3.1.

Mathematics of Incorrectly Centered Targets

The perfect sinusoidal Siemens star with a center at $r=0$ is described by Eq. (1), but the measured profiles are not the same for an image with a misaligned center. The coordinate systems for the true and offset centers are diagrammed in Fig. 2.

Fig. 2

Coordinate systems used to discuss correctly (blue) and incorrectly (red) determined centers for a sinusoidal Siemens star test target. In this diagram, $C_x>0$ and $C_y<0$.

To find the closed-form definition of the measured sinusoidal Siemens star with an arbitrary center offset, we begin with the parametric equations of a circle centered around the offset center in order to easily translate between the two polar coordinate systems:

Figure 3 shows the relationship between the assumed angle for no center offset (${\theta }_{\mathrm{m}}$) and the actual angle ($\theta$) due to various offset centers. Because the polar angle is $2\pi$ periodic, an offset center modulates $\theta$ with various degrees of severity due to the magnitude of the offset.

Fig. 3

The relationship between $\theta$ and ${\theta }_{\mathrm{m}}$ as a function of $C_x/r_{\mathrm{m}}$ offsets. Note that increasing the offset or decreasing the sampled radius will increase the difference between $\theta$ and ${\theta }_{\mathrm{m}}$.

The actual sampling angle in the case of an offset center [Eq. (2)] is substituted into Eq. (1), yielding the equation for the offset sinusoidal Siemens star

This equation reflects the impact of a constant offset center on the extracted intensity values and the introduction of a radial dependence that is due to the offset alone. Therefore, any circular profile extracted from an image of a sinusoidal Siemens star target with an incorrectly found center will not yield a pure sinusoidal profile, but rather a more complex sinusoidal profile modulated with an inverse tangent component. Furthermore, the deviation from a pure sinusoidal function increases nonlinearly as the radius of the extracted profile decreases, as shown in Fig. 3. Restated, this effect is much stronger at high-spatial frequencies than at low-spatial frequencies.

Because of the modulation in the $\theta$, the effective angular frequency, denoted as ${\omega }_{\mathrm{m}}$, for a circular profile extracted for an offset center also exhibits a one cycle modulation. The effective angular frequency of Eq. (3) is calculated as

To aid in understanding the implications of Eqs. (3) and (4), an example circular profile taken about the true center is compared to an equivalent profile taken about an offset center in Fig. 4. Figure 4(a) shows a 10 cycle sinusoidal Siemens star target, with a blue dot showing the correct center of the target ($C_x=C_y=0$) and a red dot showing an offset center in the $\pi /4$ direction ($C_x/r_{\mathrm{m}}=C_y/r_{\mathrm{m}}=0.1$). The intensity patterns extracted in the polar coordinate system by the correct (blue) and offset (red) centered circles are shown in Fig. 4(b), where phase misalignment of the profile from the offset center appears to be greatest at the angles orthogonal to the offset angle of $\pi /4$. This misalignment is also apparent in Fig. 4(c), where the extracted profiles are more clearly shown as a function of ${\theta }_{\mathrm{m}}$. Figure 4(d) highlights this error in the polar angle, showing ${\theta }_m$ versus the difference of ${\theta }_{\mathrm{m}}$ and $\theta$.

Fig. 4

(a) A Sinusoidal Siemens star with $\omega =10$ cycles and two circular outlines. The blue circle is correctly centered with $C_x=0$ and $C_y=0$, and the red circle represents data shifted from the true center by $C_x/r_{\mathrm{m}}=0.1$ and $C_y/r_{\mathrm{m}}=0.1$. The blue and red dots represent the centers of the blue and red circles. (b) A polar coordinate representation of the extracted circular profiles for true (blue) and offset (red) centers. (c) Extracted circular profiles from the correctly (blue) and incorrectly (red) centered sinusoidal Siemens star. (d) The error in the polar angle induced by the offset center. (e) The change in the angular frequency (number of target cycles) induced by the offset center.

