12 Figure 2 – Geometry for exposing the DIGITAL X-RAY IMAGING DEVICE in order to determine the CONVERSION FUNCTION, the NOISE POWER SPECTRUM and the MODULATION TRANSFER FUNCTION behind t
Operating conditions
The DIGITAL X-RAY IMAGING DEVICE shall be stored and operated according to the
Follow the manufacturer's recommendations for warm-up time, ensuring that the operating conditions align with those intended for clinical use, including maintaining the specified frame rate throughout the evaluation process as required for the specific tests outlined.
Ambient climatic conditions in the room where the DIGITAL X-RAY IMAGING DEVICE is operated shall be stated together with the results.
X- RAY EQUIPMENT
For all tests described in the following subclauses, a CONSTANT POTENTIAL HIGH-VOLTAGE GENERATOR shall be used (IEC 60601-2-7) The PERCENTAGE RIPPLE shall be equal to, or less than, 4
The NOMINAL FOCAL SPOT VALUE (IEC 60336) shall be not larger than 1,2
For the measuring of AIR KERMA, calibrated RADIATION METERS shall be used The uncertainty (coverage factor 2) [2] of the measurements shall be less than 5 %
NOTE 1 “Uncertainty” and “coverage factor” are terms defined in the ISO/IEC Guide to the expression of uncertainty in measurement [ 2 ]
NOTE 2 R ADIATION METERS to read AIR KERMA are, for instance, calibrated by many national metrology institutes.
R ADIATION QUALITY
The RADIATION QUALITIES shall be one or more out of four selected RADIATION QUALITIES specified in IEC 61267:1994 (see Table 1) If only a single RADIATION QUALITY is used,
RADIATION QUALITY RQA5 should be preferred
For the application of the RADIATION QUALITIES, refer to IEC 61267:1994
The current edition of IEC 61267 will continue to reference IEC 61267:1994 to maintain harmonization within the IEC 62220 family, despite the availability of a more recent version Furthermore, the 2005 edition imposes stringent requirements on the practical implementation of radiation qualities, which are deemed unnecessary for the intended application of this standard.
According to IEC 61267:1994, radiation qualities are characterized by a specific additional filtration and a half-value layer achieved through this filtration, which is adjusted by an appropriate adaptation of the X-ray tube voltage, beginning from the approximate X-ray tube voltage as outlined in Table 1.
Table 1 – R ADIATION QUALITY (IEC 61267:1994) for the determination of DETECTIVE QUANTUM EFFICIENCY and corresponding parameters
Approximate X- RAY TUBE VOLTAGE kV
H ALF - VALUE LAYER (HVL) mm Al
NOTE 3 The additional filtration is the filtration added to the inherent filtration of the X- RAY TUBE
X-ray generators may struggle to produce adequate low air kerma levels for RQA9 To address this issue, it is advisable to increase the distance between the focal spot and the detector surface.
T EST DEVICE
The TEST DEVICE designed to determine the MODULATION TRANSFER FUNCTION will feature a tungsten plate that is 1.0 mm thick, with a purity exceeding 90% The plate will measure 100 mm in length and at least 75 mm in width If the tungsten purity is insufficient, the thickness of the plate will be increased to compensate.
The tungsten plate serves as an edge test device, requiring the edge used for test irradiation to be meticulously polished to a straight finish at a 90° angle to the plate When X-rays irradiate the edge in contact with a screenless film, the resulting image must not display ripples larger than 5 μm on the edge.
The tungsten plate must be securely attached to a 3 mm thick lead plate, as illustrated in Figure 1 This configuration is ideal for measuring the Modulation Transfer Function (MTF) of the digital X-ray imaging device in a single direction.
NOTE The TEST DEVICE consists of a 1,0 mm thick tungsten plate (1) fixed on a 3 mm thick lead plate (2)
Dimension of the lead plate: a : 200 mm, d : 70 mm, e : 90 mm, f : 100 mm
Dimension of the tungsten plate: 100 mm × 75 mm
The region of interest (ROI) used for the determination of the MTF is defined by b × c , 50 mm × 100 mm (inner long dashed line)
The irradiated field on the detector (outer dashed line) is at least 160 mm × 160 mm
Geometry
The measuring arrangement must adhere to the geometric configuration illustrated in Figure 2 The X-ray equipment operates in this setup similarly to its use in standard diagnostic applications The distance from the focal spot of the X-ray tube is a critical factor in this arrangement.
The DETECTOR SURFACE must be at least 1.50 m If technical constraints prevent achieving this distance, a shorter distance may be selected, but it must be clearly stated when reporting the results.
