideal observer *SNR*^{2} for a SKE/BKE detection task involving the absorption signal (*x, y*). However, the equations for
${\mathit{SNR}}_{\phi}^{2}$ and
${\mathit{SNR}}_{cc}^{2}$ (Eqns. (41) and (42) or Eqns. (46) and (47)) describe higher-order detection tasks that involve a preferential weighting of the higher frequency components of the NEQ as prescribed by *W _{}*(

For propagation-based imaging, the NEQ can be expressed as

$$\text{NEQ}(u,v)=\frac{{\text{MTF}}_{\mathit{det}}^{2}(u,v){\text{MTF}}_{\mathit{coh}}^{2}(u,v)}{N(u,v)},$$

(48)

where MTF* _{det}*(

Equations (35)–(37) were implemented numerically to gain quantitative insights into the relative contributions of phase- and absorption-contrast to the squared ideal observer *SNR*. The signal corresponded to a uniform rod of radius *R* = 20 *μ*m composed of breast tumor tissue embedded in a uniform layer of fat tissue of thickness 3 cm. The energy-dependent refractive index values *n* = 1 − (*E*) + *jβ*(*E*) of the tissues were computed from knowledge of the breast tissue composition and density as described in [49–51]. The rod was assumed to be of infinite extent with its long axis oriented perpendicular to the optical axis of the imaging system. The quantities *Ã _{d}*(

$${\stackrel{\sim}{A}}_{d}(u,v)=\pi k{R}^{2}\beta (E)[{J}_{2}(2\pi Ru)+{J}_{0}(2\pi Ru)]\delta (v),$$

(49a)

and

$${\stackrel{\sim}{\phi}}_{d}(u,v)=-\pi k{R}^{2}\mathrm{\Delta}(E)[{J}_{2}(2\pi Ru)+{J}_{0}(2\pi Ru)]\delta (v),$$

(49b)

where *J _{m}*(·) and

Figures 3 and and44 display the value of each *SNR*^{2} component and its fraction of the total *SNR*^{2} value, respectively, as a function of beam energy. In this case, the full width at half maximum (FWHM) of the system blur *h*(*x, y*) was fixed at 20 *μ*m. For energies below approximately 10 keV, the absorption-contrast contribution to *SNR*^{2} is larger than the phase-contrast contribution. However, for higher beam energies the phase-contrast contribution dominates. This behavior is expected due to the fact that, in the diagnostic X-ray energy regime, phase-contrast diminishes less rapidly than absorption-contrast.

A plot of the values of the *SNR*^{2} components as a function of beam energy in propagation-based imaging.

A plot of the relative values of the *SNR*^{2} components as a function of beam energy in propagation-based imaging.

Figure 5 displays the fraction of each *SNR*^{2} component of the total *SNR*^{2} value as a function of the FWHM of the system blur *h*(*x, y*) for an illuminating beam of energy 20 keV. For the particular object under consideration, the phase-contrast contribution to *SNR*^{2} is larger than the absorption-contrast contribution when the FWHM of the system blur is less than approximately 120.757 *μ*m. As expected, when the FWHM of the PSF increases the relative *SNR*^{2} contribution associated with phase-contrast decreases monotonically while the relative absorption-contrast contribution increases monotonically. Similarly, Eqns. (35)–(37) could be employed readily to quantify the *SNR*^{2} components for different objects and imaging geometries.

As a second example, we apply the findings described in Section 4 to examine the squared ideal observer *SNR* for analyzer-based X-ray phase-contrast imaging [3, 52]. We refer the reader to Refs. [53] and [54] for Fourier optics descriptions of analyzer-based X-ray phase-contrast imaging.

We assume that the wavefield incident on the object is planar and monochromatic and that the analyzer crystal is defect-free. In this case, the impulse response *G _{m}*(

$${\stackrel{\sim}{G}}_{m}^{a}(u)=\frac{1}{2}\left[\frac{{\stackrel{\sim}{G}}_{0}(u+{\omega}_{m})}{{\stackrel{\sim}{G}}_{0}({\omega}_{m})}+\frac{{\stackrel{\sim}{G}}_{0}^{\ast}({\omega}_{m}-u)}{{\stackrel{\sim}{G}}_{0}^{\ast}({\omega}_{m})}\right]$$

(50a)

and

$${\stackrel{\sim}{G}}_{m}^{p}(u)=\frac{j}{2}\left[\frac{{\stackrel{\sim}{G}}_{0}(u+{\omega}_{m})}{{\stackrel{\sim}{G}}_{0}({\omega}_{m})}-\frac{{\stackrel{\sim}{G}}_{0}^{\ast}({\omega}_{m}-u)}{{\stackrel{\sim}{G}}_{0}^{\ast}({\omega}_{m})}\right],$$

(50b)

where
${\omega}_{m}\equiv {\scriptstyle \frac{\delta {\theta}_{m}}{\lambda}}$. On substitution from Eqn. (50) into Eqn. (31) the components of *SNR*^{2} can be obtained.