Supporting the initial observation of profile misalignment, the phase angle error for this offset center is near zero in the angles aligned with and opposed to the offset direction, and the largest absolute error is near the angles orthogonal to the offset direction. Interestingly, the error caused by the one cycle modulation of the phase angle does not directly translate to the effective spatial frequency. Figure 4(e) shows that ${\omega }_{\mathrm{e}}$ of the profile from the offset center is most different from the true $\omega$ of the target in the directions aligned with and opposed to the offset direction.

This difference in the frequency of the extracted profile has significant implications when assuming a known angular frequency for calculations of the image contrast at that known frequency. While this effect may at first seem innocuous, problems arise when fitting a pure sinusoidal function to Eq. (3), a function with an effective frequency modulation. The ISO 12233 standard recommends extracting a circular profile from an image and performing a least squares fit of the intensity signal with a sine function.

Figure 5 shows the best fit amplitude for a correctly centered, 72 cycle sinusoidal Siemens star for a range of spatial frequencies (i.e., a range of radii), and the best fit amplitude for an incorrectly centered target with a center offset of $(C_x,C_y)=(1\mathrm{px},0\mathrm{px})$. Significant degradation is possible in this ideal case, with decreases in best fit amplitude at all spatial frequencies, and the introduction of contrast inversions at higher frequencies.

Fig. 5

Spatial frequency response (SFR) extracted from a least squares best fit of Eq. (1) for a correctly centered sinusoidal Siemens star, and an SFR measured for a center offset of $(C_x,C_y)=(1\mathrm{px},0\mathrm{px})$. The sinusoidal Siemens star target has $\omega =72$ cycles, and a $\varphi =0$.

3.2.

Simulated Images

To examine these effects in a more realistic setting, an 8-bit image of a sinusoidal Siemens star was generated mathematically, convolved with a known point spread function (PSF) to simulate an optical aberration, and SFR data was collected by fitting different radial profiles with Eq. (1). A Dirac delta function was chosen to further examine an optically perfect simulated sinusoidal Siemens star, and a nearly diffraction limited Cooke triplet PSF, simulated within Zemax optical design software, was chosen to examine how offset centers affect a realistic, optically degraded image.

The PSFs and sinusoidal Siemens stars convolved with each PSF are shown in Fig. 6. The true center location of each image is known in the generation of these images. We again choose to fit data to the entire radial profile (i.e., one segment is used for the analysis in this subsection). This was chosen so that a comparison could be made between the purely mathematical analysis of Sec. 3.1, and the more realistic case of processing an image of a sinusoidal Siemens star target.

Fig. 6

Discrete point spread functions (PSFs) of: (a) a delta function, (b) a nearly diffraction limited Cooke triplet lens system, (c) and (d) the sinusoidal Siemens star after convolution with the PSFs from (a) and (b), respectively.

Figure 7 shows best fit contrast (i.e., SFR) versus cycles per pixel for the two PSFs shown in Fig. 6 and with an increasing pixel offset value. Also shown is the area under curve (AUC) for each pixel offset, normalized by the zero-offset center AUC. Examining Fig. 7(a) for a pixel offset of 1 shows a curve that is nearly equivalent to the purely mathematical analysis in Fig. 5, showing differences only due to sampling of the simulated image. In Fig. 7(c), SFR measurements from the Cooke triplet PSF show less degradation for the same pixel offset compared to the Dirac PSF. This is due to realistic optical systems typically yielding low-contrast values at high frequencies, which effectively attenuates the potential errors due to incorrect center locations. However, a single pixel shift reduces the AUC by approximately 17% for the Cooke PSF example.

Fig. 7

(a) The measured contrast for a perfect (i.e., Dirac PSF convolved) sinusoidal Siemens star with an increasing center offset error. (b) The AUC measured for (a) from each center offset error, normalized to the correctly centered AUC value. (c) Measured contrast from a sinusoidal Siemens star convolved with a realistic Cooke triplet point spread function. (d) The normalized AUC measured for each offset error from (c).

