When discussing the resolution (objective physical limit) of an imaging system, as well as sharpness and clarity (subjective visual perception), MTF/SFR are the core metrics for quantifying these objective and subjective dimensions. Through the MTF curve, they intuitively reveal the system's contrast transfer capability at different detail scales.
I. Core Definitions: What are MTF and SFR?
MTF (Modulation Transfer Function) is the modulus of the Optical Transfer Function (OTF), characterizing the amplitude (i.e., contrast) attenuation characteristics of sinusoidal waves at various frequencies in an imaging system. Its core mathematical expression is: $M = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}}$ (i.e., calculating the modulation based on the maximum and minimum luminance of line pairs), with a value range from 0 to 1. The closer the value is to 1, the stronger the contrast transfer capability.

SFR (Spatial Frequency Response) describes the degree of attenuation of the amplitude (contrast) of an imaging system's response to a sinusoidal input signal as spatial frequency changes, used to quantify the contrast transfer capability at different frequencies; its essence is consistent with MTF, and they can be used interchangeably in engineering applications.

Differentiation of Related Concepts: Resolution, Contrast, Sharpness, Clarity

• Resolution: Measures the limiting capability of an imaging system to distinguish spatial details, typically measured by the number of “line pairs” resolvable per unit length (e.g., lp/mm). It is the physical “ceiling” of the imaging system's ability to capture details.

• Contrast: Refers to the degree of light and dark difference between bright and dark areas in an image, reflecting the distinctness of black-and-white boundaries, which is the foundation for details to be perceived.

• Sharpness: Refers to the steepness of edge transitions in an image. The steeper the edge change, the higher the sharpness, making it visually “sharper and clearer”.

• Clarity: Refers to the subjective perception result of the overall clearness of an image by the human eye, under the combined effects of resolution, sharpness, noise, contrast, and other factors.

II. Decoding the “Eye Chart”: MTF Curve Information
2.1 Axis Definitions:
Horizontal Axis (X-axis): Spatial Frequency
Represents the fineness of details. Common units: lp/mm (line pairs per millimeter), LW/PH (line widths per picture height), or cycles/pixel. It is like the progressively smaller optotypes on an eye chart.

Unit Conversion for Spatial Frequency:

Vertical Axis (Y-axis): MTF Value
Ranges from 0 to 1, representing the contrast transfer capability (1 indicates perfect reproduction, 0 indicates complete loss). It is like the clarity with which the human eye can see optotypes of different sizes.

2.2 Interpretation of Frequency Zones:
The spatial frequency components of an image include: low frequency, mid frequency, and high frequency. The MTF curve above intuitively demonstrates the law that modulation (MTF) gradually attenuates as spatial frequency increases:

• Low-frequency zone: Represents large-area, slowly changing regions in the image, reflecting the overall light and dark tone and macroscopic contours of the scene. In the curve, the sine wave amplitude in this zone is large, and the MTF value is close to 1, corresponding to strong overall image contrast and rich tonal gradation.
  
• Mid-frequency zone: Represents the transition of medium-scale spatial features in the image, reflecting the main shapes and structures of the scene. It is the key frequency band that determines the visual clarity of the imaging system. In the curve, the sine wave amplitude in this zone decreases, and the MTF value drops, corresponding to a decrease in image edge sharpness.
  
• High-frequency zone: Represents rapidly changing fine details in the image, such as complex textures and minute features. It is the key frequency band that determines the imaging system's ability to reproduce fine details. In the curve, the sine wave amplitude in this zone attenuates significantly. When the MTF value attenuates to the human eye's contrast sensitivity threshold (usually referring to MTF10, i.e., MTF=0.1), the visual limiting resolution of the system is reached.
  

In digital imaging systems, the transfer of high frequencies does not extend infinitely and is also physically limited by the sensor's pixel pitch. This limit is defined by the Nyquist frequency ($ f_{N}= 1 / (2 x PixelPitch)$), which is the theoretical boundary of digital sampling, representing the highest theoretical spatial frequency the system can reproduce. When the frequency of scene details exceeds this limit, the MTF not only drops to extremely low levels but also triggers aliasing phenomena, producing false colors or moiré patterns.