Simplified expressions can be obtained if one assumes that *Ã _{d}*(

$${\stackrel{\sim}{G}}_{0}(u+{\omega}_{m})\approx {{\stackrel{\sim}{G}}_{0}({\omega}_{m})+u\frac{d{\stackrel{\sim}{G}}_{0}(u)}{du}|}_{u={\omega}_{m}}$$

(51a)

$${\stackrel{\sim}{G}}_{0}^{\ast}({\omega}_{m}-u)\approx {{\stackrel{\sim}{G}}_{0}^{\ast}({\omega}_{m})-u\frac{d{\stackrel{\sim}{G}}_{0}^{\ast}(u)}{du}|}_{u={\omega}_{m}}.$$

(51b)

Accordingly, the ATF and PTF reduce to

$${\stackrel{\sim}{G}}_{m}^{a}(u)\approx 1+{jb}_{m}u$$

(52a)

and

$${\stackrel{\sim}{G}}_{m}^{p}(u)\approx {j\frac{u}{2R({\omega}_{m})}\frac{dR(u)}{du}|}_{u={\omega}_{m}}.$$

(52b)

Here, *R*(*u*) |_{0}(*u*)|^{2} defines the rocking curve of the analyzer and

$${b}_{m}\equiv \frac{1}{R({\omega}_{m})}Im\left\{{{\stackrel{\sim}{G}}_{0}^{\ast}({\omega}_{m})\frac{d{\stackrel{\sim}{G}}_{0}^{\ast}(u)}{du}|}_{u={\omega}_{m}}\right\},$$

(53)

where Im{·} denotes the imaginary-valued component of a complex-valued number. The difference signal in Eqn. (28) can accordingly be expressed as

$$\stackrel{\sim}{\mathrm{\Delta}{I}_{m}}(u,v)\approx -2{I}_{i}R({\omega}_{m})exp(-2{A}_{b})\times \left\{{(1+{jb}_{m}u){\stackrel{\sim}{A}}_{d}(u,v)+j\frac{u}{2R({\omega}_{m})}\frac{dR(u)}{du}|}_{u={\omega}_{m}}{\stackrel{\sim}{\phi}}_{d}(u,v)\right\}$$

(54)

or, in the spatial domain,

$$\mathrm{\Delta}{I}_{m}(x,y)\approx -2{I}_{i}R({\omega}_{m})exp(-2{A}_{b})\left\{{(1+\frac{{b}_{m}}{2\pi}\frac{\partial}{\partial x}){A}_{d}(x,y)+\frac{1}{4\pi R({\omega}_{m})}\frac{dR(u)}{du}|}_{u={\omega}_{m}}\frac{\partial}{\partial x}{\phi}_{d}(x,y)\right\}.$$

(55)

Note that these expressions are similar to what would be obtained by use of a geometrical optics approximation [26,27,54].

On substitution from Eqns. (52a) and (52b) into Eqn. (31), it can be verified readily that the *SNR*^{2} components take on particularly simple forms:

$${\mathit{SNR}}_{A}^{2}=4{I}_{i}^{2}{R}^{2}({\omega}_{m})exp(-4{A}_{b}){\iint}_{\infty}\mathit{dudv}{\mid {\stackrel{\sim}{A}}_{d}(u,v)\mid}^{2}(1+{b}_{m}^{2}{u}^{2})\text{NEQ}(u,v),$$

(56)

$${\mathit{SNR}}_{\phi}^{2}={I}_{i}^{2}exp(-4{A}_{b})){\left(\frac{dR({\omega}_{m})}{du}\right)}^{2}{\iint}_{\infty}\mathit{dudv}{\mid {\stackrel{\sim}{\phi}}_{d}(u,v)\mid}^{2}{u}^{2}\text{NEQ}(u,v),$$