The REFERENCE AXIS shall be aligned with the CENTRAL AXIS
To ensure accurate measurements, position the TEST DEVICE directly in front of the DETECTOR SURFACE, aligning the center of its edge with the REFERENCE AXIS of the X-ray beam Any displacement from this REFERENCE AXIS will result in a decreased measured Modulation Transfer Function (MTF) The optimal location of the REFERENCE AXIS can be determined by maximizing the MTF in relation to the displacement of the TEST DEVICE.
For optimal results, it is essential to center the TEST DEVICE within the X-ray field on the detector If centering is not achieved, it is important to specify the positions of both the center of the X-ray field and the TEST DEVICE.
In the set-up of Figure 2, the DIAPHRAGM B1 and the ADDED FILTER shall be positioned near the
The focal spot of the X-ray tube is crucial for accurate measurements While diaphragms B2 and B3 are recommended for use, they can be omitted if it is demonstrated that their absence does not affect the measurement outcomes.
The diaphragms B1 and, if applicable, B2, along with the added filter, must maintain a fixed position relative to the focal spot Additionally, diaphragm B3, if applicable, and the detector surface should also be consistently aligned at each distance from the focal spot Specifically, the square diaphragm B3, when applicable, should be positioned 120 mm in front of the detector surface and sized to ensure an irradiated field of at least 160 mm × 160 mm at the detector surface.
The RADIATION APERTURE of DIAPHRAGM B2 can be adjusted to ensure that the beam remains tightly collimated with varying distances Additionally, the irradiated field at the DETECTOR SURFACE must measure a minimum of 160 mm × 160 mm.
The diaphragms must possess attenuating properties that prevent their transmission into shielded areas from affecting measurement results Additionally, the radiation aperture of diaphragm B1 should be sufficiently large to ensure that the penumbra of the radiation is adequately managed.
BEAM will be outside the sensitive volume of the monitor detector R1 and the RADIATION APERTURE of DIAPHRAGM B2 – if applicable
To ensure the accuracy of the X-ray generator, a monitor detector is essential The monitor detector R1 should ideally be positioned within the beam that irradiates the detector surface, provided it is sufficiently transparent and free of obstructions If not, it must be placed outside the beam area that passes through aperture B3 The monitor detector's precision, indicated by a standard deviation of 1σ, must exceed 2% Additionally, the correlation between the monitor reading and the air kerma at the detector surface needs to be calibrated for each radiation quality utilized.
To minimize the effect of back-scatter from layers behind the detector, a minimum distance of
500 mm to other objects should be provided
The calibration of the monitor detector is highly sensitive to the placement of the added filter and the adjustment of the shutters in the X-ray source Consequently, any changes to these components necessitate a re-calibration of the monitor detector to ensure accurate readings.
This geometry is used either to irradiate the DETECTOR SURFACE uniformly for the determination of the CONVERSION FUNCTION and the NOISE POWER SPECTRUM or to irradiate the
The DETECTOR SURFACE located behind a TEST DEVICE must be consistently irradiated during all measurements It is essential to document the center of this irradiated area in relation to either the center or the border of the digital X-ray detector.
All measurements shall be made using the same geometry
For the determination of the NOISE POWER SPECTRUM and the CONVERSION FUNCTION, the TEST DEVICE shall be moved out of the beam
NOTE The TEST DEVICE is not used for the measurement of the CONVERSION FUNCTION and the NOISE POWER SPECTRUM
Figure 2 – Geometry for exposing the DIGITAL X- RAY IMAGING DEVICE in order to determine the CONVERSION FUNCTION , the NOISE POWER SPECTRUM and the MODULATION TRANSFER
FUNCTION behind the TEST DEVICE
I RRADIATION conditions
General conditions
Before conducting any tests, it is essential to calibrate the digital X-ray detector, ensuring that all necessary corrections outlined in Clause 5 are implemented All measurements must be completed without re-calibration, although offset calibrations are permitted and can be performed as they would be in standard clinical practice.
The AIR KERMA level for a digital X-ray detector should be set to the "normal" level used in clinical practice Additionally, two other AIR KERMA levels must be established: one at 3.2 times the normal level and another at 1/3.2 of the normal level It is important to note that no adjustments to the settings of the DIGITAL X-RAY IMAGING DEVICE, such as gain, are permitted when changing AIR KERMA levels within a single Imaging Mode.
NOTE A factor of three in the AIR KERMA above and below the “normal” level approximately corresponds to the bright and dark parts within one clinical radiation image
Depending on the intended clinical use of the digital X-ray detector, one or more of the following Imaging Modes with their corresponding “normal” levels shall be chosen:
Imaging Mode1, Fluoroscopy “normal” level 20 nGy ± 10 %
Imaging Mode2, Cardiac imaging “normal” level 200 nGy ± 10 %
Imaging Mode3, Series exposures “normal” level 2 000 nGy ± 10 %
In each Imaging Mode, the settings of the DIGITAL X-RAY IMAGING DEVICE must remain constant When switching to a different Imaging Mode, new settings can be selected, but they should also be maintained consistently within that mode Additionally, "normal" levels may be selected as needed.
