Gaussian Function Model for Task-Specific Evaluation in Medical Imaging: A Theoretical Investigation

Abstract

In medical image diagnosis, understanding image characteristics is crucial for selecting and optimizing imaging systems and advancing their development. Objective image quality assessments, based on specific diagnostic tasks, have become a standard in medical image analysis, bridging the gap between experimental observations and clinical applications. However, conventional task-based assessments often rely on ideal observer models that assume target signals have circular shapes with well-defined edges. This simplification rarely reflects the true complexity of lesion morphology, where edges exhibit variability. This study proposes a more practical approach by employing a Gaussian distribution to represent target signal shapes. This study explicitly derives the task function for Gaussian signals and evaluates the detectability index through simulations based on head computed tomography (CT) images with low-contrast lesions. Detectability indices were calculated for both circular and Gaussian signals using non-prewhitening and Hotelling observer models. The results demonstrate that Gaussian signals consistently exhibit lower detectability indices compared to circular signals, with differences becoming more pronounced for larger signal sizes. Simulated images closely resembling actual CT images confirm the validity of these calculations. These findings quantitatively clarify the influence of signal shape on detection performance, highlighting the limitations of conventional circular models. Thus, it provides a theoretical framework for task-based assessments in medical imaging, offering improved accuracy and clinical relevance for future advancements in the field.

Introduction

In medical image diagnosis, clarifying the physical characteristics of images is essential for selecting imaging systems, optimizing operational conditions, and driving further development [1–3]. Objective image quality assessments, particularly those based on specific diagnostic tasks, play a critical role in bridging the gap between diagnostic performance estimated by experimental observational studies, such as receiver operating characteristic analysis, and actual clinical applications [4–11].

Task-based assessments commonly focus on detection performance metrics derived from ideal observer models. This approach is also significant in applying newer technologies, including deep learning [12]. In the context of recent medical imaging, these assessments have contributed practically to selecting imaging systems tailored to specific clinical tasks and optimizing imaging and processing conditions across various modalities [13–17]. The established history and ongoing development of suitable model observers underscore the continued need for research in this crucial field of medical image diagnosis [3, 18–21].

These models often treat images as mathematical functions for computational analysis, assuming that the target signal is circular with well-defined edges. However, in medical imaging, particularly when dealing with internal structures such as tumors or calcifications, target signals rarely exhibit such ideal characteristics (forming a perfect circle on the image) [22–24]. Their edges often display variability and irregularity, making the Gaussian distribution a more realistic representation of target signal shapes [25, 26]. Despite its potential advantages, the Gaussian signal model has not been widely adopted in task-based assessments, and there is a lack of literature explicitly deriving and validating task functions for Gaussian signals, limiting the availability of approaches based on this more realistic model.

This study addresses this gap by focusing on image evaluation based on the task characteristics of model observers. It aims to explicitly derive the task function for Gaussian signals and theoretically calculate detection performance metrics to clarify the differences between the Gaussian and conventional circular signal models. By comparing these two models, we demonstrate the usefulness and validity of assuming Gaussian signals in task-specific assessments in medical image analysis. Specifically, this study makes the following contributions:

1. By applying Gaussian signals, it provides a novel theoretical framework for task-based evaluations in medical imaging.

2. It clarifies the limitations of conventional approaches based on circular signal models and addresses them by refining medical image processing techniques and evaluation criteria through the introduction of a more realistic Gaussian signal model.

3. This framework enables highly accurate image quality evaluations for clinical applications.

Materials and Methods

Measurement Theory of Detectability Index

In medical image analysis, model observers used to evaluate image quality and detection performance through computer-based simulation analysis. These mathematical models provide a theoretical framework or understanding observer behavior in signal detection tasks, eliminating or mitigating the need for time-consuming and potentially subjective human observer studies [4, 20]. Several types of model observers exist. Several types of model observers exist, each with different assumptions and applications. This study focuses on two commonly used models in medical imaging: the non-prewhitening (NPW) and the Hotelling observers [20]. The performance of these models is quantified by the detectability index $d\prime$, a metric that indicates how accurately a specific signal can be distinguished from background noise. It represents the statistical differences between the signal and background distributions, effectively describing the difficulty of the signal detection task.