(57)

and

$${\mathit{SNR}}_{cc}^{2}={4{I}_{i}^{2}R({\omega}_{m})exp(-4{A}_{b})\frac{dR(u)}{du}|}_{u={\omega}_{m}}{\iint}_{\infty}\mathit{dudv}Im\left\{{\stackrel{\sim}{A}}_{d}(u,v){\stackrel{\sim}{\phi}}_{d}^{\ast}(u,v)(1+{jb}_{m}u)\right\}u\phantom{\rule{0.16667em}{0ex}}\text{NEQ}(u,v).$$

(58)

The *SNR*^{2} components in Eqns. (56)–(58) are expressed in the spatial domain as

$${\mathit{SNR}}_{A}^{2}=4{I}_{i}^{2}{R}^{2}({\omega}_{m})exp(-4{A}_{b}){\iint}_{\infty}\mathit{dxdy}{\left|\left(1+\frac{{b}_{m}}{2\pi}\frac{\partial}{\partial x}\right){A}_{d}(x,y)\ast {\mathit{neq}}^{{\scriptstyle \frac{1}{2}}}(x,y)\right|}^{2},$$

(59)

$${\mathit{SNR}}_{\phi}^{2}=\frac{{I}_{i}^{2}exp(-4{A}_{b})}{4{\pi}^{2}}{\left(\frac{dR({\omega}_{m})}{du}\right)}^{2}{\iint}_{\infty}\mathit{dxdy}{\left|\frac{\partial}{\partial x}{\phi}_{d}(x,y)\ast {\mathit{neq}}^{{\scriptstyle \frac{1}{2}}}(x,y)\right|}^{2},$$

(60)

and

$${\mathit{SNR}}_{cc}^{2}={\frac{2}{\pi}{I}_{i}^{2}R({\omega}_{m})exp(-4{A}_{b})\frac{dR(u)}{du}|}_{u={\omega}_{m}}\times {\iint}_{\infty}\mathit{dxdy}Re\left\{\left(\left(1+\frac{{b}_{m}}{2\pi}\frac{\partial}{\partial x}\right){A}_{d}(x,y)\ast {\mathit{neq}}^{{\scriptstyle \frac{1}{2}}}(x,y)\right){\left(\frac{\partial}{\partial x}{\phi}_{d}(x,y)\ast {\mathit{neq}}^{{\scriptstyle \frac{1}{2}}}(x,y)\right)}^{\ast}\right\}.$$

(61)

Equations (56)–(58) describe *SNR*^{2}s for higher-order detection tasks that prescribe a preferential weighting of the higher frequency components of the NEQ. The forms of these tasks, however, is distinct from those corresponding to the propagation-based method discussed in the previous section. Equations (56) and (59) reveal that
${\mathit{SNR}}_{A}^{2}$ is determined by *A _{d}*(

Numerical studies similar to those described in Section 5.C were conducted to demonstrate the relative contributions of phase- and absorption-contrast to the squared ideal observer *SNR* in analyzer-based phase-contrast imaging.

The transfer function _{0}(*u*) of the analyzer crystal was computed by use of Eqn. (13) in Ref. [56]. A silicon analyzer crystal using a [111] reflection was assumed. The ATF
${\stackrel{\sim}{G}}_{m}^{a}(u)$ and PTF
${\stackrel{\sim}{G}}_{m}^{p}(u)$ were determined by use of Eqn. (50) with
${\omega}_{m}\equiv {\scriptstyle \frac{\delta {\theta}_{m}}{\lambda}}$. The angular position *δθ _{m}* of the analyzer was set at the angular location where the magnitude of the rocking curve was one half of its maximum value. The energy-dependent angular Darwin widths of the crystal were calculated assuming no crystal absorption [57]. The object model described in Section 5.C was utilized, but the radius of the uniform rod was set at 1 mm. As before, the quantities

Figures 6 and and77 display the value of each *SNR*^{2} component and its fraction of the total *SNR*^{2} value, respectively, as a function of beam energy. In this case, the FWHM of *h*(*x, y*) was fixed at 2 *μ*m. For energies below approximately 21.31 keV, the absorption-contrast contribution to *SNR*^{2} is larger than the phase-contrast contribution. However, for higher beam energies the phase-contrast contribution dominates. Figure 8 displays the relative value of the *SNR*^{2} components as a function of the FWHM of *h*(*x, y*) for an illuminating beam energy of 20 keV. For the particular object under consideration, the phase-contrast contribution to *SNR*^{2} is larger than the absorption-contrast contribution for all FWHMs of the system blur in the considered range. This result is different than observed for the case of propagation-based imaging (Fig. 5), in which the phase-contrast contribution rapidly diminishes as the amount of system blur is increased. This is consistent with the fact that analyzer-based imaging does not rely on spatial coherence to realize phase-contrast.