The variation of AIR KERMA shall be carried out by variation of the X-RAY TUBE CURRENT or the
IRRADIATION TIME or both The IRRADIATION TIME level shall be similar to the conditions for clinical application of the digital X-ray detector
The IRRADIATION conditions shall be stated together with the results (see Clause 7)
The RADIATION QUALITY shall be assured when varying the X-RAY TUBE CURRENT or the
IRRADIATION TIME and shall be checked at the lowest AIR KERMA level.
AIR KERMA measurement
The AIR KERMA at the DETECTOR SURFACE is assessed using a suitable RADIATION METER, with the digital X-ray detector temporarily removed from the beam The RADIATION DETECTOR of the RADIATION METER is positioned behind APERTURE B3 in the DETECTOR SURFACE plane, ensuring minimal back-SCATTERED RADIATION It is essential to note the correlation between the RADIATION METER readings and any monitoring detector used, as this data will aid in calculating the AIR KERMA at the DETECTOR SURFACE for determining the CONVERSION FUNCTION, NOISE POWER SPECTRUM, and MODULATION TRANSFER FUNCTION Given the requirement for numerous image exposures, it is advisable to measure the total AIR KERMA, including stabilization images, and divide this figure by the number of images exposed.
NOTE 1 To reduce back- SCATTERED RADIATION , a lead screen of 4 mm in thickness may be placed 450 mm behind the RADIATION DETECTOR It has been proven by experiments that, under these conditions, the back- SCATTERED RADIATION is not more than 0,5 % If the lead screen is at a distance of 250 mm, the back- SCATTERED RADIATION is not more than 2,5 %
If it is not possible to remove the digital X-ray detector out of the beam, the AIR KERMA at the
The DETECTOR SURFACE can be determined using the inverse square distance law, which involves measuring the AIR KERMA at various distances from the FOCAL SPOT in front of the DETECTOR.
SURFACE For this measurement, radiation, back-scattered from the DETECTOR SURFACE, shall be avoided Therefore, a minimum distance between the DETECTOR SURFACE and the
RADIATION DETECTOR of 450 mm is recommended
When utilizing a monitoring detector, it is essential to plot the equation as a function of the distance \( d \) between the focal spot and the radiation detector This relationship is crucial for accurately interpreting the radiation readings obtained from the detector.
By extending the nearly linear curve to the distance between the focal spot and the detector surface, denoted as \( r_{SID} \), we can determine the ratio of readings at \( r_{SID} \) This allows for the calculation of the air kerma at the detector surface based on any monitoring detector reading.
When a monitoring detector is not utilized, the square root of the inverse RADIATION METER reading is graphed against the distance from the FOCAL SPOT to the RADIATION DETECTOR The extrapolation process follows the same method as described previously.
NOTE 2 To reduce back- SCATTERED RADIATION , a lead shield of 4 mm thickness may be placed in front of the DETECTOR SURFACE
LAG EFFECTS
LAG EFFECTS influence the measurement of the NOISE POWER SPECTRUM They therefore, influence the measurement of the DETECTIVE QUANTUM EFFICIENCY
During standard clinical use, LAG EFFECTS are inherently present in digital X-ray detectors These effects will be assessed, and the estimated NOISE POWER SPECTRUM will be adjusted accordingly to produce a LAG EFFECT corrected NOISE POWER SPECTRUM Importantly, separate image acquisitions for measuring the LAG EFFECT are not required, as they will be integrated with the necessary image acquisitions for determining the NOISE POWER SPECTRUM For additional background information, refer to sources [11, 12, and 13].
I RRADIATION to obtain the CONVERSION FUNCTION
The DIGITAL X-RAY IMAGING DEVICE settings must match those utilized for the TEST DEVICE exposure IRRADIATION should be performed following the geometry depicted in Figure 2, ensuring no TEST DEVICE is present in the beam AIR KERMA measurements are to be conducted as outlined in section 4.6.2.
CONVERSION FUNCTION shall be determined from AIR KERMA level zero up to four times the normal AIR KERMA level
The conversion function for air kerma level zero is established using a dark image taken under identical conditions as an X-ray image Additionally, the minimum X-ray air kerma level must not exceed one-fifth of the standard air kerma level.
The number of exposures required depends on the form of the conversion function For checking linearity, five uniformly distributed exposures within the desired range are adequate However, to fully determine the conversion function, the air kerma must be varied so that the maximum increments of logarithmic air kerma (base 10) do not exceed 0.1 Additionally, the radiation quality must be consistent across all measurements.