The NPW observer model assumes that the observer responds without considering the spatial correlation of noise in the background. This assumption closely approximates human observer behavior in signal detection, making the NPW observer suitable for representing human performance characteristics. The AAPM TG233 guideline recommends using this model for measuring the detectability index $\left({d\prime }_{\mathit{NPW}}\right)$ [3].

In contrast, the Hotelling observer model assumes that the observer accounts for the spatial correlation of the noise, potentially leading to higher accuracy. As an ideal observer that utilizes all statistical information within the image, Hotelling observer provides a theoretical upper limit of detection performance. Therefore, the detectability index $\left({d\prime }_H\right)$ based on the Hotelling observer is always higher than that of the NPW observer. This model is particularly useful for comparing imaging system performance or guiding system development.

The detectability index $\left(d,\prime \right)$ represents the separability of the probability density distributions of responses in the presence and absence of a signal. Although it is typically defined using signal vectors and covariance matrices [20], $d\prime$ can be calculated more efficiently by treating images as functions, avoiding complex matrix calculations. This approach utilizes the following image characteristics: the signal characteristics, determined by the spatial frequency characteristics of the signal $W\left(u,,,v\right)$ and the imaging system’s point spread function, represented by the modulation transfer function (MTF),$MTF\left(u,,,v\right)$, and the noise characteristics, expressed by the noise power spectrum (NPS),$NPS\left(u,,,v\right)$, which is the Fourier transform of the autocorrelation function. Based on these properties, the detectability index $\left(d,\prime \right)$ for each model is calculated as follows [4, 20].

$${d\prime }_{\mathit{NPW}}=\frac{\int \int {\left|W,\left(u,,,v\right)\right|}^2{\mathit{MTF}}^2\left(u,,,v\right)dudv}{\sqrt{\int \int {\left|W,\left(u,,,v\right)\right|}^2{\mathit{MTF}}^2\left(u,,,v\right)NPS\left(u,,,v\right)dudv}}$$ 1

$${d\prime }_H=\sqrt{\int \int \frac{{\left|W,\left(u,,,v\right)\right|}^2{\mathit{MTF}}^2\left(u,,,v\right)}{NPS\left(u,,,v\right)}dudv}$$ 2

where $W\left(u,,,v\right)$ is referred to as the task function because it relates to the specific characteristics of the detection task. In medical image analysis, it is common to assume that the target be detected has a circular shape with well-defined edges.

Derivation of Task Functions

Circular Signal with Well-Defined Edges

First, the derivation process of the task function for a circular signal, commonly used as a target in medical image analysis, is explained. The 2D Fourier transform $F\left(u,,,v\right)$ of a 2D signal $f\left(x,,,y\right)$ is expressed as follows:

$$F\left(u,,,v\right)=\int {\int }_{-\infty }^{\infty }f\left(x,,,y\right)e^{-j2\pi \left(u,x,+,v,y\right)}dxdy$$ 3

When converting Cartesian coordinates $\left(x,,,y\right)$ to polar coordinates $\left(r,,,\theta \right)$, $x=r\text{cos}\theta$ and $y=r\text{sin}\theta$. Here, $r$ is the distance from the origin in the spatial domain, $r=\sqrt{x^2+y^2}$, and the angle $\theta ={\text{tan}}^{-1}\left(\frac{y}{x}\right)$. To consider the integral in polar coordinates, we convert the infinitesimal area $\mathit{dxdy}$ in the Cartesian to the polar coordinates as follows:

$$dxdy=rdrd\theta$$ 4

In the spatial frequency domain, when converting from Cartesian to polar coordinates, we have $u=\rho \text{cos}\varphi$ and $v=\rho \text{sin}\varphi$. Here, $\rho$ is the distance from the origin in the spatial frequency domain, $\rho =\sqrt{u^2+v^2}$, and the $\varphi ={\text{tan}}^{-1}\left(\frac{v}{u}\right)$. From the above, the Fourier transform $F\left(\rho ,,,\theta \right)$ of $f\left(x,,,y\right)$ in polar coordinates is expressed as follows:

$$F\left(\rho ,,,\theta \right)={\int }_0^{2\pi }{\int }_0^{\infty }f\left(r,,,\theta \right)e^{-j2\pi \rho r\text{cos}\left(\theta ,-,\varphi \right)}rdrd\theta$$ 5

where $f\left(r,,,\theta \right)$ is a circular signal with diameter $D$ (radius $R=\frac{D}{2}$) and contrast (signal intensity) $C$,

$$f\left(r,,,\theta \right)=\left\{\begin{array}{c}C\left(0,\le ,r,\le ,R\right)\\ 0\left(r,>,R\right)\end{array}\right)$$ 6

