2D Modulation Transfer Function Research Articles

Three-dimensional transfer function of optical microscopes in reflection mode.

Three-dimensional (3D) transfer functions build the basis for a comprehensive characterization of optical imaging systems in the spatial frequency domain. Utilizing the projection-slice theorem, the 2D modulation transfer function of an incoherent imaging system can be derived from a 3D transfer function by integration with respect to the axial spatial frequency. For a diffraction limited microscope with homogeneous incoherent pupil illumination, the modulation transfer function equals the 2D autocorrelation function of a circular disc. However, until now to the best of our knowledge no 3D transfer function has been published, which exactly leads to the 2D modulation transfer function of a diffraction limited microscope in reflection mode. In this article, we derive a formula, which after integration with respect to the axial spatial frequency coordinate perfectly fits to the diffraction limited 2D modulation transfer function. The inverse three-dimensional Fourier transform of the 3D transfer function results in a complex-valued 3D point spread function, from which the depth of field, the lateral resolution and, in addition, the corresponding 3D point spread function of both, a conventional and an interference microscope, can beobtained.

In-plane image quality and NPWE detectability index in digital breast tomosynthesis

A rigorous 2D analysis of signal and noise transfer applied to reconstructed planes in digital breast tomosynthesis (DBT) is necessary for system characterization and optimization. This work proposes a method for assessing technical image quality and system detective quantum efficiency (DQEsys) for reconstructed planes in DBT. Measurements of 2D in-plane modulation transfer function (MTF) and noise power spectrum (NPS) were made on five DBT systems using different acquisition parameters, reconstruction algorithms and plane spacing. This work develops the noise equivalent quanta (NEQ), DQEsys and detectability index (d’) calculated using a non-prewhitening model observer with eye filter (NPWE) for reconstructed DBT planes. The images required for this implementation were acquired using a homogeneous test object of thickness 40 mm poly(methyl) methacrylate plus 0.5 mm Al; 2D MTF was calculated from an Al disc of thickness 0.2 mm and diameter 50 mm positioned within the phantom. The radiant contrast of the MTF disc and the air kerma at the system input were used as normalization factors. The NPWE detectability index was then compared to the in-plane contrast-detail (c-d) threshold measured using the CDMAM phantom.The MTF and NPS measured on the different systems showed a strong anisotropy, consistent with the cascaded models developed in the literature for DBT. Detectability indices calculated from the measured MTF and NPS successfully predicted changes in c-d detectability for details between 0.1 mm and 2.0 mm, for DBT plane spacings between 0.5 mm and 10 mm, and for air kerma values at the system input between 157 µGy and 1170 μGy. The linear Pearson correlation between the detectability index and threshold gold thickness of the CDMAM phantom was −0.996. The method implements a parametric means of assessing the technical image quality of reconstructed DBT planes, providing valuable information for optimization of DBT systems.

The Minimum Modulation Curve as a tool for specifying optical performance: application to surfaces with mid-spatial frequency errors.

There are a variety of common situations in which specification of a one-dimensional modulation transfer function (MTF) or two orthogonal profiles of the 2D MTF are not adequate descriptions of the image quality performance of an optical system. These include systems with an asymmetric on-axis impulse response, systems with off-axis aberrations, systems with surfaces that include mid-spatial frequency errors, and freeform systems. In this paper, we develop the concept of the Minimum Modulation Curve (MMC). Starting with the two-dimensional MTF in polar form, the minimum MTF for any azimuth angle is plotted as a function of the radial spatial frequency. This can be presented in a familiar form similar to an MTF curve and is useful in the context of guaranteeing that a given MTF specification is met for any possible orientation of spatial frequencies in the image. In this way, an MMC may be of value in specifying the required performance of an optical system. We illustrate application of the MMC using profile data for surfaces with mid-spatial frequency errors.

Low-dose cone-beam CT via raw counts domain low-signal correction schemes: Performance assessment and task-based parameter optimization (Part I: Assessment of spatial resolution and noise performance).