3.3.

Error Mitigation Through Angular Segmentation

The ISO 12233 standard states an image of a sinusoidal Siemens star target should be angularly divided into a user defined number of segments, typically eight segments, though 24 segments are mentioned later in the document. Segmentation is stated as a technique derived to assist the processing of distorted images without requiring modifications to the original image. However, angular segmentation can also serve the secondary purpose of minimizing errors in the SFR associated with incorrectly determined centers. Conceptually, an offset center can be thought of as generating a nonuniform modulation of the measured signal, as defined by Eq. (3), similar to that of optical distortion. In this subsection we investigate the effect of angular segmentation of data on the normalized AUC given an image with an offset center, and the minimum number of segments needed to reach at least 95% of the nominal AUC.

An image of a 144-cycle sinusoidal Siemens star was analyzed by performing a least squares fit given an offset center ranging from 1 to 10 pixels, in 1 pixel increments, and analyzing the sinusoidal Siemens star in angular segments ranging from 1 to 72 segments. The best fit contrast was averaged over all segments, integrated, and compared to the AUC from the correctly determined center case. The AUC for the zero-offset center is referred to as the nominal AUC. Figure 8 shows the results of this analysis given the previous two PSFs convolved with images of a sinusoidal Siemens star in Fig. 6.

Fig. 8

(a) Normalized AUC versus segments for a Dirac PSF convolved target. (b) Segments needed to maintain 95% nominal AUC for the Dirac PSF convolved target. (c) Normalized AUC versus segments using the Cooke PSF convolved target. (d) Segments needed to maintain 95% nominal AUC using the Cooke PSF convolved target. Test target images are the same used in the analysis discussed in Fig. 7, and used a 144-cycle sinusoidal Siemens star target. These results show that a significant number of segments must be used when the test target center is not correctly identified. Small shifts can yield large decreases in AUC if the appropriate number of segments is not utilized in the SFR analysis.

Normalized AUC increases as more segments are used to analyze a sinusoidal Siemens star image, regardless of the level of aberration present in the optical system, and larger offset centers require increased segmentation to reduce errors. To achieve a 95% AUC measurement, the number of segments needed scales approximately linearly with the magnitude of the offset center error. The Dirac PSF convolved image requires more segments than the Cooke PSF convolved image, but both have approximately linear relationships between the number of segments needed to attain 95% AUC and pixel offset. Higher quality optical systems are more influenced by offset centers, as shown in the Dirac PSF case, and require approximately 22 segments to achieve 95% of the nominal AUC for a 3-pixel offset case. The realistic optical aberration of the Cooke PSF convolved image requires at least 10 segments to achieve 95% of the AUC of the nominal case for a 3-pixel offset. Note that the maximum shift, $C_x=10$, is only 2.22% of the largest radial sample used to analyze this image.

Repeating the analysis above, but for a 72 cycle sinusoidal Siemens star target, shows the Dirac PSF image requires half the number of segments compared to the 144-cycle sinusoidal Siemens star to achieve the same 95% nominal AUC given a 3-pixel center offset. Using the Cooke PSF, maintaining a 95% nominal AUC enables an approximate 15% reduction in the number of segments required compared to the 144-cycle sinusoidal Siemens star. This analysis appears to indicate that a more complex relationship exists between required segments and target cycles.

Based on the assumption that the center of the sinusoidal Siemens star target is found within 3 pixels, using 22 segments appears to minimize errors induced by incorrectly determined centers. However, if an error of greater than 3 pixels is expected from the center finding algorithm, the appropriate number of segments should be increased to reduce error.

3.4.