By substituting Eq. (6) into Eq. (3), we obtain the following:

$$F\left(\rho ,,,\theta \right)=C{\int }_0^{R}r\left\{{\int }_0^{2\pi },e^{-j2\pi \rho r\text{cos}\left(\theta ,-,\varphi \right)},d,\theta \right\}dr$$ 7

Here, the zeroth-order Bessel function of the first kind is defined as follows:

$$J_0\left(a\right)=\frac{1}{2\pi }{\int }_0^{2\pi }{e}^{-ja\text{cos}\left(\theta ,-,\varphi \right)}d\theta$$ 8

If $J_0\left(2,\pi ,\rho ,r\right)$, then

$$F\left(\rho \right)=2\pi C{\int }_0^{R}r{J}_0\left(2,\pi ,\rho ,r\right)dr$$ 9

Setting $2\pi \rho r=r^{\prime }$, gives $r=\frac{r^{\prime }}{2\pi \rho }$, which can be expressed as follows:

$$F\left(\rho \right)=2\pi C{\int }_0^{2\pi R\rho }\frac{r^{\prime }}{2\pi \rho }{J}_0\left(r^{\prime }\right)d\frac{r^{\prime }}{2\pi \rho }=\frac{C}{2\pi {\rho }^2}{\int }_0^{2\pi R\rho }{r}^{\prime }{J}_0\left(r^{\prime }\right)d{r}^{\prime }$$ 10

The integral theorem of Bessel functions gives the following relationship:

$${\int }_0^{2\pi R\rho }{r}^{\prime }{J}_0\left(r^{\prime }\right)d{r}^{\prime }=2\pi R\rho \bullet J_1\left(2,\pi ,R,\rho \right)$$ 11

By substituting this into Eq. (10), we obtain the following:

$$F\left(\rho \right)=\frac{C}{2\pi {\rho }^2}{\int }_0^{2\pi R\rho }{r}^{\prime }{J}_0\left(r^{\prime }\right)d{r}^{\prime }=\frac{C}{2\pi {\rho }^2}\bullet 2\pi R\rho \bullet J_1\left(2,\pi ,R,\rho \right)=\frac{\mathit{CR}}{\rho }\bullet J_1\left(2,\pi ,R,\rho \right)=\frac{\mathit{CD}}{2\rho }\bullet J_1\left(\pi ,D,\rho \right)$$ 12

This is the task function for a circular signal with diameter $D$ and contrast $C$. However, if we consider the condition of $\rho =0$, the denominator of Eq. (12) becomes 0, making it impossible to calculate directly. Owing to the nature of the Fourier transform, $\rho =0$ corresponds to the direct current component. Therefore, it can be calculated as the product of the area and contrast of the circular signal as follows:

$$F\left(0\right)={\left(\frac{D}{2}\right)}^2\bullet \pi \bullet C=\frac{C\pi D^2}{4}$$ 13

Gaussian Signal

While it is common to assume a circular shape for the detection target, anatomical structures within the human body rarely exhibit such clearly defined edges. To address this, it is more appropriate to consider a task function with smoother edges, such as a Gaussian function. This section derives the task function based on this more realistic assumption.