A plot of the values of the *SNR*^{2} components as a function of beam energy in analyzer-based imaging.

A plot of the relative values of the *SNR*^{2} components as a function of beam energy in analyzer-based imaging.

The ideal observer *SNR* is a summary measure of optimal signal detection performance based on Bayesian decision theory and represents a meaningful metric for optimizing imaging hardware and data-acquisition protocols. In this work we have applied concepts from statistical detection theory to analyze the components of the ideal observer *SNR*^{2} in phase-contrast imaging for a SKE/BKE detection task involving a weak signal. This quantified the inherent interplay between the imaging physics and spatial resolution response of the imaging system that determines the relative contribution of phase- and absorption-contrast to ideal observer signal detectability. Propagation-based and analyzer-based X-ray phase-contrast imaging were considered as special cases of our analysis, and the relative magnitudes the squared *SNR* components under different physical conditions were numerically quantified. The theoretical framework we employed may also be useful for designing and optimizing new methods for phase-contrast imaging.

Our analysis assumed that the measured intensity data were continuous and therefore ignored the discrete nature of the data that would be recorded in a digital imaging system. Unless the detection system is film-based or the discrete detector elements are sufficiently small, sampling effects can influence signal detection measures. An interesting discussion of this within the context of conventional radiography can be found in Ref. [58]. Nevertheless, the continuous analysis in this paper based on Fourier optics provides fundamental insights into the interplay between the phase-contrast imaging physics and an imaging system’s pre-sampling PSF as it relates to signal detectability.

Because relatively few works [26, 27, 29, 30] have utilized task-based measures of image quality in studies of phase-contrast imaging, there remain many important avenues for future research. In this study, we considered a very simple detection task in which the background structure was assumed to be known. It will be interesting and important to investigate the effects of randomness in the background structure on signal detectability. Fully digital measures of signal detectability should also be systematically investigated. Additionally, the framework we presented can be applied to analyze and compare the relative contributions of absorption- and phase-contrast to signal detectability for a variety of existing and future phase-contrast imaging systems.

This work was supported in part by National Science Foundation (NSF) awards CBET-0546113 and CBET-0854430 and National Institutes of Health (NIH) award EB009715. AMZ was supported by NIH award CA136102. JGB was supported by NIH award HL091017.

1. Gabor D. A new microscopic principle. Nature. 1948;161(4098):777–778. [PubMed]

2. Nugent KA, Gureyev TE, Cookson D, Paganin D, Barnea Z. Quantitative phase imaging using hard x-rays. Physical Review Letters. 1996;77:2961–2964. [PubMed]

3. Ingal VN, Beliaevskaya EA. X-ray plane-wave topography observation of the phase contrast from a non-crystalline object. Journal of Physics D: Applied Physics. 1995;28:2314–2317.

4. Chapman D, Thomlinson W, Johnston R, Washburn D, Pisano E, Gmur N, Zhong Z, Menk R, Ardeli F, Sayers D. Diffraction enhanced x-ray imaging. Physics in Medicine and Biology. 1997;42:2015–2025. [PubMed]

5. Wernick MN, Wirjadi O, Chapman D, Zhong Z, Galatsanos NP, Yang Y, Brankov JG, Oltulu O, Anastasio MA, Muehleman C. Multiple-image radiography. Physics in Medicine and Biology. 2003;48(23):3875–3895. [PubMed]

6. Zernike F. Phase contrast, a new method for the microscopic observation of transparent objects. Physica. 1942;9:686–698.

7. Wilhein T, Kaulich B, Fabrizio ED, Romanato F, Cabrini S, Susini J. Differential interference contrast X-ray microscopy with submicron resolution. Applied Physics Letters. 2001;78:2082.

8. Tanaka T, Honda C, Matsuo S, Noma K, Ohara H, Nitta N, Ota S, Tsuchiya K, Sakashita Y, Yamada A, Yamasaki M, Furukawa A, Takahashi M, Murata K. The first trial of phase contrast imaging for digital full-field mammography using a practical molybdenum x-ray tube. Investigative Radiology. 2005;40:385–396. [PubMed]

9. Wu X, Liu H. Clinical implementation of x-ray phase-contrast imaging: Theoretical foundations and design considerations. Medical Physics. 2003;30:2169–2179. [PubMed]

10. Donnelly EF, Price RR, Pickens DR. Characterization of the phase-contrast radiography edge-enhancement effect in a cabinet x-ray system. Medical Physics. 2003;30:2292–2296. [PubMed]

11. Wilkins SW, Gureyev TE, Gao D, Pogany A, Stevenson AW. Phase-contrast imaging using polychromatic hard X-rays. Nature (London) 1996;384:335–338.

