1. INTRODUCTION

The spatial frequency response (SFR) is a parameter used to specify image resolution. The term SFR refers to the resolution of the overall camera system, from the optical lenses to the image sensor. A similar term, the modulation transfer function (MTF), is sometimes used to represent the spatial resolution of the optical lenses only. In the following discussion, SFR and MTF are used interchangeably.

The SFR represents the amplitude attenuation characteristics of sinusoidal patterns in an object image. In the Michelson contrast, the SFR is defined as the ratio of the amplitude of a sinusoidal pattern to the bias component [1]. This method requires repeated measurements using various sinusoidal test charts from low to high frequencies.

The most efficient method for measuring the SFR is to analyze the camera image of a point-light source. The camera image is subject to a blur owing to the optical diffraction limit and aberrations, even if it is in focus. The blurred image corresponds to the point spread function (PSF). The SFR can be computed using the Fourier transform of the PSF. Because a point-light source has limited intensity and the camera image is distributed over a small area, the digital image is inevitably degraded because of the low signal-to-noise ratio (SNR), and spatial features smaller than the pixel spacing are lost. To accurately estimate the SFR, it is not sufficient to just use a single point-light source; instead, an image including multiple point-light sources is required. To improve the resolution and accuracy, a method using the dot array pattern on a liquid crystal display has been proposed [2].

The slanted-edge method, also called the edge-based method, has been developed as an alternative to the aforementioned method, i.e., specified in the international standard, ISO 12233 [3]. It is based on an analysis of a camera image of a slanted-edge chart, i.e., the border between white and black regions has a slightly tilted straight line, just like a knife edge. The image blur occurs in a direction perpendicular to the edge, for which the intensity distribution is called an edge spread function (ESF). It is uniformly distributed in each row, which involves a subpixel shift corresponding to the tilt angle. The oversampled ESF with a subpixel density was obtained by projecting each row of pixel data onto the axial direction perpendicular to the edge and interleaving them [4]. The resultant distribution has high-density information compared to the original image. The discrete Fourier transform (DFT) of the line spread function (LSF) that is obtained based on the derivative of the ESF is used to derive the SFR. The calculated SFR has high-frequency information over the Nyquist frequency [5]. Such high-resolution measurement is increasingly important for the evaluation of high-definition cameras. The technology used for the measurement of camera vibrations [6] has also been developed based on these considerations.

The slanted-edge method has gone through several changes since ISO 12233 was established [7], and various software tools based on the slanted-edge method have been developed, e.g., Imatest [8], SFRMAT3 [9], Quick MTF [10], and MTF mapper [11], etc. They are used for testing not only consumer cameras but also aerial imaging systems using retro-reflection [12]. Various improvements based on the original ISO 12233 have also been investigated.

The estimation accuracy of the edge angle significantly affects the calculation result of the SFR, especially in a noisy environment [13]. The angle estimation specified in ISO 12233 is based on a simple average of the shift between each row. Therefore, the accuracy and the noise robustness are comparatively low. In some practical tools, least square estimation is employed for line-fitting to the sampled edge slope to improve the results. In this regard, the approximation of the edge slope to a quadratic curve has also been proposed [14]. The edge slope is necessary for resampling in subpixel intervals and for computing the SFR over the Nyquist frequency. However, it causes the SFR to be degenerated depending on the edge angle. The correct SFR is obtained by expanding the degenerated SFR to correspond with a direction perpendicular to the edge [15].

The subpixel data projected from each row of images are unequally spaced. Therefore, they must be rearranged in equally spaced subpixel bins and uniformized, i.e., called binning. Because this process involves non-uniform averaging in each subpixel bin, it causes instability in the SFR estimation. The analysis of this instability has been investigated, and compensation methods have been examined [16–18]. A method based on the non-uniform Fourier transform that does not require binning has also been proposed [19]. In addition, the problem is ill-posed because the derivative operation used to obtain the LSF is sensitive to noise during SFR computation, especially at the higher frequencies. This instability has been identified since the early phase of edge-based measurement. An edge spectrum ratio method that does not require derivative operation has also been proposed [20]. To address the instability associated with noise, a regularization method [21] and parametric modeling of the transfer function [22] have been utilized. An approach that is adaptable to the image distortion that is encountered in a practical lens system [23,24] and full-field MTF analysis of an image [25] have been investigated. The edge detection from natural scenes [26] and the SFR estimation based on the detected edges have also been presented [27].