2.3 Common MTF Evaluation Metrics:
MTF50: Refers to the spatial frequency value when MTF is 50%. It has a strong correlation with the human eye's subjective perception of clarity and is the core metric for evaluating overall sharpness.
MTF50P: Refers to the spatial frequency when MTF drops to 50% of its peak value. Due to sharpening effects, its peak may exceed 1.
MTF10: Refers to the spatial frequency value when MTF is 10%. It is close to the lower limit of human eye detail recognition and is often used to evaluate “limiting resolution”.

2.4 Directional Analysis: Tangential and Sagittal
Tangential direction (Tangential, denoted as T): The intersection of the tangential plane (containing the optical axis and the image point) and the image plane is radial. Sampling in the tangential (T) direction moves along the radius, used to measure tangential lines. It is sensitive to astigmatism and field curvature, and the MTF curve attenuates faster.
Sagittal direction (Sagittal, denoted as S): The sagittal plane (perpendicular to the tangential plane) corresponds to the tangential direction. Sampling in the sagittal (S) direction moves along the circumference, used to measure radial lines. The MTF curve is flatter and better reflects the system's basic resolution.
Interpretation of T/S Curves:
Axis Definitions:
Horizontal axis: Represents the position on the image plane (from center 0mm to edge 20mm), reflecting imaging performance at different fields of view.
Vertical axis: Represents the MTF value (contrast transfer capability), ranging from 0 to 1. The higher the value, the clearer the image.
By measuring the MTF curve in both tangential (T) and sagittal (S) directions, the magnitude of astigmatism and the contrast transfer characteristics across the full field of view can be intuitively reflected: the higher the overlap of T/S, the smaller the astigmatism; high-frequency contrast transfer is weaker than low-frequency, and contrast gradually decreases from the center field to the edge field. The T curve attenuates faster, while the S curve is flatter. Bidirectional testing can comprehensively evaluate imaging quality and accurately diagnose directional aberrations.

III. Diagnosing “Poor Vision”: Causes of Imaging Blur
Image blur in digital cameras is the result of multiple factors, which can be mainly divided into three categories:
3.1 Imaging Hardware Level
(1) Geometric Aberrations and Diffraction of the Lens

Geometric schematic diagrams of third-order monochromatic aberrations. (a) No aberration (b) Spherical aberration © Coma (d) Astigmatism (e) Field curvature (f) Distortion

Among them, spherical aberration, coma, and astigmatism all prevent a point object from converging into an ideal point image, causing imaging blur. In addition, the different refractive indices of optical media for light of different frequencies cause differences in the imaging positions of different colored lights. This aberration is called chromatic aberration, which also leads to unclear imaging. Using lenses with complex surface shapes, lenses made of special materials, combining multiple lenses into a lens group, and more reasonable positions of the aperture stop can correct some aberrations. However, at the present stage, aberrations still cannot be completely eliminated.

Schematic diagram of the causes and phenomena of diffraction

As shown in Figure (a), when a plane wave passes through the aperture of an imaging system (such as a lens aperture), each point within the aperture can be regarded as a secondary wave source. When these secondary waves propagate to the image plane, they interfere and superimpose, failing to converge into an ideal geometric point. Under far-field (Fraunhofer) conditions, this interference forms the one-dimensional intensity distribution in Figure (b) and the Airy disk (circular aperture diffraction pattern) in Figure ©. The Airy disk causes the image of a point target to spread, with light energy diffusing into the geometric shadow area, ultimately producing diffraction-limited imaging blur.

(2) Pixel Aperture and Photoelectric Crosstalk of the Image Sensor

Schematic diagram of image blur caused by the aperture of the photosensitive element in a solid-state imager

Assuming the English character “W” is imaged on an 8×8 pixel array, each pixel can only output a single luminance value. This spatially averages the light intensity information within its aperture area, leading to the smoothing of character edges and details, thereby producing image blur.