12. Snigirev A, Snigireva I, Kohn V, Kuznetsov S, Schelokov I. On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation. Review of Scientific Instruments. 1995;66:5486–5492.

13. Cloetens P, Barrett R, Baruchel J, Guigay JP, Schlenker M. Phase objects in synchrotron radiation hard x-ray imaging. Journal of Physics D. 1996;29:133–146.

14. Pfeiffer F, Weitkamp T, Bunk O, David C. Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources. Nature physics. 2006;2:258–261.

15. Kotre CJ, Birch IP, Robson KJ. Anomalous image quality phantom scores in magnification mammography: evidence of phase contrast enhancement. The British Journal of Radiology. 2002;75:170–173. [PubMed]

16. Paganin D, Barty A, Mcmahon PJ, Nugent KA. Quantitative phase-amplitude microscopy III. The effects of noise. Journal of Microscopy. 2003;214:51–61. [PubMed]

17. Arhatari BD, Mancuso AP, Peele AG, Nugent KA. Phase contrast radiography: Image modelling and optimization. Review of Scientific Instruments. 2004;75:5271–5276.

18. Wu X, Liu H, Yan A. Optimization of X-ray phase-contrast imaging based on in-line holography. Nuclear Instruments and Methods in Physics Research B. 2005;234:563–572.

19. Nesterets YI, Wilkins SW, Gureyev TE, Pogany A, Stevenson AW. On the optimization of experimental parameters for x-ray in-line phase-contrast imaging. Review of Scientific Instruments. 2005;76:093706.

20. Gureyev TE, Nesterets YI, Stevenson AW, Miller PR, Pogany A, Wilkins SW. Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging. Opt Express. 2008;16(5):3223–3241. [PubMed]

21. Barrett HH, Myers K. Wiley Series in Pure and Applied Optics. 2004. Foundations of Image Science.

22. Barrett HH. Objective assessment of image quality: effects of quantum noise and object variability. Journal of the Optical Society of America A. 1990;7:1266–1278. [PubMed]

23. Barrett HH, Yao J, Rolland JP, Myers KJ. Model observers for assessment of image quality. Proc Natl Acad Sci. 1993;90:9758–9765. [PubMed]

24. Wagner RF, Brown DG. Unified SNR analysis of medical imaging systems. Physics in Medicine and Biology. 1985;30(6):489–518.

25. Insana MF, Hall TJ. Visual detection efficiency in ultrasonic imaging: A framework for objective assessment of image quality. The Journal of the Acoustical Society of America. 1994;95:2081.

26. Brankov JG, Saiz-Herranz A, Wernick MN. Noise analysis for diffraction enhanced imaging. Proc IEEE/NIH Intl Symp Biomed Imaging. 2004;2:1428–1431.

27. Brankov JG, Saiz-Herranz AA, Wernick MN. Task-based evaluation of diffraction-enhanced imaging. 2005 IEEE Nuclear Science Symposium Conference Record; 2005. p. 3.

28. Majidi K, Brankov JG, Wernick MN. Sampling strategies in multiple-image radiography. Proceedings of the IEEE/NIH International Symposium on Biomedical Imaging.

29. Chou CY, Anastasio MA. Influence of imaging geometry on noise texture in quantitative in-line x-ray phase-contrast imaging. Optics Express. 2009;17(17):14 466–14 480. [PubMed]

30. Chou CY, Anastasio MA. Noise texture and signal detectability in propagation-based x-ray phase-contrast tomography. Medical Physics. 2010;37(1):270–281. [Online] [PubMed]

31. Medical imaging - the assessment of image quality, ICRU Report 54. International Commission on Radiation Units and Measurements; 7910 Woodmontt Ave., Bethesda, MD, 20814: 1996.

32. Paganin DM. Coherent X-Ray Optics. Oxford University Press; 2006.

33. Paganin DM, Gureyev TE. Phase-contrast, phase retrieval and aberration balancing in shift-invariant linear imaging systems. Optics Communications. 2008:965–981.

