1. Background
The EMVA 1288 standard was initiated by the European Machine Vision Association (EMVA) in 2004 to address the pain points of inconsistent metrics and incomparable data among manufacturers in the industrial imaging field. Since the release of its first edition in 2005, the standard has translated complex camera performance into quantifiable physical parameters, providing a unified and objective measurement and analysis method for image sensor and camera performance testing.

Version Evolution:
The current latest version is Release 4.0, which officially took effect in June 2021, comprehensively replacing version 3.1 that focused on linear response devices.
Based on sensor response characteristics, Release 4.0 is divided into two core modules:
- Linear (Linear Model): The direct successor to version 3.1, applicable to traditional sensors with linear response and no built-in preprocessing (e.g., no interpolation, no noise reduction, no sharpening). It uses the Photon Transfer Method to derive quantum efficiency $\eta$, system gain $K$, and various noise parameters from the relationship between signal mean and variance.
- General (General Model): A brand-new branch applicable to various image sensors and cameras, compatible with both linear and non-linear imaging systems. Based on black-box theory, this model no longer relies on the linear response assumption. It achieves standardized performance characterization (e.g., SNR) of complex camera systems solely by analyzing the statistical properties of the output images.

Figure 1: Schematic diagram of the Linear Model and General Model

International Recognition:
Machine vision industry organizations such as AIA (USA), CMVU (China), EMVA (Europe), JIIA (Japan), and VDMA (Germany) have signed the G3 agreement for mutual recognition of standards, making EMVA 1288 the only internationally and domestically recognized standard for image sensor and camera performance testing in the industry, widely adopted by mainstream global sensor chip and camera manufacturers.

This article will follow the EMVA 1288 Release 4.0 standard to systematically analyze the testing methods for the Linear and General models.

2. Definitions of Core Parameters

1. Quantum Efficiency: The efficiency with which a sensor converts incident photons into photoelectrons, i.e., the photoelectric conversion efficiency. A higher value indicates a stronger response to light and better imaging performance under low-light conditions.
2. Absolute Sensitivity Threshold: The minimum average number of incident photons required to achieve a signal-to-noise ratio (SNR) of 1. It is a key metric for evaluating low-light performance. A smaller value indicates a stronger ability to extract useful information in extremely dark environments.
3. Saturation Capacity: The maximum number of electrons a single pixel can store before reaching charge saturation. It determines the maximum light-bearing capacity of the camera. A larger saturation capacity means the camera is less prone to overexposure under strong light and can achieve a higher maximum SNR.
4. System Gain (K): The scaling factor that converts one photoelectron (e⁻) into a digital number (DN), i.e., $K = \frac{\text{DN}}{\text{e}^-}$. It establishes a quantitative link between physical charge and the camera's output gray values, reflecting the combined characteristics of the camera's A/D converter and readout circuitry.
5. Maximum SNR: The extreme value of the signal-to-noise ratio when the signal reaches the saturation capacity, usually expressed in dB or bits. This value is determined by the square root of the saturation capacity and defines the upper limit of image quality under ideal lighting. It is the decisive metric for measuring the ability to distinguish minute contrast differences in high-brightness areas (highlight detail discrimination).
6. Dynamic Range (DR): The ratio of the saturation capacity to the absolute sensitivity threshold, defining the span of light intensities that the system can effectively distinguish. A larger dynamic range results in richer image tonal gradations in high-contrast scenes, allowing for simultaneous highlight suppression and shadow detail extraction.
7. Temporal Dark Noise: The root mean square (RMS) value of the temporal random noise generated jointly by the fluctuation of the photodiode dark current and the readout circuitry in sensor pixels under no-light conditions. A smaller value yields a higher SNR and effective contrast in low-light imaging (e.g., fluorescence microscopy, low-illumination inspection).
8. Dark Current: The average number of electrons generated per unit time by the photodiode (PD) due to thermal excitation under no-light conditions.
9. Dark Signal Non-Uniformity (DSNU): The spatial variation of dark signals among pixels under dark-field conditions. It originates from the inherent differences in pixel dark signals, primarily caused by deviations in sensor manufacturing processes, wafer material inhomogeneity, and individual differences in pixel circuit parameters (e.g., reset noise, dark current).
10. Photo-Response Non-Uniformity (PRNU): The spatial variation of photo-responsivity among pixels under uniform illumination, caused by inherent differences in pixel sensitivity. It is primarily due to process inconsistencies in photodiode quantum efficiency, pixel gain, and fill factor.