Schematic diagram of the crosstalk generation mechanism in a solid-state imager

In CMOS imagers, electrical crosstalk caused by carrier diffusion in the quasi-neutral region and optical crosstalk caused by photons mistakenly entering adjacent pixels both lead to image blur. Both optical and electrical crosstalk can be improved through physical methods. For example, using a back-illuminated design can improve optical crosstalk, and adding isolation trenches between pixels can reduce electrical crosstalk. However, the influence of the geometric characteristics of the photosensitive element aperture always exists.

3.2 Image Processing Software Level

Blur caused by image processing is generally intentional (such as beauty filters and background blur for shallow depth of field effects) or a side effect of a certain process (such as blur caused by removing some details in the image while denoising).

3.3 Environmental Factors Level
When shooting distant targets, atmospheric turbulence and aerosols can cause atmospheric blur. Relative motion between the scene and the imager during exposure introduces motion blur. Stray light (non-imaging light) generated by strong light inside and outside the camera's field of view forms ghosting and glare, which reduces light and dark contrast and thereby causes image blur.

IV. Practicing the “Eye Exam”: MTF/SFR Testing Methods
4.1 Using Sinusoidal Patterns
Input a sinusoidal grating pattern (such as logarithmic frequency stripes or a Siemens star target) with luminance continuously changing according to a sinusoidal function into the imaging system. In the spatial domain, directly extract the peak and valley values at each frequency in the linearized image, calculate its modulation depth, and normalize it by taking the ratio with the original contrast (or low-frequency baseline) at the input end, thereby fitting a complete MTF curve.

Logarithmic frequency stripe target	Siemens star target

Advantages: The measurement principle is most consistent with the physical definition of MTF and is less affected by post-processing such as image sharpening; the Siemens star chart can measure MTF in all directions at once, making it suitable for astigmatism diagnosis and full-angle resolution evaluation. In addition, using aliasing phenomena such as “reverse contrast” or “false colors” appearing in the central area of the star chart can accurately detect the Nyquist frequency limit of the sensor.
Limitations: Low spatial utilization of the target, highly sensitive to focus accuracy, light source uniformity, and image noise, slow calculation speed, and usually requires noise subtraction or multi-frame averaging algorithms to ensure accuracy.
Applications: Mainly used for laboratory-grade optical baseline calibration, lens image quality evaluation, slanted-edge method result verification, and evaluating the true detail retention capability of the imaging system after strong ISP algorithm processing.

4.2 Using Edge Spread Function
4.2.1 Slanted-Edge Method
According to the latest ISO 12233 standard, a straight edge with a slight tilt angle (usually 5°) is photographed. The tilt of the edge can create different “phases”, forming an oversampled edge spread function to counteract the uncertainty introduced by the shift-variant nature of the discrete pixel array on the sensor. Then, the first derivative of the edge spread function is calculated to obtain the line spread function, which is then Fourier transformed and its modulus is taken to obtain the MTF curve. The calculation process is shown in the figure below:
Advantages: Complies with the ISO 12233 standard. It is recommended to use a 4:1 low-contrast edge to effectively suppress the artificially high MTF caused by post-processing sharpening; high spatial utilization, strong noise resistance, and high computational efficiency, making it suitable for mass production scenarios.
Limitations: Easily affected by lens distortion, causing the edge to change from a straight line to a curve, or making the actual tilt angle deviate from the set value, thereby reducing the accuracy of oversampling reconstruction.
Applications: Widely used in automated production line MTF testing and quality control for products such as mobile phones, automotive cameras, and industrial cameras, as well as for evaluating the resolution performance and algorithm verification of imaging systems in laboratories.