To ensure compliance with AIR KERMA levels, it is essential to verify these levels at their minimum threshold If any deviations occur, it may be necessary to increase the distance from the focal spot to the detector surface.
I RRADIATION for determination of the NOISE POWER SPECTRUM and LAG
The DIGITAL X-RAY IMAGING DEVICE settings must match those utilized for the TEST DEVICE exposure IRRADIATION should follow the geometry outlined in Figure 2, but without the presence of the TEST DEVICE in the beam AIR KERMA is measured in accordance with section 4.6.2.
A central square area measuring approximately 125 mm × 125 mm, positioned behind the 160 mm square diaphragm, is utilized to evaluate an estimate of the noise power spectrum, which will subsequently be used to calculate the DQE.
To evaluate the NOISE POWER SPECTRUM, a minimum of N IM consecutive non-exposed images and N IM consecutive exposed images are required, with each image containing at least 256 PIXELS in both spatial directions All images must be captured under the same RADIATION QUALITY and AIR KERMA conditions, as illustrated in the image acquisition sequence in Figure 3.
N IM is defined as the number of images It shall be at least 64 and shall always be a power of
To prevent transient effects in digital X-ray imaging, both non-exposed and exposed images are preceded by additional images that are not retained for analysis The number of frames skipped is determined by the LAG EFFECT of the digital X-ray detector It is recommended that the mean PIXEL value of the first valid stored frame should not differ by more than 2% from the average value of the entire sequence of stored images.
Images saved Images not saved
Images Acquisition : NOISE POWER SPECTRUM and LAG EFFECTS
Figure 3 – Image acquisition sequence to determine the NOISE POWER SPECTRUM and LAG EFFECTS
NOTE The minimum number of stored images is determined by two requirements:
To achieve lag effects with an accuracy exceeding 5%, the number of images, \( N_{IM} \), must be sufficiently high to ensure the required frequency resolution It is important to avoid zero-padding in the Fourier transform Therefore, when using the Fast Fourier Transform (FFT), \( N \) should be a power of 2, and a minimum of 64 images is necessary to meet this criterion.
The required accuracy for the two-dimensional noise power spectrum dictates the minimum number of independent image pixels, necessitating at least 960 overlapping regions of interest (ROIs) for a 5% accuracy level This translates to 16 million independent image pixels based on the specified ROI size However, after applying the averaging and binning process to achieve a one-dimensional cut, the minimum requirement is reduced to four million independent image pixels while still maintaining the necessary accuracy To meet this criterion, a total of 64 images is sufficient.
No change of system setting is allowed when making the IRRADIATIONS
Images for determining the NOISE POWER SPECTRUM and LAG EFFECTS will be captured at three AIR KERMA levels for each Imaging Mode: the standard mode and two additional modes, each differing by a factor of 3.2 from the standard Refer to Table 2 in section 4.6.7 for further details.
I RRADIATION with TEST DEVICE in the RADIATION BEAM
The irradiation process will utilize the geometry illustrated in Figure 2, with the test device positioned directly on the detector surface It is essential that the test device is tilted at an angle α, ranging from 1.5° to 3°, in relation to the axis of the pixel columns or pixel rows.
NOTE 1 The method of tilting the TEST DEVICE relative to the rows or columns of the IMAGE MATRIX is common in other standards (ISO 15529 and ISO 12233) and reported in numerous publications when the pre-sampling MODULATION TRANSFER FUNCTION has to be determined
The TEST DEVICE must be positioned perpendicular to the REFERENCE AXIS of the RADIATION BEAM, ensuring that its edge is aligned as closely as possible with the REFERENCE AXIS.
NOTE 2 Deviations from this ideal set-up will result in a lower measured MTF
Two IRRADIATIONS shall be made with the TEST DEVICE in the RADIATION BEAM, one with the
The TEST DEVICE should be oriented along the columns, while another TEST DEVICE should be aligned with the rows of the IMAGE MATRIX The positions of the other components must remain unchanged A new adjustment of the TEST DEVICE will be necessary for its new position.
The Modulation Transfer Function (MTF) will be assessed using images captured at one of the three specified AIR KERMA levels for the selected Imaging Mode, with the MTF calculated individually for each Imaging Mode.
To ensure accuracy, it is advisable to average a sufficient number of images, particularly those obtained at lower AIR KERMA levels Additionally, the MTF value at the Nyquist frequency should remain consistent, with no more than a 5% variation upon repeated measurements.