34. Paganin D, Gureyev TE, Pavlov KM, Lewis RA, Kitchen M. Phase retrieval using coherent imaging systems with linear transfer functions. Optics Communications. 2004;234(1–6):87–105.

35. Faulkner HML, Allen LJ, Oxley MP, Paganin D. Computational aberration determination and correction. Optics Communications. 2003;216(1–3):89–98.

36. Pavlov KM, Gureyev TE, Paganin D, Nesterets Y, Morgan MJ, Lewis RA. Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging. J Phys D Appl Phys. 2004;37:2746–2750.

37. Goodman JW. Introduction to Fourier Optics. 3. Roberts & Company Publishers; 2004.

38. Born M, Wolf E. Principles of Optics. Pergamon Press; New York: 1980.

39. Guigay JP. Fourier transform analysis of Fresnel diffraction patterns and in-line holograms. Opitk. 1977;49:121–125.

40. Shi D, Anastasio MA. Relationships between smooth-and small-phase conditions in X-ray phase-contrast imaging. IEEE Transactions on Medical Imaging. 2009;28(12):1969. [PubMed]

41. Cloetens P, Ludwig W, Boller E, Helfen L, Salvo L, Mache R, Schlenker M. Quantitative phase-contrast tomography using coherent synchrotron radiation. Developments in X-ray Tomography III, Proceedings of the SPIE. 2002;4503:82–91.

42. Gureyev TE, Myers GR, Nesterets YI, Paganin D, Pavlov KM, Wilkins SW. Stability and locality of amplitude and phase contrast tomographies. Proc of SPIE. 2006;6318:63180V.

43. Green DM, Swets JA. Signal detection theory and psychophysics. Wiley; New York: 1966.

44. Cunningham IA, Shaw R. Signal-to-noise optimization of medical imaging systems. Journal of the Optical Society of America A. 1999;16(3):621–632.

45. Hanson KM. Variations in task and the ideal observer. SPIE conference on application of optical instrumentation to medicine; 1983.

46. Pogany A, Gao D, Wilkins SW. Contrast and resolution in imaging with a microfocus x-ray source. Review of Scientific Instruments. 1997;68:2774–2782.

47. Nesterets Y, Gureyev T, Wilkins S. Polychromaticity in the combined PB/AB phase-contrast imaging. Journal of Physics D: Applied Physics. 2005;38:4259–4271.

48. Cloetens P, Ludwig W, Baruchel J, Dyck D, Landuyt J, Guigay JP, Schlenker M. Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays. Applied physics letters. 1999;75:29132.

49. Phelps M, Hoffman E, Ter-Pogossian MM. Attenuation coefficients of various body tissues, fluids, and lesions at photon energies of 18 to 136 KeV. Radiology. 1975;117:573–583. [PubMed]

50. Kalef-Ezra JA, Karantanas AH, Koligliatis T, Boziari A, Tsekeris P. Electron density of tissues and breast cancer radiotherapy: a quantitative ct study. International Journal of Radiation Oncology Biology Physics. 1998;41(5):1209–1214. [PubMed]

51. Ullman G, Sandborg M, Hunt R, Dance DR, Carlsson GA. Tech Rep. Department of Radiation Physics, Linkoping University; Dec, 2003. Implementation of pathologies in the monte carlo model chest and breast.

52. Lewis R. Medical phase contrast x-ray imaging: current status and future prospects. Physics in Medicine and Biology. 2004;49:3573–3583. [PubMed]

53. Nesterets Y, Gureyev T, Paganin D, Pavlov K, Wilkins S. Quantitative diffraction-enhanced x-ray imaging of weak objects. Journal of Physics D: Applied Physics. 2004;37(8):1262–1274.

54. Guigay J, Pagot E, Cloetens P. Fourier optics approach to x-ray analyser-based imaging. Optics Communications. 2007;270(2):180–188.

55. Gureyev TE, Wilkins SW. Regimes of X-ray phase-contrast imaging with perfect crystals. Nuovo Cimento D. 1997;19:545–552.

56. Guigay J, Pagot E, Cloetens P. Fourier optics approach to X-ray analyser-based imaging. Optics communications. 2007;270(2):180–188.

57. Als-Nielsen J, McMorrow D. Elements of Modern X-ray Physics. Wiley; Chichester: 2001.

58. Pineda AR, Barrett HH. What does DQE say about lesion detectability in digital radiography? Proc of SPIE. 2001;4320:561–569.

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