As indicated, various studies have been conducted to improve the original slanted-edge method based on ISO 12233. The common aim is to improve the estimation accuracy of the SFR. For example, the estimation accuracies for various edge angles and SNRs have been investigated [28]. Comparative evaluations using a reference dataset have been performed for various implementations of the slanted-edge method for on-orbit MTF measurements [29]. The next version of ISO 12233 corresponding to image distortion is also planned at an early date [30]. However, the method or tool that yields the most reliable result has not been investigated. Thus, the fundamental question of whether the slanted-edge method under ideal conditions yields a true curve of the SFR remains unanswered.

This study examines the accuracy of the estimated SFR obtained using the slanted-edge method and compares it to the theoretical value. To evaluate the estimation accuracy, a blurred image must be produced based on a known PSF for use with the slanted-edge method. The study proposes a numerical method to synthesize the blurred slanted-edge image with subpixel accuracy. The SFR was computed using the synthesized image with the slanted-edge method. The accuracy can be estimated based on a comparison of the experimentally obtained value to the theoretical value of the SFR. The comparison reveals that under ideal conditions, the slanted-edge method yields a true value for the SFR, but the actual cases with background noise inevitably cause some estimation errors. This analysis also facilitates the identification of parameters used for the SFR computation that dominantly influence the estimation accuracy.

In Section 2, the principle of the original slanted-edge method and some corrections for improving the estimation accuracy are reviewed. An additional correction to the binning process is also proposed. Furthermore, the influence of additive noise on the estimation of the SFR is discussed. Based on a theoretical analysis using Gaussian noise, a higher spatial frequency involves more estimation error owing to the image noise. In Section 3, a numerical method for evaluating the estimation accuracy of the SFR is presented. In Section 4, numerical simulation confirms that the SFR estimated based on the slanted-edge method converges to the true value, and the distortion of the SFR owing to noise coincides with the theoretical analysis. The application to an actual camera image is also presented.

2. MODIFIED SLANTED-EDGE METHOD

A. Basic Formulation based on ISO 12233

First, the original slanted-edge method based on ISO 12233 is reviewed.

A slanted-edge image that contains a slightly tilted straight line at the boundary of white and black regions is prepared as described in this section. As shown in Fig. 1, the axis ${x_\theta}$ is selected to be in a direction perpendicular to the slanted edge, and ${x_\theta} = 0$ corresponds to the line passing through the origin $(0,0)$ and parallel to the slanted edge. The distance ${x_\theta}$ from the line ${x_\theta} = 0$ to an arbitrary point $(x,y)$ on the image can then be represented as follows:

 

Fig. 1. Example of a slanted-edge image and the relationship with the pixel array. Every row of data is projected onto the common ${x_\theta}$ axis.

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Generally, the blurred image under an optical system is formed by a convolution operation with the PSF in a two-dimensional coordinate system $(x,y)$. However, because the original image only consists of a straight edge, it is reduced to a one-dimensional convolution with the LSF. When the LSF of blur in a direction perpendicular to the slanted edge is described as $h({x_\theta})$, the slanted-edge image with the blur is represented as

The denominator in Eq. (4) means the total count of the sampling data belonging to the $i$th bin, and the index $i$ represents a subpixel number $0 \le i \le N{B_{{\rm{in}}}}$ where $0$ and $N{B_{{\rm{in}}}}$ correspond to the left end of the ${x_\theta}$ axis and the total number of bins or subpixels. The distribution ${g_{{\rm{ESF}}}}(i/{B_{{\rm{in}}}})$ represents the approximation of the ESF ${u_{{\rm{blur}}}}(i/{B_{{\rm{in}}}})$ that is over-sampled at ${B_{{\rm{in}}}}$ times the initial sampling density. Equation (4) corresponds to Eq. (D.6) in the specification of ISO 12233 [3].