We consider a 2D signal $f\left(x,,,y\right)$ that is a circular Gaussian signal, characterized by a maximum contrast $C$, full width at half maximum (FWHM) $d$, and standard deviation (SD) $\sigma$. To distinguish it from circular signals, $f\left(x,,,y\right)$ is expressed as $g\left(x,,,y\right)$. This Gaussian signal is defined as follows:

$$g\left(x,,,y\right)=C\bullet e^{\left(-,\frac{x^2+y^2}{2{\sigma }^2}\right)}$$ 14

The relationship between $d$ and $\sigma$ is expressed as follows:

$$d=2\sqrt{2\text{ln}2}\bullet \sigma$$ 15

When $g\left(x,,,y\right)$ is expressed in polar coordinates, we obtain the following:

$$g\left(r\right)=C\bullet e^{\left(-,\frac{r^2}{2{\sigma }^2}\right)}$$ 16

The 2D Fourier transform $G\left(\rho \right)$ of $g\left(r\right)$ is as follows:

$$G\left(\rho \right)={\int }_0^{2\pi }{\int }_0^{\infty }g\left(r\right)e^{-j2\pi \rho r\text{cos}\left(\theta ,-,\varphi \right)}rdrd\theta$$ 17

Because $g\left(r\right)$ is a circularly symmetric signal, $G\left(\rho \right)$ can be expressed using Bessel functions as follows:

$$G\left(\rho \right)=2\pi {\int }_0^{\infty }g\left(r\right)J_0\left(2,\pi ,\rho ,r\right)rdr$$ 18

Substituting $g\left(r\right)$ in this equation, we obtain the following:

$$G\left(\rho \right)=2\pi {\int }_0^{\infty }C\bullet e^{\left(-,\frac{r^2}{2{\sigma }^2}\right)}\bullet J_0\left(2,\pi ,\rho ,r\right)rdr$$ 19

Owing to the relationship between the particular solution of the Gaussian integral and the integral of the Bessel function, we obtain the following:

$${\int }_0^{\infty }{e}^{\left(-,a,r^2\right)}\bullet J_0\left(b,r\right)rdr=\frac{1}{2a}\bullet e^{\left(-,\frac{b^2}{4a}\right)}$$ 20

By matching this result with that obtained from Eq. (19), we obtain $a=\frac{1}{2{\sigma }^2}$ and $b=2\pi \rho$, which are ${\sigma }^2=\frac{1}{2a}$ and $\frac{b^2}{4a}=2{\pi }^2{\rho }^2{\sigma }^2$. Substituting these variables into the right-hand side of Eq. (20) gives

$$G\left(\rho \right)=2\pi C{\sigma }^2\bullet e^{\left(-,2,{\pi }^2,{\rho }^2,{\sigma }^2\right)}$$ 21

To express this in terms of the FWHM of a circular Gaussian signal, which can be directly measured from the image pixel values, we use the relationship in Eq. (15) as follows:

$$G\left(\rho \right)=2\pi C\bullet \frac{d^2}{8\text{ln}2}\bullet e^{\left(-,2,{\pi }^2,\bullet ,{\rho }^2,\bullet ,\frac{d^2}{8\text{ln}2}\right)}=\frac{\pi C{d}^2}{4\text{ln}2}\bullet e^{\left(-,\frac{{\pi }^2\bullet {\rho }^2\bullet d^2}{4\text{ln}2}\right)}$$ 22

This represents the task function for the circular Gaussian signal. It is important to consider the case where $\rho =0$, then $e^{\left(0\right)}=1$,

$$G\left(0\right)=\frac{\pi C{d}^2}{4\text{ln}2}$$ 23

Derivation of Detectability Index

When the target is a circularly symmetric signal, the calculations for the detectability index can be simplified by expressing the task function $W\left(u,,,v\right)$, $MTF\left(u,,,v\right)$, and $NPS\left(u,,,v\right)$ in polar coordinates. That is, by transforming from Cartesian to polar coordinates, the detectability index $d\prime$ in Eqs. (1) and (2) can be calculated as follows:

$${d\prime }_{\mathit{NPW}}=\frac{\int \int {\left|W,\left(u,,,v\right)\right|}^2{\mathit{MTF}}^2\left(u,,,v\right)dudv}{\sqrt{\int \int {\left|W,\left(u,,,v\right)\right|}^2{\mathit{MTF}}^2\left(u,,,v\right)NPS\left(u,,,v\right)dudv}}=\frac{2\pi \int {\left|W,\left(\rho \right)\right|}^2{\mathit{MTF}}^2\left(\rho \right)\rho d\rho }{\sqrt{2\pi \int {\left|W,\left(\rho \right)\right|}^2{\mathit{MTF}}^2\left(\rho \right)NPS\left(\rho \right)\rho d\rho }}$$ 24

$${d\prime }_H=\sqrt{\int \int \frac{{\left|W,\left(u,,,v\right)\right|}^2{\mathit{MTF}}^2\left(u,,,v\right)}{NPS\left(u,,,v\right)}dudv}=\sqrt{2\pi \int \frac{{\left|W,\left(\rho \right)\right|}^2{\mathit{MTF}}^2\left(\rho \right)}{NPS\left(\rho \right)}\rho d\rho }$$ 25