Low-signal correction (LSC) in the raw counts domain has been shown to effectively reduce noise streaks in CT because the data inconsistency associated with photon-starved regions may be mitigated prior to the log transformation step. However, a systematic study of the performance of these raw data correction methods is still missing in literature. The purpose of this work was to provide such a systematic study for two well-known low-signal correction schemes using either the adaptive trimmed mean (ATM) filter or the anisotropic diffusion (AD) filter in the raw counts domain. Image data were acquired experimentally using an anthropomorphic chest phantom and a benchtop cone-beam CT (CBCT) imaging system. Phantom scans were repeated 50 times at a reduced dose level of 0.5 mGy and a reference level of 1.9 mGy. The measured raw counts at 0.5 mGy underwent LSC using the ATM and AD filters. Two relevant parameters were identified for each filter and approximately one hundred operating points in each parameter space were analyzed. Following LSC and log transformation, FDK reconstruction was performed for each case. Noise and spatial resolution properties were assessed across the parameter spaces that define each LSC filter; the results were summarized through 2D contour maps to better understand the trade-offs between these competing image quality features. 2D noise power spectrum (NPS) and modulation transfer function (MTF) were measured locally at two spatial locations in the field-of-view (FOV): a posterior region contaminated by noise streaks and an anterior region away from noise streaks. An isotropy score metric was introduced to characterize the directional dependence of the NPS and MTF (viz., ϵNPS and ϵMTF , respectively), with a range from 0 for highly anisotropic to 1 for perfectly isotropic. The noise magnitude and coarseness were also measured. (a) Both the ATM and AD LSC methods were successful in reducing noise streaks, but their noise and spatial resolution properties were found to be highly anisotropic and shift-variant. (b) NPS isotropy scores in the posterior region were generally improved from ϵNPS = 0.09 for the images without LSC to the range ϵNPS = (0.11, 0.67) for ATM and ϵNPS = (0.06, 0.67) for AD, depending on the filter parameters used. (c) The noise magnitude was reduced across the parameter space of either LSC filter whenever a change along the axis of the controlling parameter led to stronger raw data filtration. Changes in noise magnitude were inversely related to changes in spatial resolution along the direction perpendicular to the streaks. No correlation was found, however, between the contour maps of noise magnitude and the NPS isotropy. (d) Both filters influenced the noise coarseness anisotropically, with coarser noise occurring along directions perpendicular to the noise streaks. The anisotropic noise coarseness was intrinsically and directly related to resolution losses in a given direction: coarseness plots mimic the topography of the 2D MTF, i.e., the coarser the noise, the lower the resolution. Both AD and ATM LSC schemes enable low-dose CBCT imaging. However, it was found that noise magnitude and overall spatial resolution vary considerably across the parameter space for each filter, and more importantly these image quality features are highly anisotropic and shift-variant.

Three-dimensional cascaded system analysis of a 50 µm pixel pitch wafer-scale CMOS active pixel sensor x-ray detector for digital breast tomosynthesis

High-resolution, low-noise x-ray detectors based on the complementary metal-oxide-semiconductor (CMOS) active pixel sensor (APS) technology have been developed and proposed for digital breast tomosynthesis (DBT). In this study, we evaluated the three-dimensional (3D) imaging performance of a 50 µm pixel pitch CMOS APS x-ray detector named DynAMITe (Dynamic Range Adjustable for Medical Imaging Technology). The two-dimensional (2D) angle-dependent modulation transfer function (MTF), normalized noise power spectrum (NNPS), and detective quantum efficiency (DQE) were experimentally characterized and modeled using the cascaded system analysis at oblique incident angles up to 30°. The cascaded system model was extended to the 3D spatial frequency space in combination with the filtered back-projection (FBP) reconstruction method to calculate the 3D and in-plane MTF, NNPS and DQE parameters. The results demonstrate that the beam obliquity blurs the 2D MTF and DQE in the high spatial frequency range. However, this effect can be eliminated after FBP image reconstruction. In addition, impacts of the image acquisition geometry and detector parameters were evaluated using the 3D cascaded system analysis for DBT. The result shows that a wider projection angle range (e.g. ±30°) improves the low spatial frequency (below 5 mm−1) performance of the CMOS APS detector. In addition, to maintain a high spatial resolution for DBT, a focal spot size of smaller than 0.3 mm should be used. Theoretical analysis suggests that a pixelated scintillator in combination with the 50 µm pixel pitch CMOS APS detector could further improve the 3D image resolution. Finally, the 3D imaging performance of the CMOS APS and an indirect amorphous silicon (a-Si:H) thin-film transistor (TFT) passive pixel sensor (PPS) detector was simulated and compared.