The approximated LSF can be obtained by calculating the difference between neighboring subpixels, i.e.,

Although a double-sided difference is used in ISO 12233 [3], a single-sided one is employed in this investigation because a phase shift of a half-pixel does not affect the absolute value in the computation of the SFR and the high-pass attenuation owing to the difference operation is reduced rather than the double-sided one. Finally, the SFR to be estimated can be obtained by applying the DFT to the LSF, calculating the modulus, and normalizing it based on the DC component, i.e.,

It should be noted that the upper limit of the frequency is expanded to ${B_{{\rm{in}}}}/2$ from the initial 0.5 cycles/pixel because the sampling density is increased to ${B_{{\rm{in}}}}$ times the original one. In other words, the binning factor ${B_{{\rm{in}}}}$ is selected so that the LSF satisfies the sampling theorem. For ISO 12233, the window function is applied to the LSF for noise reduction. In the following analysis, however, the measures to address image noise are not introduced to investigate the intrinsic performance of the slanted-edge method.

B. Corrections for Accuracy Improvement

As previously indicated, processing occurs in the discrete domain because the input image is digital data. Therefore, the implementation requires corrections for improvement of the estimation accuracy.

One of the corrections is owing to the binning for uniformizing the sampling data. If it is assumed that the projected data in Eq. (A2) are uniformly distributed on each bin slot (see Fig. 2), Eq. (4) can approximately be rewritten as follows:

 

Fig. 2. Binning process and its approximate representation. The projected pixel data from each image row have a non-uniform distribution (upper left). They are sectioned into each bin (bottom left). Equation (4) represents the direct averaging of non-uniform data in each bin and yields uniform subpixel data (upper right). Against this procedure, Eq. (7) means that the non-uniform data are approximately formed at a regular interval before averaging (bottom right). This approximation allows for the binning process to be expressed as a convolution operation as in Eq. (7).

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The discrete domain can be computed using a PC; however, the difference operation, as shown in Eq. (5), must be employed as an alternative to differentiation.

Combined with the averaging effect based on Eq. (7), Eq. (5) can be rewritten as

C. Noise Analysis

The actual camera image is inevitably subject to background noise. The image noise causes the SFR estimation to be significantly distorted. The effect of noise on the SFR estimation has been investigated by some researchers; however, the detailed formulation of the effect of image noise on the estimation of SFR has not been studied. In the following discussion, the SFR for the slanted-edge image with background noise is precisely formulated.

Image noise is added independently in each pixel of the image sensor. Therefore, the noisy image version of Eq. (3) can be represented as

The second term inside the square root indicates the effect of noise. Because the term ${H_{{\rm{bin}}}}(k/N)$ is approximately equal to 1, the preceding equation shows that the image noise increases the LSF in proportion to the frequency $f = k/N$ and destabilizes the estimation for high frequencies. It also shows that, while the larger height of the region of interest (ROI) $M$ reduces the effect of noise more, the binning factor ${B_{{\rm{in}}}}$ is not dependent on the noise term. These properties will also be shown in numerical simulations and experiments discussed later in the report.

3. NUMERICAL SYNTHESIS OF SLANTED-EDGE IMAGE

To verify the aforementioned formulation and for comparison with some tools that implement the slanted-edge method, a slanted-edge image with a known blur is numerically synthesized. To enable SFR estimation over the Nyquist frequency, the slanted-edge image to be synthesized must have an adequate fineness under the pixel interval. Two types of PSF models for synthesizing the slanted-edge image with an arbitrary subpixel resolution are presented. Although a similar image synthesis method has been proposed [31], the following computation methods yield blurred images with the exact pixel values required to evaluate the estimation error of the SFR with very high accuracy.