Mitigation Through Proper Location of Target Center

Regardless of the level of aberration or number of segments used in analysis, an accurate determination of the true center reduces error. We propose a two-step method whereby an initial coarse estimate is made of the center of the target, a low-frequency single circular profile is extracted from the image of the sinusoidal Siemens star, fit to Eq. (3), and constants $C_x$ and $C_y$ are solved for and used to correct the initial guess. A singular circular profile, as visualized in Fig. 4(a), will exhibit a changing spatial frequency dependent upon the magnitude of the center offset error. By extracting this singular circular profile and fitting Eq. 3, the center offsets can be measured.

A Monte-Carlo simulation using 10,000 iterations was performed, where each iteration of the simulation generated an incorrect pixel offset, chosen via Gaussian weighted random number generation using $({\sigma }_x,{\sigma }_y)=(2\mathrm{px},2\mathrm{px})$. Values were centered about the true center of the test target (i.e., $\mu =0$). The simulated image parameters used for the Cooke triplet PSF analysis in Sec. 3.2 were reused. Figure 9 shows the offset center locations in red and the corrected center locations in blue after the two-step correction has been applied. After correction, the mean center error is 0.026 pixels, with a standard deviation of 0.014 pixels.

Fig. 9

Offset centers (subpixel) compared to the center of the target, defined as the origin, with red points as the offsets from the Monte-Carlo simulation, and blue points the corrected offsets after applying a least squares fit of Eq. (3) to a single circular profile. Pixels are represented by the grid, and a view of the center pixel is shown on the right.

The results from corrected center locations are compared to the results without correction by computing the normalized AUC as a function of pixel offset magnitude. Figure 10 shows these results given a single segment analysis and an eight segment analysis, identified by the ISO 12233 standard as the typical segmentation number. Using one segment fitting and the two-step center estimating process, the maximum relative error compared to the nominal AUC is 0.136%, and the maximum deviation from the true SFR at any spatial frequency is 0.37%. Similar results are seen when eight segments are used, with the maximum relative error compared to the nominal AUC being 0.026%, and the maximum deviation from the true SFR at any spatial frequency being 0.39%. These results clearly show the benefits of using a two-step center finding method. An additional point to note is that the number of segments can be reduced if confidence is sustained in the accuracy of the center finding algorithm.

Fig. 10

Normalized AUC versus pixel offset magnitude. The red data shows normalized AUC using a single center estimation step that contains some error and no center correction feedback process, while the blue data shows the results from the proposed two-step center correction method.

3.5.

Analysis of a Real Image

The previous examples have utilized images created from the mathematical definition of a sinusoidal Siemens star test target, as shown in Eq. (1). A similar analysis is performed using a real image of an ISO 12233:2014 compliant 144-cycle sinusoidal Siemens star test target taken with a Cannon EOS 5D Mark III digital camera. The image of the test target used in this analysis is shown in Fig. 11(a). The center of the test target is found via human analysis. An error of 10 pixels in the $x$-direction is then added to the true center value, and SFR is measured using 8 segments. Using the techniques described in Sec. 3.4, the incorrect center is mitigated and the SFR is calculated for this corrected case. Figure 11(b) shows the SFR for the nominal, uncorrected, and corrected cases. Before correction, the AUC is 21% less than the nominal AUC, and the spatial frequency measured with 30% contrast, called the SFR30, is 17.3% less than the nominal SFR30. After correction, the AUC is 0.15% greater than the nominal AUC, and the SFR30 value is 0.43% greater than the nominal SFR30.

Fig. 11

(a) Real image taken of an ISO 12233:2014 compliant sinusoidal Siemens star target using a digital camera. The blue x denotes the true center of the test target, as determined by human analysis. The red circle and dot denote a user added error of 10 pixels in the $x$-direction. The green square and dot represent the corrected center using the technique described in Sec. 3.4. (b) Measured SFR for the nominal, correctly centered case (blue), offset center case (red), and corrected case (green). The dashed line shows the SFR10 value. The red circle and dot, green square and dot, and blue x show the SFR10 values for the offset center measurement, corrected center measurement, and nominal case, respectively.