Overview of all necessary IRRADIATIONS
Table 2 gives an overview on all necessary IRRADIATIONS A tolerance of ± 10 % applies to all specified AIR KERMA levels
Imaging Mode1 Imaging Mode2 Imaging Mode3
System Settings 1 System Settings 2 System Settings 3
Conversion function 0 80 nGy 0 800 nGy 0 8 000 nGy
The following linear and image-independent corrections of the RAW DATA are allowed in advance of the processing of the data for the determination of the CONVERSION FUNCTION, the
NOISE POWER SPECTRUM, and the MODULATION TRANSFER FUNCTION
All the following corrections if used shall be made as in normal clinical use:
– replacement of the RAW DATA of bad or defective pixels by appropriate data;
• correction of the non-uniformity of the RADIATION FIELD;
• correction for the offset of the individual pixels; and
• gain correction for the individual pixels;
NOTE 1 Some detectors execute linear image processing due to their physical concept As long as this image processing is linear and image-independent, these operations are allowed as an exception
NOTE 2 Image correction is considered image-independent if the same correction is applied to all images independent of the image contents
6 Determination of the DETECTIVE QUANTUM EFFICIENCY
Definition and formula of DQE(u,v)
The equation for the frequency-dependent DETECTIVE QUANTUM EFFICIENCY DQE(u,v) is:
The source for this equation is the Handbook of Medical Imaging Vol 1 equation 2.153 [4],
In this standard, the NOISE POWER SPECTRUM at the output W out (u, v) has to be corrected for
LAG EFFECTS resulting in W outcorrected (u, v) (according to subclause 6.3.2) The NOISE POWER
SPECTRUM at the output W outcorrected (u, v) and the MODULATION TRANSFER FUNCTION MTF(u,v) of the DIGITAL X-RAY IMAGING DEVICE shall be calculated on the LINEARIZED DATA The LINEARIZED
DATA are calculated by applying the inverse CONVERSION FUNCTION to the ORIGINAL DATA
According to subclause 6.3.1, the gain \( G \) of the detector at zero spatial frequency, as defined in equation (1), is included in the conversion function and does not require separate determination This gain is expressed in terms of the number of exposure quanta per unit area.
Therefore the working equation for the determination of the frequency-dependent DETECTIVE
QUANTUM EFFICIENCY DQE(u,v) according to this standard is:
MTF(u,v) is the pre-sampling MODULATION TRANSFER FUNCTION of the DIGITAL X-RAY IMAGING
DEVICE,determined according to subclause 6.3.3;
W in (u,v) is the NOISE POWER SPECTRUM of the radiation field at the DETECTOR SURFACE, determined according to subclause 6.2;
W outcorrected (u,v) is the NOISE POWER SPECTRUM at the output of the DIGITAL X-RAY IMAGING
DEVICE, corrected for LAG EFFECTS as determined according to subclause 6.3.2.
Parameters to be used for evaluation
For the determination of the DETECTIVE QUANTUM EFFICIENCY, the value of the input NOISE
POWER SPECTRUM W in (u,v)shall be calculated: in 2 a in(u,v) K SNR
K a is the measured A IR KERMA , unit: μGy;
SNR in 2 is the squared signal-to- NOISE ratio per AIR KERMA , unit: 1/(mm 2 ⋅μGy) as given in column 2 of Table 3
The values for SNR in 2 in Table 3 shall apply for this standard
Table 3 – Parameters mandatory for the application of this standard
R ADIATION QUALITY No SNR in 2
Background information on the calculation of SNR in 2 is given in Annex B.
Determination of different parameters from the images
Linearization of data
The LINEARIZED DATA are calculated by applying the inverse CONVERSION FUNCTION to the
The original data is analyzed on an individual pixel basis, where the conversion function represents the output level as a function of exposure quanta per unit area Consequently, the linearized data is expressed in terms of exposure quanta per unit area.
NOTE In case of a linear CONVERSION FUNCTION this calculation reduces to the multiplication by a conversion factor
The CONVERSION FUNCTION is determined from the images generated according to 4.6.4
The output is determined by averaging the pixel values from a 100 × 100 pixel area at the center of the exposed region, using the ORIGINAL DATA, which consists of RAW DATA values corrected as per Clause 5 This output is then plotted against the input signal, represented by the number of exposure quanta per unit area \( Q \), calculated by multiplying the AIR KERMA with the corresponding value from column 2 of Table 3 (refer to section 6.2).
The experimental data points will be fitted using a model function If the conversion function is assumed to be linear, based on the five exposures conducted as per section 4.6.4, only a linear function will be applied for fitting The fitting results must meet specific requirements.
– Final R 2 ≥ 0,99 (R 2 being the correlation coefficient); and
– no individual experimental data point deviates from its corresponding fit result by more than 2 %.