3. Core Test Items
The Linear Model is a white-box model that can measure electron-level physical parameters such as absolute quantum efficiency and system gain, and is only applicable to linear sensors without preprocessing. The General Model is based on black-box theory and cannot measure internal physical parameters such as absolute quantum efficiency and system gain; it can only characterize overall performance in units of photons, and is applicable to various non-linear cameras. Taking the General Model as an example, its test items are:
1. Sensitivity and Noise

• Mandatory: Dark noise, maximum SNR $\mathrm{SNR_{p,max}}$, absolute sensitivity threshold, saturation capacity, dynamic range (DR);
• Optional: Relative quantum efficiency (QE)

2. Dark Current

• Mandatory: Dark current calculated from the mean or variance at a single temperature;
• Optional: Temperature dependence of dark current (doubling temperature difference $T_d$).

3. Spatial Non-Uniformity

• Mandatory: Dark signal non-uniformity (DSNU), photo-response non-uniformity (PRNU), including full-frame, row, column, and pixel-level metrics.

4. Defective Pixels

• Optional: Characterization of outlier pixels via logarithmic histograms and cumulative histograms.

3. Test Methods
1. Quantum Efficiency
(1) Linear Model: Absolute Quantum Efficiency $\eta$
Absolute quantum efficiency $\eta$ (Release 4.0 Linear 6.6)
Refers to the physical proportion of incident photons converted into effective photoelectrons by the sensor at a specific wavelength:
$$ \eta = \frac{\mu_e}{\mu_p} \tag{1}$$ where $\mu_p$ is the mean number of photons incident on a single pixel during the exposure time; $\mu_e$ is the mean number of corresponding photoelectrons generated.
Based on the ideal linear photoelectric conversion assumption, the output gray values of the camera satisfy a linear relationship with the number of incident photons:
$$\mu_y = \mu_{y,\text{dark}} + K \eta \mu_p \tag{2}$$ where: $\mu_y$ is the mean output gray value in the bright field (DN); $\mu_{y,\text{dark}}$ is the mean output gray value in the dark field (DN); $K$ is the system gain (unit: $\text{DN}/e^-$).
Thus, the calculation formula for the absolute quantum efficiency $\eta$ can be derived as:$$\eta = \frac{\mu_y - \mu_{y,\text{dark}}}{K \mu_p} \tag{3}$$ Test Equipment: (Release 4.0 Linear 9.1)
Broadband light source + monochromator, or a multi-band light source with switchable wavelengths. The light source must meet the specified geometric requirements: a light source with diameter D must be placed at a distance of d=8D.

Test Conditions: (Release 4.0 Linear 9.2)

1. Sensor area: The sensor area used for testing must be clearly specified;
2. Bandwidth: The FWHM of the light source < 10 nm; only monochrome sensors may relax this to < 50 nm under technical constraints. Color sensors cannot be relaxed, and the actual bandwidth used must be noted in the report. For monochrome cameras, measurement is only required within a single band with a bandwidth ≤ 50 nm; for color cameras, tests must be performed separately for all color channels, and the absolute quantum efficiency for all corresponding bands must be reported.
3. Wavelength range: The scanned wavelength range should cover the entire response band of the sensor. Typically, this range includes at least 350 nm to 1100 nm; for UV-sensitive sensors, the wavelength range must be extended to shorter bands, typically down to 200 nm. Measurement points require an adjacent wavelength interval ≤ 2 × FWHM to ensure gap-free coverage across the entire band.
4. Signal range: The exposure time should be set to ensure a sufficiently large effective signal at all wavelengths. If the source irradiance fluctuates significantly at different wavelengths, multiple spectral scans with different exposure times may be required. A dark image must be captured before and after each wavelength scan, and for each exposure time used.