Estimation of the two‐dimensional presampled modulation transfer function of digital radiography devices using one‐dimensional test objects

The modulation transfer function (MTF) of medical imaging devices is commonly reported in the form of orthogonal one-dimensional (1D) measurements made near the vertical and horizontal axes with a slit or edge test device. A more complete description is found by measuring the two-dimensional (2D) MTF. Some 2D test devices have been proposed, but there are some issues associated with their use: (1) they are not generally available; (2) they may require many images; (3) the results may have diminished accuracy; and (4) their implementation may be particularly cumbersome. This current work proposes the application of commonly available 1D test devices for practical and accurate estimation of the 2D presampled MTF of digital imaging systems. Theory was developed and applied to ensure adequate fine sampling of the system line spread function for 1D test devices at orientations other than approximately vertical and horizontal. Methods were also derived and tested for slit nonuniformity correction at arbitrary angle. Techniques were validated with experimental measurements at ten angles using an edge test object and three angles using a slit test device on an indirect-detection flat-panel system [GE Revolution XQ∕i (GE Healthcare, Waukesha, WI)]. The 2D MTF was estimated through a simple surface fit with interpolation based on Delaunay triangulation of the 1D edge-based MTF measurements. Validation by synthesis was also performed with simulated images from a hypothetical direct-detection flat-panel device. The 2D MTF derived from physical measurements yielded an average relative precision error of 0.26% for frequencies below the cutoff (2.5 mm(-1)) and approximate circular symmetry at frequencies below 4 mm(-1). While slit analysis generally agreed with the results of edge analysis, the two showed subtle differences at frequencies above 4 mm(-1). Slit measurement near 45° revealed radial asymmetry in the MTF resulting from the square pixel aperture (0.2 mm × 0.2 mm), a characteristic which was not necessarily appreciated with the orthogonal 1D MTF measurements. In simulation experiments, both slit- and edge-based measurements resolved the radial asymmetries in the 2D MTF. The average absolute relative accuracy error in the 2D MTF between the DC and cutoff (2.5 mm(-1)) frequencies was 0.13% with average relative precision error of 0.11%. Other simulation results were similar to those derived from physical data. Overall, the general availability, acceptance, accuracy, and ease of implementation of 1D test devices for MTF assessment make this a valuable technique for 2D MTF estimation.

Characterization of the 3D resolution of topometric sensors based on fringe and speckle pattern projection by a 3D transfer function

The increasing importance of optical 3D measurement techniques and the growing number of available methods and systems require a fast and simple method to characterize the measurement accuracy. However, the conventional approach of comparing measured coordinates to known reference coordinates of a test target faces two major challenges: the precise fabrication of the target and – in case of pattern projecting systems – finding the position of the reference points in the obtained point cloud. The modulation transfer function (MTF) on the other hand is an established instrument to describe the resolution characteristics of 2D imaging systems. Here, the MTF concept is applied to two different topometric systems based on fringe and speckle pattern projection to obtain a 3D transfer function. We demonstrate that in the present case fringe projection provides typically 3.5 times the 3D resolution achieved with speckle pattern projection. By combining measurements of the 3D transfer function with 2D MTF measurements the dependency of 2D and 3D resolutions are characterized. We show that the method allows for a simple comparison of the 3D resolution of two 3D sensors using a low cost test target, which is easy to manufacture.