A. Circular Blur

First, the PSF of blurring is assumed to have a uniform distribution inside a circle, i.e., corresponding to the circle of confusion. The PSF of the circular blur can be formulated as follows:

Using the convolution based on Eq. (3), the slanted-edge image with the circular blur can be represented as

The calculation process is illustrated in Fig. 3(a). Finally, the pixel-sampled image is generated from ${g_{{\rm{circle}}}}(n,m) = {[{{u_{{\rm{circle}}}}({x_\theta})}]_{x = n,y = m}}$, and one of the examples is shown in Fig. 3(b). Notably, the image synthesis ensures that the accuracy of each pixel is within the quantization error, e.g., 8-bit or 16-bit, because each pixel value is assigned directly from the analytical solution. Furthermore, the theoretical SFR can be obtained using the Fourier transform of Eq. (14), i.e.,

 

Fig. 3. Synthesis of the slanted-edge image with a circular blur. (a) Formation process. (b) An example of the synthesized image. The tilt angle is $5^\circ$, and the radius $r$ of the circular blur is 0.7 pixels. An area of $20 \times 20$ pixels around the edge is enlarged. The grid mesh is drawn together with the image data to reveal borders between each pixel.

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B. Gaussian Blur

As another example, the PSF of blurring is assumed to have a Gaussian distribution with circular symmetry and a variance of ${\sigma ^2}$, i.e.,

As with the case of a circular blur, the slanted-edge image with the Gaussian blur is represented as

The calculation process is illustrated in Fig. 4(a). To facilitate numerical integration, it is rewritten so that the integral range has a finite interval. Numerical integration is performed using Simpson’s rule, in which the integration is divided into several subintervals, and each subinterval is approximated using a quadratic curve. The pixel-sampled image is obtained from ${g_{{\rm{gauss}}}}(n,m) = {[{{u_{{\rm{gauss}}}}({x_\theta})}]_{x = n,y = m}}$, and one of the examples is shown in Fig. 4(b). Compared to Fig. 3(b), differences in the blur are observed around the edge. It is confirmed that no error occurs in each pixel value that is quantized to 16-bit by selecting the number of subintervals at more than 50 per pixel for numerical integration. Using the Fourier transform of Eq. (18), the theoretical SFR is written as

 

Fig. 4. Synthesis of the slanted-edge image with a Gaussian blur. (a) Formation process. (b) An example of the synthesized image. The standard deviation $\sigma$ of the Gaussian blur is 0.7 pixels. The other configuration is the same as Fig. 3.

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This equation and Eq. (16) are used for comparison to the SFR estimated using the slanted-edge method.

4. NUMERICAL SIMULATION AND EXPERIMENT

A. Behavior in Ideal Conditions

The slanted-edge method has been implemented in practical tools equipped with graphical user interfaces, e.g., Imatest [8], SFRMAT3 [9], Quick MTF [10], and MTF mapper [11], etc. In this case, Imatest and SFRMAT3 are used to perform quantitative comparisons with the modified method that is described in Section 2.B. In the following simulations, the edge angle is set to $5^\circ$, which is the value used for ISO 12233 and known to be relatively stable compared to other angles [17,18]. The radius of the circular blur and the standard deviation of the Gaussian blur added to the slanted-edge image are commonly set to 0.7 pixels because the image blur seen in many measurements [7,14,16,17,22,29–31,33] and a real camera experiment shown in Section 4.D is close to this value. The quantization rate of the image is 16-bit.