4.2.2 Circular Edge Method
As described in 4.2.1, although the slanted straight-edge method is suitable for mass production testing, it has two core limitations: first, a single straight edge can only measure the one-dimensional SFR in the direction perpendicular to the edge, and cannot simultaneously obtain the imaging performance in the two key directions of tangential and sagittal; second, it is easily affected by the distortion of wide-angle cameras, leading to edge bending or angle deviation, which in turn invalidates oversampling and makes it difficult to accurately characterize the true resolution in distorted areas.
To address these issues, the circular edge MTF testing method provides an effective solution. This method replaces the straight-edge target with a circular edge target. First, an ROI containing a complete circle is selected in the image and pixel values are linearized. Then, the center of the circle is located and a sector area in the target direction is selected. Subsequently, the arc edge is extracted and fitted. An oversampled edge spread function is constructed by calculating the distance from pixels to the edge curve. After derivation and windowed smoothing, a discrete Fourier transform is performed on the obtained line spread function, and its modulus is taken and normalized to obtain the circular edge SFR in the specified direction. The calculation process is shown in the figure below:
Applications: Suitable for mass production of wide-angle/fisheye automotive cameras, full-field multi-directional MTF measurement, and upstream/downstream measurement benchmarking.

4.3 Using Random Patterns
The commonly used test target is the dead leaves chart, which has the characteristics of a known spectrum and full spatial frequency coverage. This target is composed of randomly overlapping circular “coins” whose radii, grayscale, and positions follow specific probability distributions, possessing a 1/f scale-invariant spectral characteristic. Its design typically adopts a 3:1 low contrast (compliant with standards such as CPIQ) to simulate the weak texture features of natural scenes and avoid signal saturation.

• Testing Principle: The imaging system photographs the “dead leaves chart” target with known spectral characteristics, and a Fourier transform is performed on the captured image to extract its actual power spectral density (PSD). By calculating the ratio of this actual spectrum to the predefined theoretical spectral model of the target, the modulation transfer function (MTF) of the imaging system at different spatial frequencies can be inversely derived.
• Advantages: A single shot can obtain the full-band MTF curve, resulting in high efficiency; the statistical characteristics of the low-contrast random texture are extremely close to natural scenes, which can effectively bypass the interference of edge sharpening algorithms, and the evaluation results are more aligned with the human eye's visual perception of true detail fidelity.
  
• Limitations: The spatial utilization of the target is lower than that of the slanted-edge method; the calculation highly relies on accurate noise power spectrum (PSD) subtraction and texture extraction algorithms; it is extremely sensitive to noise, requiring multi-frame averaging to ensure stability; and it is easily affected by target manufacturing accuracy and image plane uniformity, with absolute measurement accuracy slightly lower than the high-contrast slanted-edge method.
  
• Applications: The black-and-white version focuses on the basic resolution of the system's luminance channel; the color version is the core means for diagnosing the loss of color details caused by non-linear denoising algorithms and evaluating color texture fidelity.
  

V. From Objective SFR to Subjective Perception: Acutance
SFR/MTF is an objective physical metric measuring the contrast transfer capability of an imaging system for details at different spatial frequencies, but it cannot directly characterize subjective visual sensations such as clarity and sharpness perceived by the human eye. Images are ultimately perceived by the human eye, which is not a linear receiving device. Under different viewing conditions, the sharpness perceived by the human eye may be completely different. Therefore, it is necessary to introduce an objective quantitative metric that aligns with the human eye's subjective perception of sharpness — Acutance.
5.1 Principle
According to the method in Annex L of ISO 12233:2023, Acutance uses the human eye's contrast sensitivity function (CSF) as a weighting basis to weight and integrate the objective spatial frequency response (SFR) data, ultimately obtaining a quantitative value correlated with subjective sharpness perception (represented by the Q value).

Its mathematical model is:
$$Q = \frac{\sum_{i=1}^{N} SFR_i \, CSF_i}{\sum_{i=1}^{N} CSF_i}, \quad i = 1, 2, \dots, N \tag{L.2}$$ Where \(SFR_i\) is the spatial frequency response value at the \(i\)-th spatial frequency; \(CSF_i\) is the human eye contrast sensitivity weight at the \(i\)-th spatial frequency; \(N\) is the number of spatial frequency points involved in the calculation.