Test Method:

• Camera settings: Reach thermal equilibrium, use the highest bit depth, use the lowest gain, set the offset to < 0.5% pixel underflow (see Release 4.0 Linear 6.5 for details);
• Light source calibration: Replace the sensor with a calibrated photodiode, measure the irradiance E at each wavelength, calculate the number of incident photons $\mu_p$, and establish a spectral calibration curve (see Release 4.0 Linear 9.3 for details).
• Image acquisition: Acquire bright and dark field images with graded exposures at a single wavelength (for calculating system gain K); acquire a set of bright and dark field images at each selected test wavelength (for calculating absolute quantum efficiency).
• Data analysis: For each test wavelength, after subtracting the dark signal, divide sequentially by the system gain K (calibrated at the same operating point) and the spectrally calibrated number of incident photons $\mu_p(\lambda)$ to calculate the absolute quantum efficiency (see Equation (3)); for color cameras, all color channels must be evaluated separately.
  

Spectral sensitivity curve of a 12-bit monochrome camera	Spectral sensitivity curve of a 12-bit color camera

(2) General Model: Relative Quantum Efficiency $\eta$
Constrained by the black-box model and the unknown system gain K, it is impossible to directly measure the number of accumulated charges ${\mu_e}$ within the pixel, and thus the absolute quantum efficiency cannot be obtained. Instead, only the relative quantum efficiency relative to a reference wavelength can be measured through spectral scanning. Except for the data analysis method, the test conditions, test equipment, and test procedures for this model are identical to those of the Linear Model.
Data Analysis:
For each selected wavelength, a cubic B-spline regression must be performed on the inverse characteristic curve of the reference wavelength to obtain the conversion relationship between gray values and the number of photons: $$\mu_p = R^{-1}\left( \mu_y - \mu_{y.\text{dark}} \right) \tag{4}$$ $\mu_p$ is the mean number of incident photons; $R^{-1}$ is the inverse response characteristic function; $\mu_y$ is the mean bright field gray value; $\mu_{y.\text{dark}}$ is the mean dark field gray value.
Calculate the spectral gray value average $\mu_y(\lambda)$ at that wavelength, use the aforementioned inverse characteristic curve to convert the gray values into a linearized input signal, and substitute into Equation (5) to calculate the relative quantum efficiency relative to the reference wavelength $\lambda_{\text{ref}}$: $$\frac{\eta(\lambda)}{\eta(\lambda_{\text{ref}})} = \frac{R^{-1}\left(\mu_y(\lambda) - \mu_{y.\text{dark}}\right)}{R^{-1}\left(\mu_y(\lambda_{\text{ref}}) - \mu_{y.\text{dark}}\right)} \cdot \frac{\mu_p(\lambda)}{\mu_p(\lambda_{\text{ref}})}\tag{5}$$

2. Dark Current
Dark current ($\boldsymbol{\mu_I}$) refers to the number of thermally generated electrons per unit time per pixel in the sensor under completely dark and constant temperature conditions. It increases linearly with exposure time and exponentially with temperature; therefore, dark current is highly temperature-dependent and must be measured at different ambient temperatures.
Test Conditions:

• Number of measurement points: Acquire at least 6 different exposure times (an equal-interval exposure time scheme is recommended); acquire at least 2 dark images at each exposure time to reduce the impact of temporal random noise;
  
• Exposure time: Since dark current increases linearly with exposure time, to obtain stable and reliable values, prioritize using the maximum exposure time recommended by the manufacturer;
  
• Temperature-controlled environment: When measuring the temperature dependence of dark current, the camera must be placed in a high-precision temperature control system (such as a climate chamber) for precise temperature control; the test must be conducted in a completely light-tight dark box environment to ensure zero illumination conditions;
  
• Temperature range: Must cover the operating temperature range of the camera;
  
• Thermal equilibrium requirement: At each temperature point, wait until the camera reaches thermal stability before starting acquisition to avoid the impact of temperature drift on the results.
  