First, this study investigates whether the slanted-edge method yields the true value of SFR, i.e., the main theme of this paper. The SFR based on Eq. (10) is computed under various binning factors ${B_{{\rm{in}}}}$, where $N$, corresponding to the width of the ROI, is fixed at 100 pixels, but $M$, corresponding to the height, is selected to be $20{B_{{\rm{in}}}}$ pixels to increase the proportion of the binning factor to avoid producing empty bins even for a larger binning factor. To evaluate the deviation between the theoretical values based on Eqs. (16) or (20) and the estimated values of SFR based on Eq. (10), the root-mean-square (RMS) errors of the SFR values for the normalized frequencies 0 to 1 cycles/pixel are calculated for each binning factor (see Fig. 5). It can be verified that the SFR estimations in both cases of circular and Gaussian blurs converge to the theoretical SFR with an increase of the binning factor. To identify the cause of the residual error for a larger binning factor, an 8-bit quantization image is also evaluated in addition to a 16-bit image. Although the 16-bit exhibits a very small error, the 8-bit image exhibits a relatively large error, even for a large bin factor. This means that the accuracy limitation for SFR estimation using the slanted-edge method is owing to the quantization rate of the image under ideal conditions, such as a noiseless image and a large binning factor. This is the first consequence that has not been clarified in any previous study. However, SFRMAT3 remains at the same accuracy as the 8-bit quantization image, even when using 16-bit quantization. The same evaluation with respect to Imatest could not be performed because it is not possible to compute using other binning factors besides ${B_{{\rm{in}}}} = 4$.

 

Fig. 5. Error convergence of the SFR estimation by the slanted-edge method. (a) Circular blur. (b) Gaussian blur.

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Fig. 6. Comparison of SFRs estimated using the slanted-edge methods for ${B_{{\rm{in}}}} = 4$ and $M = 1000$. (a) Circular blur. (b) Gaussian blur.

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Fig. 7. Absolute deviations from the theoretical SFR to that estimated using each method. The computational conditions are the same as that in Fig. 5. (a) Circular blur. (b) Gaussian blur.

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Fig. 8. Effects of the difference in the height of ROI on the estimation accuracy of SFR for ${B_{{\rm{in}}}} = 4$. (a) Circular blur. (b) Gaussian blur.

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Fig. 9. Effects of the edge angle error on the accuracy of SFR estimation. Height of ROI: (a) $M = 100$, (b) $M = 500$; true angle: $5^\circ$; binning factor: ${B_{{\rm{in}}}} = 4$.

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Representative examples of SFR estimated using the slanted-edge methods are shown in Fig. 6. Each of these methods yields values that appear to be close to the theoretical SFR. However, as can be seen in Fig. 7, the absolute deviations from the theoretical SFR to the values estimated using each method show that the modified method based on Eq. (10) is superior to the others. In particular, the absence of the binning correction term ${H_{{\rm{bin}}}}(f)$ from Eq. (10) degrades the estimation accuracy. The correction term works well to minimize the degradation owing to the binning.

B. Effects of Each Parameter

Next, the study investigates the effect of each parameter used to implement the slanted-edge method on the SFR estimation accuracy. The height of the ROI, i.e., corresponding to the variable $M$, is one of the major factors that affect the estimation accuracy of the SFR. Figure 8 shows the effects of the difference in the heights of the ROI. The larger the height, the greater the amount of pixel data associated with each bin. Therefore, the estimation error of the SFR is reduced at a larger height. However, in the absence of the binning correction, no improvement is observed, even if the ROI height is large. The heights above a certain threshold saturate the estimation accuracy because the finite binning factors, especially the low values such as ${B_{{\rm{in}}}} = 4$ used in the simulation, cause an aliasing distortion for higher-frequency components.

 

Fig. 10. Variation rate of the SFR estimation error caused by the edge angle error. Every binning factor leads to approximately the same result.

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The angle of the slanted edge is also an important factor in determining the estimation accuracy of the SFR. The edge chart is allowed to arbitrarily set the tilt angle when the chart is fabricated or embedded in the experimental system. However, it is difficult to determine the correct edge angle even if premeasurement is implemented. Even a slight error in the angle estimation is expected to cause a computation error in the SFR. Figure 9 shows the extent to which the angle error affects the accuracy of SFR estimation. The estimation error of the SFR is amplified with an increase in the angle error. The small height of the ROI becomes comparatively insensitive to the edge angle error. However, the large ones exhibit sensitivity. For $M = 100$, if the angle error falls within 2%, the estimation error of the SFR is held to a minimum level [see Fig. 9(a)]. For $M = 500$, however, the angle error is required to be within 0.2% to minimize the SFR error [see Fig. 9(b)]. It shows that the sensitivity to the edge angle error is highly dependent on the height of the ROI. Figure 10 shows the effect of the height of the ROI on the estimation error of the SFR. The vertical axis indicates the increasing rate of RMS error caused by a deviation of $0.1^\circ$ from the true angle of $5^\circ$. A larger height of ROI increases the estimation accuracy’s sensitivity to the angle error. The difference in binning factor does not affect the variation rate of the SFR estimation error caused by the angle error. These criteria provide a design guide for the edge angle estimation method.