This study investigates the differences in detectability index when applying Gaussian signals to the task functions compared to when circular signals are assumed.

Calculation Conditions of Detectability Index

To calculate the detectability index $\left(d^{\prime }\right)$, we conducted simulations based on head computed tomography (CT) images, based on following the task-based assessment approach recommended in AAPM TG233 [3]. The MTF and NPS of the CT images were determined based on the image quality of routine head CT examinations reported in previous studies [27]. Specifically, the NPS was obtained by analyzing images acquired under various imaging conditions at different facilities, and the average value was used in this study. Figure 1 shows the graphs of the MTF and NPS used in the calculations.

Fig. 1.

MTF (a) and NPS (b) used to calculate the detectability index. The solid line in the NPS graph represents the modeled NPS value, the circles represent the multicenter average NPS for routine head CT examinations used to determine the parameters, and the crosses represent the NPS for the specific system with characteristics closest to the average

Calculating the detectability index requires quantitative values of MTF and NPS at each spatial frequency. The MTF was modeled using the following Gaussian function-based formula [28], and the MTF value at each frequency was calculated. In this explanation, the spatial frequency is expressed in polar coordinates as $\rho$ [cycles/mm] because the detection target is a circularly symmetric signal.

$$MTF\left(\rho \right)=\text{exp}\left(-,\frac{{\rho }^2}{2{\sigma }^2}\right)$$ 26

$\sigma$ is a parameter that determines the shape of the MTF. It is determined by the frequency ${\rho }_{50}$ at which the MTF value becomes 50%, as follows:

$$\sigma =\frac{{\rho }_{50}}{\sqrt{2\text{ln}2}}$$ 27

In this case, we used ${\rho }_{50}=0.3$ cycles/mm for the calculation [29], resulting in $\sigma =0.2548$.

Next, the NPS was modeled using the following equation [30], and the NPS value at each frequency was calculated:

$$NPS\left(\rho \right)=a\bullet \rho \bullet {\left[\text{exp},\left(-,b,\bullet ,{\rho }^c\right)\right]}^2$$ 28

where $a,b$, and $c$ are parameters that determine the shape of the NPS. These are determined by the NPS peak frequency $\left({\rho }_{\text{peak}}\right)$ and the NPS peak value $\left({\mathit{NPS}}_{\text{peak}}\right)$ as follows:

$$a=\frac{{\mathit{NPS}}_{\text{peak}}}{{\rho }_{\text{peak}}\bullet {\left[\text{exp},\left(-,b,\bullet ,{{\rho }_{\text{peak}}}^c\right)\right]}^2}$$ 29

$$b=\frac{1}{2\bullet c\bullet {{\rho }_{\text{peak}}}^c}$$ 30

We used the average NPS shown in Fig. 1 as a reference and set ${\rho }_{\text{peak}}=0.23$ cycles/mm, ${\mathit{NPS}}_{\text{peak}}=11$ HU² mm², and $c=1.8$, obtaining $a=83.36$ and $b=3.91$.

The diameter of the circular signal for the detection task was set to four different values: 20, 10, 5, and 3 mm. The Gaussian signal was adjusted such that the FWHM was approximately equivalent to the diameter of the corresponding circular signal. The SD of the Gaussian distribution was set to 7.5, 4.0, 2.0, and 1.25. Figure 2 shows the noise-free images of each evaluated signal. The maximum contrast for each signal was set to 5 and 10 HU, simulating low-contrast signals representative of early stroke and microbleeds in head CT images [31, 32].

Fig. 2.