5.2 Contrast Sensitivity Function CSF
The core of Acutance is the contrast sensitivity function CSF. Its model originates from the S-CIELAB perceptual color space and has been adopted by the ISO 12233 standard for perceptual weighting of the luminance channel. The CSF curve (see figure below) describes the human eye's sensitivity to contrast at different spatial frequencies:
Horizontal axis: Spatial frequency f, in units of cycles/degree, representing the fineness of image details.
Vertical axis: Contrast sensitivity. The higher the value, the easier it is for the human eye to distinguish contrast differences at that frequency.
The curve shows that the human eye is most sensitive to mid-to-low frequency details of about 4 cycles/degree, and the perception ability for extremely high and extremely low frequency details decreases significantly.
Its corresponding mathematical model is:
$$csf_{\text{lum}}(f) = \frac{(a \cdot f^c) e^{-bf}}{K} \tag{L.1}$$ Where the values of each parameter are explicitly specified in Annex L of ISO 12233:2023: $a = 75$, the amplitude coefficient; $b = 0.2$, the high-frequency attenuation coefficient; $c = 0.8$, the frequency power coefficient; $K = 102.16$, the normalization constant (making the peak of the curve 1.0); $f$ is the spatial frequency, in units of cycles/degree.

5.3 Viewing Conditions
The Acutance result is strongly correlated with the viewing conditions of the image: the same image may appear clear on a small display but blurry when enlarged or viewed on a large display. To ensure that Acutance values calculated by different devices and different laboratories are comparable, Annex L of ISO 12233:2023 explicitly defines three sets of standard viewing conditions (VC1/VC2/VC3) (see table below) to unify key parameters such as image display size, viewing distance, and pixel pitch. Before applying the CSF model to the SFR data, the data units must first be converted from cycles per pixel (cycles/pixel) to cycles per degree (cycles/degree). According to the Nyquist sampling theorem, the maximum frequency $f_{cut} $ that an image can represent is 0.5 cycles/pixel. To convert the frequency in units of “cycles per pixel” to the “cycles per degree” unit used for human eye contrast sensitivity calculations, the pixel size (i.e., pixel pitch p) and viewing distance D when the observer finally views the image must be known. This unit conversion is completed through formula (L.3): $$f_{\text{cycles/degree}} = \frac{\pi D}{180 p} f_{\text{cycles/pixel}} = \frac{\pi D N_H}{180 H} f_{\text{cycles/pixel}} \tag{L.3}$$

Where $N_H$ is the number of pixels in the vertical direction of the image; $H$ is the image height; $p$ is the pixel pitch of the display device; $D$ is the viewing distance.

VI. From Laboratory to Production Line: Test Equipment Solutions
6.1 Testing Process:

1. Environment Setup: According to the testing requirements of the Device Under Test (DUT), configure a standard light source with continuously adjustable color temperature, illuminance, and optional bands in a dark room. Set parameters according to testing needs (e.g., color temperature 6500K, chart center illuminance 800 lx) to simulate the target test scenario. Simultaneously, select standardized targets that comply with image quality testing specifications (such as the 4:1 contrast, 5° slanted-edge chart recommended by the ISO 12233 international standard) to provide an accurate theoretical reference for subsequent MTF calculations.
   
2. Image Acquisition: Securely fix the DUT using a dedicated fixture, connect supporting equipment, and turn on the power to bring it into normal working condition. Complete parameter configuration and locking according to the DUT output type to ensure stable and reproducible test conditions (e.g., RAW modules need to be switched to linear mode, hardware noise reduction, HDR, and other non-linear processing turned off, and key parameters such as exposure time and gain manually configured and locked; YUV modules need to complete 3A (auto exposure, auto focus, auto white balance) convergence, then lock 3A core parameters and related settings such as sharpening and noise reduction).
   
3. Software Analysis: Use professional image quality analysis software (such as RIQA software) to process the captured target images, calculate, and generate the MTF curves corresponding to the selected targets.
   
4. Result Interpretation: Analyze the MTF curves and key metrics (such as MTF50, MTF10, etc.), compare them with preset standards and customer technical requirements, and determine whether the DUT's imaging resolution and sharpness meet the standards.
   

6.2 Testing Systems

See More
Spatial Frequency Response SFR Test, CMS Sharpness Test, CMS Depth of Field Test