Test Methods:
① Dark Current Evaluation at a Single Temperature: (Release 4.0 7.1)
Dark current measurement must be performed under dark conditions. It can be calculated from the mean or variance of the dark field gray values $y_{dark}$ as they increase linearly with exposure time. When dark current compensation is not applied, both methods are acceptable, but the mean method is preferred because the estimation accuracy of the mean is much higher than that of the variance. If the camera features dark current compensation, the dark current must be calculated using the variance method.

The dark current is the slope of the curve relating the exposure time t to the mean/variance of the dark values. By performing linear least squares regression on the measured mean $\bar{y}_{\text{dark}}$ or variance $\sigma^2_{y.\text{dark}}$ of the dark signal, the dark current slopes $\mu_{I,y}$ in $\text{DN/s}$ and $\mu_{I,y,\text{var}}$ in $\text{DN}^2/\text{s}$ can be obtained, respectively.

Linear Model:
Unit conversion is completed using the camera gain $K$, and the corresponding dark current is:
$\mu_I = \frac{\mu_{I,y}}{K} \quad$ (mean method, unit $\mathrm{e^-/s}$) , $\mu_{I,\text{var}} = \frac{\mu_{I,y,\text{var}}}{K^2}$ (variance method, unit $\mathrm{e^-/s}$)

General Model:
Unit conversion is completed using the zero-exposure slope of the characteristic curve $R_d$, and the corresponding dark current is:
$\mu_I = \frac{\mu_{I,y}}{R_d}$ (mean method, unit $\mathrm{p/s}$) and $\mu_{I,\text{var}} = \frac{\mu_{I,y,\text{var}}}{R_d^2}$ (variance method, unit $\mathrm{p/s}$).

Linear fitting of dark signal mean versus exposure time (mean method)	Linear fitting of dark signal variance versus exposure time (variance method)

② Dark Current Evaluation at Different Temperatures: (Release 4.0 7.2)
The doubling temperature difference ($T_d$) of the dark current is obtained by measuring the dark current at different housing temperatures and fitting an exponential model. The test range must cover the entire operating temperature range of the camera, and at least 6 equally spaced temperature points must be set within this range. Cover the camera's sensor port, place it in a temperature-controlled chamber or other temperature control equipment, and after the temperature changes and the camera parameters stabilize (which can be verified by continuously monitoring the dark values at the maximum exposure time) and thermal equilibrium is reached, measure the dark current at each temperature point according to the single-temperature method. For cameras with integrated cooling functions, no additional temperature chamber is needed, and the temperature dependence test is performed within the normal operating temperature range of the camera.
Calculation Method:
The dark current increases exponentially with temperature. Based on the $\mu_I$ data at multiple temperature points, fit the exponential model: $$\mu_I = \mu_{I,\text{ref}} \cdot 2^{(T-T_{\text{ref}})/T_d}$$ to obtain the reference temperature dark current $\mu_{I,\text{ref}}$ and the doubling temperature difference $T_d$.

3. Temporal Dark Noise
Linear Model:
Temporal dark noise must be measured at the camera's minimum exposure time to obtain the lowest baseline temporal dark noise; at higher exposure times, the temporal dark noise will increase linearly with exposure time. Two core metrics need to be reported:

• Temporal dark noise standard deviation $\sigma_{y,\text{dark}}$: Unit is DN, reflecting the actual output performance of the camera at the current bit depth, including quantization noise;
• Corrected temporal dark noise $\sigma_d$: Unit is $e^-$, representing the sensor baseline noise excluding the influence of quantization noise. The calculation formula is: $\sigma_d = \frac{\sqrt{\sigma_{y,\text{dark}}^2 - \sigma_q^2}}{K}$. If $\sigma_{y,\text{dark}}$ < 0.24, it indicates that the temporal noise is dominated by quantization noise and a reliable estimate cannot be obtained. In this case, $\sigma_{y,\text{dark}}$ should be set to 0.49, and the upper limit of the temporal dark noise is: $\sigma_d < \frac{0.40}{K}$.