The estimation accuracy of the SFR is also dependent on the choice of the edge angle even if the edge angle can be pre-estimated correctly. This effect has been investigated by two researchers [16–18], and the same result was also obtained in this investigation. As shown in Fig. 11, some specific angles cause instability or a significant error in the SFR estimation. For example, the critical edge angle of ${\tan ^{- 1}}\,1/2 \approx 26.57^\circ$ assigns just two samples per 1 pixel for the projection process of Eq. (A2), even if the height of the ROI is large, and causes empty bins for ${B_{{\rm{in}}}} \gt 2$. For the same reason, ${\tan ^{- 1}}\,1/3 \approx 18.43^\circ$ and ${\tan ^{- 1}}\, 2/3\approx 33.69^\circ$ also cause divide-by-zero errors. Some edge angles increase the SFR error for non-uniformity of the pixel data in each bin. Both the Gaussian blur and the circular one lead to approximately the same result.

 

Fig. 11. Effects of the difference in the edge angle on the accuracy of SFR estimation. Dashed circles represent singular points owing to divide-by-zero in the critical edge angles. Height of ROI: $M = 100$.

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Fig. 12. Effectiveness of averaging multiple SFRs based on different edge angles. Edge angles are randomly selected between $1^\circ$ and $44^\circ$. The blue-filled circle uses random angles generated from a seed different from the red one. The green triangle uses random angles with a deviation of $0.03^\circ$. Binning factor: ${B_{{\rm{in}}}} = 4$; height of ROI: $M = 100$.

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Figure 11 indicates the existence of the edge angles to minimize the estimation error. However, the edge angle cannot be set up precisely because the positioning accuracy of the slanted-edge chart has a physical limitation. Averaging multiple SFRs estimated under various edge angles can effectively reduce the influence of edge angles on the estimation accuracy of the SFR. The simulation result in Fig. 12 confirms the effectiveness of averaging. Each SFR is computed from the slanted-edge images with randomized angles, and the RMS error can be calculated from the averaged SFR. The slanted-edge images are generated from random angles with different seeds in each of three examples. One of them uses random angles with a deviation to examine the effect of angle error. Every case converges to a constant error with the increase in the number of SFRs in averaging. Therefore, averaging of multiple SFRs based on different angles contributes to the stabilization of the solution.

C. Behavior in Noisy Environment

Furthermore, the study investigates the effect of image noise on SFR estimation. The theoretical analysis has already been discussed in Section 2.C; this section evaluates whether the numerical simulation results match the theoretical solution based on Eq. (12). To examine the average behavior of noise on the SFR estimation, 30 image samples are prepared by adding white Gaussian noise with different seeds to the slanted-edge image as described in Section 3. The SFRs are computed using Eq. (10) for each image sample, and the average SFR is obtained by taking the ensemble mean of the results. Figure 13 shows the computed average SFR for which the peak signal-to-noise ratio is set to 40 dB by adjusting the amplitude of the additive noise, ${B_{{\rm{in}}}} = 4$ and $M = 200$. The frequency axis is extended to two cycles/pixel to examine the behavior of noise at higher frequencies. It can be confirmed that the average SFR estimated based on the noise-added images coincides with the theoretical SFR in the cases of circular and

 

Fig. 13. Effects of the image noise on the SFR estimation. The red line indicates the ensemble average of the SFRs estimated from each noise-added image. The green line indicates an application result of the window operation specified in ISO 12233. (a) Circular blur. (b) Gaussian blur.