Signal images of different sizes in a noise-free condition. For each size, the left-hand side shows a circular signal with clear edges, whereas the right-hand side shows a Gaussian signal with unclear edge

Validity of the Image Simulation

The validity of the calculated detectability indices, derived under the imaging conditions described in the previous section, was verified using simulated images. This image simulation was performed using a Python program. Figure 3 illustrates an overview of the procedure.

Fig. 3.

Flowchart of the image-generation simulation process

First, circular and Gaussian signals with specified sizes and contrasts were generated on a 100 × 100 pixel image space with a pixel size of 0.5 mm, as shown in Fig. 2. Next, to simulate the blurring inherent in the imaging process, the signal image was convolved with the MTF shown in Fig. 1a. This was achieved by performing a 2D Fourier transform on the signal image, multiplying it by a 2D map of the MTF (2D-MTF filter) in the spatial frequency domain, and then applying an inverse Fourier transform to obtain the blurred signal image.

To reproduce the noise characteristics of CT images, a noise simulation image was generated. Initially, a white Gaussian noise image (Base noise image) with a mean value of 0 and a SD of 2 was created in a 100 × 100 matrix. The SD was set 2 to normalize the average power spectrum to 1, simplifying subsequent calculations with the NPS filter. A 2D Fourier transform was applied to this Base noise image. To assign the desired NPS characteristics, a 2D map of the NPS (2D-NPS filter) in the spatial frequency domain was created and multiplied by the frequency spectrum of the Base noise image. An inverse Fourier transform was then performed to generate a noise-simulated image (noise image with arbitrary NPS) that reproduced the characteristics of a CT image. Finally, the blurred signal and the noise-simulated images were added to obtain the final simulated image (simulated image). The visibility of the simulated images for each signal was qualitatively evaluated, confirming that the calculated detectability indices were appropriately computed.

To further validate the simulation, the simulated images were compared with actual CT images acquired using the Aquilion ONE Vision Edition (Canon Medical Systems, Tochigi, Japan). The imaging target comprises circular rods with diameters of 20, 10, and 5 mm, with a contrast difference of 5 HU, contained within a low-contrast resolution evaluation module in a quality control phantom. The reconstruction algorithm used was filtered back projection with the FC21 kernel. This particular image was selected because its noise characteristics were closest to the average noise characteristics obtained from images analyzed across various devices and processing algorithms from multiple institutions [27]. The NPS characteristics of this image are indicated by cross markers in Fig. 1b.

Results

Validity of the Image Simulation

Figure 4 compares images obtained from an actual CT system with those generated through the simulation. Both images are 100 × 100 matrices. To facilitate comparison with the phantom’s low-contrast circular rods, the simulated image displays circular signals matching the size of the rods. A comparison of the images, considering factors such as signal contrast, edge representation, and noise texture, reveals that the simulated image closely resembles the actual CT image in terms of overall visibility. This confirms that the computational conditions used in this study appropriately reproduce the characteristics of images obtained in clinical practice.

Fig. 4.

Comparisons of actual CT and simulated images

Evaluation of Detectability Indices for Different Signal Shapes

Figure 5 presents the task functions for different signal sizes and contrasts. For signals of the same shape, higher contrast corresponds to higher task function values. Furthermore, a comparison between circular and Gaussian signals reveals that, even with the same signal size, the task function values differ, with Gaussian signals exhibiting lower values.

Fig. 5.

Task functions for different signal sizes and contrasts. The task function for a Gaussian signal is shown in red within the right-hand side images. The solid line represents a contrast of 10 HU, whereas the dotted line represents a contrast of 5 HU

Table 1 summarizes the detectability indices calculated for circular and Gaussian signals along with their relative differences (RDs). For ${d\prime }_{\mathit{NPW}}$ in the NPW observer model, the values vary depending on the signal size, with Gaussian signals showing 33–40% lower values than circular signals. This difference increases with increasing signal size. For ${d\prime }_H$ in the Hotelling observer model, the values for Gaussian signals are approximately 33–44% lower. This difference is more pronounced than that observed for ${d\prime }_{\mathit{NPW}}$, particularly for larger signal sizes. To verify the correspondence between the detectability index, as an objective quantitative measure, and the visibility of signal images, Fig. 6 shows simulation images under each condition.

Table 1.