General Model:

• Temporal dark noise standard deviation $\sigma_{y,\text{dark}}$: Unit is DN, including quantization noise. For the method of changing the sensor's irradiance exposure by constant illumination + variable exposure time, the temporal dark noise is obtained from the intercept of the fitting relationship of $\sigma^2_{y,\text{dark}}$ versus exposure time. For the methods of changing the sensor's irradiance exposure by variable continuous illumination + constant exposure time, or pulsed illumination + constant exposure time, an additional measurement must be performed at the minimum exposure time to estimate $\sigma_{y,\text{dark}}$.
• Corrected temporal dark noise $\sigma_d$: Unit is photons $p$, representing the dark noise metric excluding the influence of quantization noise. This model cannot be calculated in units of electrons ($e^-$) and must be converted using the dark responsivity $R_d$. The calculation formula is: $\sigma_d = \frac{\sqrt{\sigma_{y,\text{dark}}^2 - \sigma_q^2}}{R_d}$. If $\sigma_{y,\text{dark}}^2 < 0.24$, it indicates that the temporal noise is dominated by quantization noise and a reliable estimate cannot be obtained. In this case, $\sigma_{y,\text{dark}}$ should be set to $0.49$, and the upper limit of the temporal dark noise (in photon units) is: $\sigma_d < \frac{0.40}{R_d}$.

4. Absolute Sensitivity Threshold
The absolute sensitivity threshold refers to the minimum number of incident photons (or equivalent electrons) per pixel when the signal-to-noise ratio (SNR=1) of the sensor output signal is 1, characterizing the weakest light detection limit of the camera system.
Linear Model:
In units of photons:
This metric is strongly influenced by the quantum efficiency $\eta$ and is only suitable for horizontal comparison within the same wavelength and the same type of sensor. Its calculation formula is as follows:
$$\mu_{p,\text{min}} = \frac{1}{\eta} \left( \sqrt{\sigma_d^2 + \sigma_q^2/K^2 + 1/4} + \frac{1}{2} \right) = \frac{1}{\eta} \left( \sqrt{\sigma_{y,\text{dark}}^2/K^2 + 1/4} + \frac{1}{2} \right) \ge \frac{1}{\eta}.$$ In units of electrons:
This metric does not include the influence of quantum efficiency $\eta$ and directly reflects the minimum electron detection capability at the physical level of the sensor. It is preferred for performance comparison across devices and wavelengths. Its calculation formula is as follows:
$$\mu_{e,\text{min}} = \left( \sqrt{\sigma_d^2 + \sigma_q^2/K^2 + 1/4} + \frac{1}{2} \right) \ge 1.$$ The electron unit threshold can be converted to the photon unit threshold via the quantum efficiency $\eta$:
$\mu_{p,\text{min}} \equiv \frac{\mu_{e,\text{min}}}{\eta}$

General Model:
The number of photons required for a signal-to-noise ratio of 1 is determined by numerical interpolation. A uniform illumination light source (such as an integrating sphere light source) is required to provide uniform surface illumination. Bright-field and dark-field test images are captured at graded exposure levels to calculate the output signal-to-noise ratio $SNR_{y}$, which is then converted to the input photon signal-to-noise ratio $SNR_{p}$ using the derivative of the characteristic curve. A cubic B-spline fit is applied to the inverse signal-to-noise relationship $μ_{p}$($SNR_{p}$) in a log-log coordinate system, and the exposure corresponding to $SNR_{p}=1$ on the fitted curve is finally identified as the absolute sensitivity threshold $μ_{p,min}$. (See Release 4.0 General 6.8 for detailed calculations)

Input Signal-to-Noise Ratio ($\text{SNR}_p$) Curve

The curve shows the variation of $\text{SNR}_p$ with exposure after cubic B-spline fitting of the measured data. The vertical dashed lines indicate the absolute sensitivity threshold (the number of photons corresponding to $\text{SNR}_p=1$) and the saturation capacity, respectively.