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Gaussian blurring. The image noise causes the estimate of the SFR to increase at higher frequencies. In particular, in the case of the circular blur, the SFR value in the frequency near two cycles/pixel corresponding to the Nyquist frequency at ${B_{{\rm{in}}}} = 4$ has a slight deviation from the theoretical curve because the SFR component close to the Nyquist frequency includes some aliasing errors.

In each case, the image noise causes a significant error in terms of the SFR estimation compared to the effect associated with the parameters discussed in the previous section. Several improvements for reducing the effect of image noise have been discussed [13,21]. However, it is generally difficult to separate the correct solution of the SFR based on measurements contaminated with noise because the SFR to be estimated is much smaller than the noise components at higher frequencies, as shown in the simulation results, i.e., the SFR estimation from the noise image is an ill-posed problem.

To address image noise, the window operation to the LSF has been presented as Eq. (D.9) in the specification of ISO 12233 [3]. It reduces the noise components in areas away from the edge position while preserving the blur component around the edge. The application result is shown in Fig. 13 to examine the effectiveness of the window operation. The effect of noise reduction can be observed, but it has a limitation.

D. Application to Actual Camera Image

Finally, the aforementioned method is applied to an actual camera image. In the experiment, a transparent glass plate printed with a slanted-edge pattern and attached to an LED back light panel is photographed using a mirrorless camera (SONY alpha 7 R IV) mounted with a macro lens (SIGMA 105 mm F2.8 DG DN MACRO). The distance between the camera and the LED display is 50 cm, and the camera settings are as follows: ISO 100, $F$-number of 3.5, and exposure time of 1/40 s. The shutter type employs an electronic front curtain mode to avoid the shutter shock. All apparatuses are fixed on an optical bench. The size of the ROI clipped out from the captured image is $260({\rm W}) \times 680({\rm H})$ pixels. The estimated edge angle was $6.52^\circ$, which was obtained by applying the least square estimation to the sampled edge slope. The SFRs estimated using this edge angle are shown in Fig. 14. The obtained distribution is similar to the Gaussian blur. In the small height of ROI of $M = 100$, the effect of noise is more prominent at higher frequencies but becomes smaller in the large height of $M = 500$. Similar to the trend in Fig. 8, it indicates that the large height of ROI is effective in noise reduction. It also shows that the effect of noise has a marginal dependence on the binning factor. This result is consistent with the discussion of Eq. (12) and Fig. 10.

 

Fig. 14. Examples of the application to an actual camera image. Height of ROI: (a) $M = 100$, (b) $M = 500$.

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5. CONCLUSION

In this paper, the details of a procedure for the slanted-edge method for the measurement of SFR were presented, with an emphasis on the corrections for the binning process and the difference operation. Numerical simulations were performed to investigate the basic characteristic of the slanted-edge method. Gaussian and circular blur models were employed as representative examples of the blurred edge image, which facilitated numerical simulation with a quantization precision of 16-bit. The simulation results show that the larger the binning factor, the smaller the estimation error of the SFR for the absence of noise. The effect of each parameter on the accuracy of the SFR estimation was also discussed. The theoretical analysis and numerical simulations regarding the effect of image noise on the SFR estimation show that a higher spatial frequency involves more estimation errors. These results indicate the uncertainty of the slanted-edge method for estimating the SFR. To improve the estimation accuracy in a noisy environment, choosing a larger height of ROI is more effective than using a larger binning factor, although the larger height of ROI becomes sensitive to the edge angle error. The experiment also showed a similar result using a real camera image. The knowledge obtained from this study will be helpful in the implementation of the slanted-edge method for practical applications.

For future work, a noise reduction method will be developed to improve the estimation accuracy of the SFR, that is effective in actual camera images. Furthermore, the slanted-edge method will be modified to realize a high-precision measurement of the SFR even in a smaller binning factor and a smaller height of ROI.