The LAG EFFECTS corrected NOISE POWER SPECTRUM (NPS)
6.3.2.1 Determination of the NOISE POWER SPECTRUM (NPS)
The NOISE POWER SPECTRUM at the output of the DIGITAL X-RAY IMAGING DEVICE shall be determined from the images generated according to 4.6.5 resulting in two NOISE POWER SPECTRA:
W out (u,v) dark the NOISE POWER SPECTRUM at the output of the DIGITAL X-RAY IMAGING DEVICE determined from the N IM dark images;
W out (u,v) exp the NOISE POWER SPECTRUM at the output of the DIGITAL X-RAY IMAGING DEVICE determined from the N IM exposed images;
For NPS analysis in digital X-ray detectors, the area is segmented into square regions known as ROIs, each measuring 256 × 256 pixels These ROIs overlap by 128 pixels both horizontally and vertically The initial ROI is positioned in the upper left corner of the analyzed region, with subsequent ROIs created by shifting the previous area 128 pixels to the right, resulting in overlapping sections.
The process involves generating horizontal "bands" by moving 128 pixels horizontally across the image, starting from the left side This is repeated vertically, creating additional bands until the entire area of approximately 125 mm × 125 mm is filled with Regions of Interest (ROIs).
Trend removal may be performed by fitting a two-dimensional second-order polynomial to the
The linearized data from each complete image is utilized to calculate the spectra by subtracting the function \( S(x_i, y_j) \) as defined in equation (4) The two-dimensional Fourier transform is then computed for each region of interest (ROI) without the application of any windowing.
The two-dimensional Fourier transform is utilized as outlined in equation (4) Based on equation 3.44 from the Handbook of Medical Imaging Vol 1, the key equation for determining the NOISE POWER SPECTRUM in accordance with this standard is presented.
∆ x ∆ y is the product of pixel spacing in respectively the horizontal and vertical direction;
M is the number of ROIs;
S(x i ,y j ) is the optionally fitted two-dimensional polynomial
An average two-dimensional NOISE POWER SPECTRUM is obtained by averaging the samples of all the spectra measured for that AIR KERMA level
NOTE The size of the ROIs shall be n = 256
Figure 4 – Geometric arrangement of the ROIs
6.3.2.2 Determination of the LAG EFFECT correction factor
A summarized description on the determination of the LAG EFFECT correction factor is given below For detailed information refer to annex A and to [12]
• The LAG EFFECT correction factor r shall be calculated from the LINEARIZED DATA using the same images as used for the determination of the NPS (see 6.3.2.1)
To eliminate potential fluctuations between images caused by variations in input AIR KERMA, each frame in the exposed sequence should be corrected by subtracting its average value, calculated from the same region of interest (ROI) selected in the subsequent step.
To perform spectral estimation, select a central rectangular region of interest (ROI) measuring at least 256 × 256 pixels within a 125 mm × 125 mm area This ROI consists of K time signals \( g_k(n) \) with a length of \( N_{IM} \) Increasing the number of pixels K helps to reduce the variance in the averaged periodogram.
To estimate the power spectral density (PSD) for both dark and exposed sequences, apply the following procedure: For each pixel \( k \) within the region of interest (ROI), compute the PSD using the periodogram and the Fast Fourier Transform (FFT) without zero-padding The average of all periodograms serves as the estimate for the temporal power spectrum of the detector, denoted as \( P_{gg-exp}(f_T) \) for exposed sequences and \( P_{gg-dark}(f_T) \) for dark sequences, where \( f_T \) represents the temporal frequency.
The power spectral density (PSD) of exposed frames consists of electronic noise and filtered quantum noise, with only the quantum noise being influenced by lag Because these two components are uncorrelated, their power spectral densities can be combined To isolate the quantum noise component, one can subtract the average periodogram of the dark frames from that of the exposed frames.
P gg T = gg − exp T − gg − dark T (5)
The resulting spectrum is an estimate for the PSD P gg (f T ) of the quantum noise that is correlated due to lag effects
The periodogram's value at zero temporal frequency is nearly zero due to the average value subtraction, necessitating a separate determination of \( P_{gg}(0) \) When the number of frames \( N_{IM} \) is sufficiently high, the power spectral density (PSD) is oversampled, allowing for perfect reconstruction from a subsampled version with \( N_{IM}/2 \) samples This method estimates the unknown PSD at zero frequency as a weighted sum of the PSD subsampled at odd positions, yielding the true value of the PSD at frequency zero for large \( N_{IM} \).
N n gg N gg (6) where d N I M is the Fourier transform of a modified (centered) version of the discrete rectangular window of even length N IM
• The ratio “r” of the integral PSDs of filtered quantum noise and white noise represents the attenuation of quantum noise due to lag effects
In discrete spectra of the Fast Fourier Transform (FFT), integration is substituted with summation, incorporating both positive and negative branches of the spectra, along with the separately determined value at zero frequency for each spectrum.