5. Saturation Capacity

• Saturation Irradiance $\boldsymbol{\mu_{p,\text{sat}}}$ (in photons): The maximum number of incident photons when the sensor output signal shows no significant clipping and the mean/variance calculations are unbiased. This metric can be directly obtained from both the linear and general models, corresponding to the number of incident photons at the maximum measurable digital value $\mu_{y,\text{sat}}$ at the output;
  

* Saturation Capacity $\boldsymbol{\mu_{e,\text{sat}}}$ (in electrons): Can only be calculated using the linear model, converted via the quantum efficiency $\eta$ with the formula: $\mu_{e,\text{sat}} = \eta \mu_{p,\text{sat}}$, directly reflecting the physical full-well capacity of the sensor pixels.
{Note}: Saturation capacity ≠ full-well capacity. The former refers to saturation in the digital domain (limited by the camera's maximum quantization value $2^k-1$, requiring statistical unbiasedness) and is typically lower than the physical full-well capacity of the pixel (the maximum number of electrons the pixel itself can hold).

6. Maximum Signal-to-Noise Ratio $SNR_{max}$
Linear Model:
Using a uniform monochromatic light source from an integrating sphere, multiple exposure RAW images are captured from dark field to saturation to calibrate the system gain $K$, quantum efficiency $\eta$, and dark noise $\sigma_d$. The unbiased saturation point is located and the saturation capacity in electrons $\mu_{e,\text{sat}}$ is calculated. Based on the formula $\boldsymbol{\mathrm{SNR} = \frac{\mu_y - \mu_{y,\text{dark}}}{\sigma_y}}$, the measured signal-to-noise ratio over the full exposure range is calculated, and the theoretical curve is fitted using the formula $\boldsymbol{\mathrm{SNR}(\mu_p) = \frac{\eta \mu_p}{\sqrt{\sigma_d^2 + \sigma_q^2/K^2 + \eta \mu_p}}}$ to verify the sensor linearity and noise characteristics. The measured SNR at the saturation point is extracted, and the theoretical limit is calculated using the formula $\boldsymbol{\mathrm{SNR_{max}} = \sqrt{\mu_{e,\text{sat}}}}$. The measured and theoretical values are compared to finally issue a compliant result in accordance with the standard.

7. Dynamic Range $DR$
The dynamic range is the ratio of the number of photons at the saturation upper limit (saturation irradiance $\mu_{p,\text{sat}}$) to the sensitivity lower limit (absolute sensitivity threshold $\mu_{p,\text{min}}$), calculated as:$$\boldsymbol{\mathrm{DR} = \frac{\mu_{p,\text{sat}}}{\mu_{p,\text{min}}}}$$ The unit is typically expressed as a numerical ratio and in decibels ($20\log_{10}\mathrm{DR}$).

8. Spatial Non-Uniformity
Spatial non-uniformity (also known as fixed-pattern noise) originates from the inherent differences in sensitivity and gain among pixels in the image sensor pixel array, manifesting as spatial inconsistency in pixel response. Its core characteristic is that it remains fixed and consistent across different image frames, which is fundamentally different from random temporal noise that varies from frame to frame. For linear sensors, this difference manifests as each pixel's characteristic curve potentially having a different offset and slope, and can be divided into two forms: Dark Signal Non-Uniformity (DSNU) and Photo-Response Non-Uniformity (PRNU).

• Dark Signal Non-Uniformity (DSNU): The spatial variation of dark signals among pixels under dark-field conditions.
• Photo-Response Non-Uniformity (PRNU): The spatial variation of photo-responsivity among pixels under bright-field conditions.