Detectability indices for circular and Gaussian signals of varying sizes and their RDs. The RD is calculated as the percentage difference between the $d\prime$ of the Gaussian signal that of the circular signal, is with the latter taken as the reference

Signal size [mm]	Contrast [HU]	Detectability index: $\mathit{d}_{}^{\prime }$
		$\mathit{d}_{}^{\mathbf{\prime }}_{\mathit{N}\mathit{P}\mathit{W}}^{}$			$\mathit{d}_{}^{\mathbf{\prime }}_{\mathit{H}}^{}$
		Circular	Gaussian	RD [%]	Circular	Gaussian	RD [%]
20	10	98.34	58.59	− 40.42	112.39	62.58	− 44.32
	5	49.17	29.30	− 40.41	56.19	31.29	− 44.31
10	10	39.32	24.66	− 37.28	45.03	27.74	− 38.40
	5	19.66	12.33	− 37.28	22.52	13.87	− 38.41
5	10	14.71	9.29	− 36.85	16.57	10.54	− 36.39
	5	7.35	4.65	− 36.74	8.29	5.27	− 36.43
3	10	6.91	4.60	− 33.43	7.63	5.11	− 33.03
	5	3.46	2.30	− 33.53	3.81	2.55	− 33.07

Fig. 6.

Simulation images obtained under each condition

The changes in the detectability index are consistent with those in visibility, confirming that the computed detectability index aligns with visual perception. These findings support the validity of the detectability index calculation results obtained in this study.

Discussion

This study proposed a novel theoretical framework that introduces a more realistic Gaussian signal model for task-specific assessments in medical image analysis. We explicitly derived the task function for Gaussian signals and compared it to that of conventional circular signals through theoretical simulations. The results demonstrated a consistent decrease in the detectability index for Gaussian signals compared to circular signals, identified the magnitude of these differences, and revealed the influence of signal size on these differences.

In a validation test to assess the appropriateness of the calculation conditions for the detectability index, we compared actual CT images with simulated images. The MTF and NPS for the simulations were based on physical data reported in previous studies [27]. The close resemblance between the actual and simulated images suggests that the image characteristics of the actual conditions were accurately reproduced. This not only indicates that the modeling procedures for MTF and NPS were appropriate but also supports the notion that the calculation conditions employed in this study reflect actual clinical imaging scenarios. Because the primary objective of this study was to elucidate the fundamental differences between circular and Gaussian signals, we did not investigate the effects of varying physical conditions related to the resolution and noise characteristics of specific CT scanners or reconstruction kernels. However, the simulation methodology employed is expected to be broadly applicable across diverse imaging conditions. Furthermore, this approach can be adapted by remeasuring the relevant image characteristics under different conditions [5, 7, 9]. Importantly, the agreement between the appearance of the simulated images and the calculated detectability indices demonstrates the accuracy of the verification process and reaffirms the usefulness of the detectability index as a robust evaluative measure.

Our results clearly demonstrated the influence of signal shape, size, and contrast on the task function and detectability index. In particular, the consistently lower detectability index observed for Gaussian signals, compared to circular signals, objectively validated that signal shape directly impacts detection performance. This finding indicates a potential difference of approximately 30–40% when estimating observer performance using a detectability index based on a model observer. This discrepancy is likely one of the factors contributing to the difference between commonly known physical metrics and subjective assessments. Additionally, the observed differences between circular and Gaussian signals became more pronounced as signal size increased. While larger signals generally improve visibility, our results underscore the necessity of considering signal edge shape when evaluating detection performance, especially for low-contrast signals [33]. This insight may help bridge the gap that can arise between clinical image assessments and evaluations using phantom images.

Furthermore, this study revealed several important advantages of using the Gaussian signal model. First, the physical width (FWHM) of Gaussian signals can be clearly defined in terms of pixel values and directly measured. This characteristic improves the reproducibility of signal dimension analysis and reduces measurement errors. Second, the shape of the Gaussian signal is less affected by the MTF, which characterizes the frequency response of the imaging system. Therefore, Gaussian signal characteristics are less susceptible to optical factors such as blurring, compared to circular signals with sharp edges, minimizing errors in analysis outcomes and ensuring image evaluation accuracy. These properties of the Gaussian signal are particularly useful in clinical tasks. The shape of Gaussian signals and their analytical simplicity offer a robust approach for conducting efficient and objective image quality evaluations in clinical imaging. Consequently, Gaussian signal-based simulations can address specific challenges encountered in clinical environments, providing a theoretical foundation for enhancing diagnostic image quality and accuracy.