6.3.2.3 Determination of the LAG EFFECT corrected NOISE POWER SPECTRUM
Only the quantum part of the NPS is affected by lag and has to be re-scaled: r v u, W v u,
W out corrected out dark out ( ) exp out ( ) dark
6.3.2.4 Determination of the one-dimensional cut
To derive one-dimensional cuts from the two-dimensional Noise Power Spectrum along the axes of the spatial frequency plane, 15 rows or columns are utilized However, only the data from seven rows or columns on each side of the corresponding axis—totaling 14—are averaged, excluding the axes themselves The exact spatial frequencies, defined by their radial distance from the origin, are calculated for all data points Smoothing is achieved by averaging the data points within a frequency interval of \(2f_{int}\) (where \(f - f_{int} \leq f \leq f + f_{int}\)) around the reported spatial frequencies, as detailed in Clause 7 The parameter \(f_{int}\) is defined accordingly.
Adjusting the binning frequency interval based on pixel pitch ensures a consistent number of data points in the binning process, regardless of pixel pitch variations This approach guarantees uniform accuracy across different pixel pitches.
The dimension of the noise power spectral density is defined as the squared linearized data per unit of spatial frequency squared, which results in a dimension that is the inverse of the unit of length squared.
To assess the impact of quantization effects on the NOISE POWER SPECTRUM, the variance of the ORIGINAL DATA (DN) from a single image must be calculated A variance greater than 0.25 indicates that quantization NOISE is negligible, while a variance less than 0.25 suggests that the data is unsuitable for determining the NOISE POWER SPECTRUM.
The variance of the original data typically exceeds a quarter of the quantization interval, except when the number of bits used for quantization is minimal, which may result in a smaller variance The quantization variance, calculated as 1/12, is based on the assumption that the analog values being digitized follow a uniform or rectangular distribution within each quantization interval.
Determination of the MODULATION TRANSFER FUNCTION (MTF)
The pre-sampling MODULATION TRANSFER FUNCTION shall be determined along two mutually perpendicular axes which are parallel to the rows or to the columns of the IMAGE MATRIX, respectively
For the determination of the MTF, the complete length of the edge spread function (ESF) as defined by the ROI shown in Figure 1 shall be used
To determine the integer number \( N \) of lines that lead to a lateral shift of the edge in the line direction closest to the pixel sampling distance, various methods can be employed One effective approach involves calculating the angle \( \alpha \) between the edge and the rows or columns of the image matrix The value of \( N \) can then be computed using the formula \( N = \text{round}(1/\tan \alpha) \), where "round" indicates rounding to the nearest integer It is essential that \( N \) is accurate to integer precision.
NOTE 1 The range of values for the angle α means that N is between about 20 and 40
The pixel values from the linearized data of N consecutive lines are utilized to create an oversampled edge profile, known as the Edge Spread Function (ESF) The first pixel in the first line corresponds to the initial data point in the oversampled ESF, while the first pixel in each subsequent line provides the following data points This process continues for all pixels in the N lines, with the second pixel in the first line contributing to the (N + 1)th data point, and so forth, ensuring a comprehensive representation of the edge profile.
NOTE 2 Refer to [14] for more detailed background information
To determine the average Edge Spread Function (ESF), the process is conducted for various groups of N consecutive lines along the edge The overall average of these ESFs is then computed, and the Modulation Transfer Function (MTF) is derived from the averaged oversampled ESF.
The sampling distance in the oversampled ESF is assumed to be constant and is given by the
The pixel spacing, denoted as Δx, is divided by N to define the oversampled edge spread function (ESF), represented as ESF(x_n) with x_n = n(Δx/N) This oversampled ESF is then differentiated using either a [–1, 0, 1] or [–0.5, 0, 0.5] kernel to obtain the oversampled line spread function (LSF) To address the spectral smoothing effect from finite-element differentiation, a correction may be applied The Fourier transform of the LSF is computed, and its modulus provides the modulation transfer function (MTF), which is normalized to its zero frequency value Additionally, since the distance from individual pixels to the edge is measured along the line direction rather than perpendicular to the edge, a frequency axis scaling correction (scaling factor: 1/cosα) is necessary.
NOTE 3 The error of the SPATIAL FREQUENCY is ≤ 0,1 % if no correction by 1/cosα is done
To report the Modulation Transfer Function (MTF) at specific spatial frequencies, data points must be binned within a frequency interval of 2f int mm –1, where the range is defined as f – f int ≤ f ≤ f + f int (refer to Clause 6.3.2.4 for details on f int).
When stating the DETECTIVE QUANTUM EFFICIENCY, the following parameters shall be stated:
– RADIATION QUALITY according to Table 1;
– distance between FOCAL SPOT and DETECTOR SURFACE if less than 1,5 m;
– deviations from recommended centred geometry (see 4.5);
– method used for MTF determination and its validation, if a method different from the standardized edge method is used;
– frame rate used for the measured Imaging Mode;
The measurement results for Detective Quantum Efficiency (DQE) should be presented in a table, detailing values for spatial frequencies of 0.5 mm⁻¹, 1 mm⁻¹, and 1.5 mm⁻¹, extending to the highest spatial frequency just below the Nyquist frequency Additional relevant parameters may also be included in the table Furthermore, the DQE values can be illustrated as a function of spatial frequency, with AIR KERMA as a parameter, using a linear scale on both axes.