Four Typical Effects:
Spatial non-uniformity is more difficult to describe than temporal noise because it is not entirely random. To adequately describe it, the following four typical effects must be considered:
① Gradual Variations: Manufacturing defects produce low-frequency, slow variations across the entire chip, and lens vignetting and non-uniform illumination also introduce additional gradual variations. This effect is difficult to measure, imperceptible to the human eye, and has a limited impact on image quality. Therefore, for applications requiring a flat response across the entire sensor array, gradual variations must be corrected through the complete imaging system.
② Periodic Variations and Spatial Patterns: Manifests as regular spatial patterns such as columns and rows. The human eye is highly sensitive to these, and they can easily interfere with image processing operations.
③ Outliers: Refers to individual pixels or pixel clusters whose signal performance deviates significantly from the mean.
④ Random Variations: If the spatial non-uniformity exhibits a purely random distribution, meaning there is no spatial correlation, its power spectrum is flat (referred to as a white power spectrum), and its variations are uniformly distributed across all wavenumbers.

Light Source Equipment Selection:
An integrating sphere light source is preferred, meeting f=8 and irradiance non-uniformity ΔE≤3%, using a quasi-monochromatic (FWHM < 10nm) light source with a wavelength matching the sensor's peak response. If an integrating sphere is unavailable, a highly uniform area light source can be used as a substitute.

Test Method:
Multiple frames of dark-field images are captured under completely light-shielded conditions and the pixel mean is calculated to compute DSNU. Under uniform light source illumination, the exposure is adjusted so that the image output gray level is approximately 50% of the sensor's saturation gray level, and multiple frames of bright-field images are captured and the pixel mean is calculated to compute PRNU. After suppressing temporal noise through multi-frame averaging, the total spatial variance is calculated for the dark-field and bright-field mean images, respectively, and decomposed into row, column, and pixel-level spatial variances according to the standard, to comprehensively characterize the spatial response consistency of the imaging system.

9. Defect Pixels
Due to varying application requirements, the standard does not provide a unified threshold for defect pixel determination. Instead, it presents pixel response characteristics through statistical distributions (logarithmic histograms and cumulative logarithmic histograms), allowing users to define defect criteria based on their specific application scenarios.
Logarithmic Histogram:
The logarithmic histogram is used to display the distribution of pixel response values in a single or averaged image. The horizontal axis represents the pixel response value x, and the vertical axis represents the pixel count on a logarithmic scale, allowing for clear observation of anomalous pixels that account for an extremely small proportion.

Figure a shows the histogram of a single image. The Model region (cyan) represents normal pixels that conform to the expected response model; the Deviation region (yellow) represents pixels whose response values deviate somewhat from the model, which may exhibit slightly inferior performance but generally meet the requirements of basic applications; Outliers (blue vertical lines) represent pixels whose response values significantly deviate from the model, i.e., potential defective pixels. By observing the position and quantity of outliers, the severity of the defective pixels can be preliminarily assessed.
Figure b compares the averaged image with a single image. The black curve, obtained by averaging multiple images, eliminates the influence of temporal noise and better reflects the spatial response characteristics of the pixels; the curve in the cyan region encompasses the total response of both temporal and spatial noise. Comparing the differences between the two curves allows the separation of fluctuations caused by temporal noise from response deviations caused by inherent pixel defects.

Cumulative Logarithmic Histogram:
The cumulative logarithmic histogram displays the cumulative distribution of the absolute deviation of pixel response values from the mean $\mu_x$. The horizontal axis represents the absolute deviation of the pixel response from the mean $\mu_x$, and the vertical axis represents the cumulative percentage, which is used to more precisely define the defect threshold.
In the figure, the “Stop Band” is the maximum acceptable deviation threshold set by the user according to the application scenario: to its left are normal pixels (including the Model and Deviation regions), and to its right are defective pixels exceeding the deviation threshold (the Outlier region). The cumulative percentage on the vertical axis corresponding to the pixels to the right of the Stop Band represents the defective pixel rate of the imaging system; by adjusting the position of the Stop Band to the left or right, it can be flexibly adapted to the defect tolerance requirements of different scenarios such as scientific imaging, consumer imaging, and automotive vision.