Conventional task-based evaluations typically assess known signals against a uniform background using quality control phantoms. Although calculation models using circular signals with well-defined edges are suitable these idealized conditions, they would struggle with the complexity of clinical images, which feature non-uniform backgrounds and potentially distorted signals. An alternative approach involves calculating detectability indices from feature vector distributions; however, this requires a substantial number of images [34, 35]. Moreover, because clinical lesion detection is specific to individual patients and imaging conditions, feature vector evaluations can be impractical for certain applications. In contrast, a computational approach using Gaussian modeling offers the potential to address these challenges on a case-by-case basis. Specifically, by analyzing the signal intensity profile of a lesion, potentially using a Gaussian mixture model, the necessary Gaussian parameters can be estimated [36, 37]. This allows for identifying the variables in Eq. (22), deriving task functions for individual lesions, and ultimately leading the computation of detectability indices for specific clinical images. Although detailed validation of such an approach was beyond the scope of this study, evaluating detectability in clinical settings—particularly quantifying the effects of signal and background non-uniformity—remains a critical challenge. Addressing these issues represents an important direction for future research.

This study quantitatively clarified the impact of signal shape on detection performance and provides guidance for evaluation processes under various imaging conditions. However, because the simulation conditions were limited to specific devices and parameters, further investigations are required to assess the reproducibility and versatility of the results under different conditions. Research on task-specific observer models now also encompasses nuclear medicine and ultrasound imaging [12, 13, 15, 21, 38]. This trend suggests that the proposed Gaussian signal model approach holds potential for enhancing generalization across a broader range of imaging modalities and conditions. Although this study focused exclusively on circular signals in a uniform noise background, further validation should consider diverse signal shapes and sizes within complex anatomical backgrounds, similar to those encountered in actual diagnostic scenarios.

Additionally, the accuracy of the detectability index should be continuously improved by modeling of human visual characteristics and internal noise, as well as by developing precise measurement processes [39–42]. Such advancements would contribute to enhancing the accuracy of clinical image diagnosis. The simulation process established in this study allows generating images with various resolution and noise characteristics and calculating the detectability index. This approach can be applied to optimize diverse imaging conditions in clinical practice, contributing to the review and refinement of imaging protocols. Expanding the insights presented in this paper to other imaging modalities and disease-specific tasks is expected to advance research aimed at enhancing the clinical applicability of image evaluation based on the task-specific characteristics of model observers.

Conclusion

This study explicitly derived the task function for Gaussian signals and examined the differences from the commonly used circular signal model in image evaluation based on the task-specific characteristics of model observers. This examination was conducted through theoretical calculations of the detectability index. A comparison of the two signal models revealed that the Gaussian signal model consistently demonstrated lower detectability indices than the circular signal model, with the discrepancy becoming more pronounced as signal size increased. This finding objectively suggests that the low-contrast characteristics and spatial spread of the Gaussian signal reduce its visibility compared to circular signals. This result was further supported by simulated images that accurately reproduced the characteristics of clinical images.

This study quantitatively clarified the impact of signal shape on the detectability index, providing a refined theoretical framework for medical image evaluation standards. In particular, understanding the influence of signal shape, size, and contrast on the detectability index enhances optimization and accuracy in clinical image diagnosis. Furthermore, expanding this theoretical framework to other imaging modalities and disease-specific tasks holds promise for further advancements in the field of medical image analysis.

Author Contribution

The author carried out all aspects of this study, including its conception, design, execution, analysis, and manuscript preparation.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Declarations

Ethical Approval

This study was based entirely on physical phantom image or simulation data and did not involve human participants or patient data. Therefore, ethics approval was not applicable.

Consent to Participate

This study did not involve human participants. Therefore, consent to participate was not required.

Consent for Publication

This study did not involve human participants or patient data. Therefore, consent to publish was not applicable.

Competing Interests

The authors declare no competing interests.