Generally, the DQE(u,v) values shall be given for both axes, the horizontal and vertical axes
If the quotient of DQE ( u , 0 ) / DQE ( 0 , v ) u = v is within the range of 0,9 to 1,1, the DQE(u,v) values for both axes may be averaged and stated to be valid for both axes
Additionally, values of DQE may be given along a diagonal axis It shall be explicitly stated with the results that the DQE refers to the diagonal axis
The uncertainty of DQE should be determined following the instructions of GUM [2] using equation (2) as a model equation
The uncertainty (coverage factor 2 according to [2]) of the DQE values presented shall be less than Δ(DQE(u)) = ± 0,06 or Δ(DQE(u))/DQE(u) = ± 0,10, whichever is greater
The uncertainty should be stated in the data sheets
This annex outlines the method used for determining and correcting lag, including repeated sections from 6.3.2.2 for thoroughness For additional details, please refer to [12].
Residual signals from prior frames create correlations between consecutive frames in an image sequence This phenomenon can be understood as a temporal low-pass filtering of uncorrelated quantum noise, which diminishes noise power and enhances the measured Detective Quantum Efficiency (DQE) To address this effect, it is essential to estimate and correct for the variance reduction caused by the temporal low-pass filtering.
The variance of a discrete random variable is determined by its auto-covariance function (ACF) at lag zero, or alternatively, by the integral of the power spectral density (PSD) \( P_{SS}(f_T) \), which represents the Fourier transform of the ACF.
NOTE 1 A normalized temporal frequency f T norm = f T /f T sample is used Therefore the integration limits are from –0,5 to 0,5 where 0,5 corresponds to the Nyquist frequency in the temporal frequency domain
If s is uncorrelated, i.e white noise, the power spectral density is constant:
Lag creates temporal correlation represented by the correlated random variable \( g \), which has a power spectral density \( P_{gg}(f_T) \) characterized by a low-pass filter This correlation leads to a reduction in variance, effectively decreasing noise power.
The correlated signal g(n) describes the detector signal at homogeneous exposure and therefore the power spectral density P gg (fT) can be estimated from measurements
Assuming that lag does not affect the mean signal, the power spectral density at frequency zero does not change by the filtering:
Inserting equations (A.2) and (A.4) into equation (A.3), the variance reduction factor (LAG EFFECT correction factor) can be estimated from the power spectral density P gg (fT) only:
Figure A.1 illustrates the effect of temporal correlation in the temporal frequency domain, with 0.5 on the x-axis representing the Nyquist frequency It also demonstrates the calculation of the noise reduction factor, known as the LAG EFFECT correction factor, as outlined in Equation A.5.
Figure A.1 – Power spectral density of white noise s and correlated signal g
(only positive frequencies are shown)
To practically implement (A.5), it is essential to estimate the spectral density from measurements A widely recognized nonparametric estimator for the power spectral density derived from a single time signal \( g_k(n) \) of length \( N \) is the periodogram.
Here, g k (n) denotes the grey level of pixel k in frame n after subtraction of the frame average
N denotes the number of images N IM
The variance of the estimate can be reduced by averaging the periodogram for all pixels within a selected region of interest:
The Power Spectral Density (PSD) of exposed frames consists of electronic noise and filtered quantum noise, with only the quantum noise being influenced by lag As these two components are uncorrelated, their PSDs combine additively To isolate the quantum noise component, one can subtract the averaged periodogram of the dark frames from that of the exposed frames.
A key challenge in signal processing is accurately determining the Power Spectral Density (PSD) at zero frequency The average periodogram at this frequency relies solely on the average of the squared sample signals, which tends to be close to zero due to the subtraction of frame averages Consequently, the periodogram does not provide information about \( P_{gg}(0) \), necessitating a separate determination of the value at zero frequency The PSD is computed using \( N_{IM} \) frames, and when \( N_{IM} \) is sufficiently large, the PSD becomes oversampled, allowing for perfect reconstruction from a subsampled version with \( N_{IM}/2 \) samples.
NOTE 2 This theorem is referred to in mathematical literature as the Whittaker-Shannon-Kotel'nikov theorem (or WSK theorem) Note however that this theorem relates to continuous functions whereas in this standard we deal with discrete signals While the sinc kernel is the Fourier transform of a continuous rectangular function, the Dirichlet kernel is the Fourier transform of a discrete rectangular function See [15]