Noise and filtration in magnetic resonance imaging

Noise in two-dimensional Fourier transform magnetic resonance images has been investigated using noise power spectra and measurements of standard deviation. The measured effects of averaging, spatial filtering, temporal filtering, and sampling have been compared with theoretical calculations. The noise of unfiltered images is found to be white, as expected, and the choice of the temporal filter and sampling interval affects the noise in a manner predicted by sampling theory. The shapes of the imager's spatial frequency filters are extracted using noise power spectra.     

The noise power spectrum in computed X-ray tomography

An expression is derived showing that the two-dimensional noise power spectrum of computed X-ray tomography is proportional to [G(k)]2/k where k is the radial spatial frequency and G(k) is the one-dimensional corrective filter used in the filtered back-projection reconstuction technique. It is shown that predicted noise power spectra compare well with those estimated from CT reconstructions of simulated noise for both the ramp filter and the Hanning-weighted ramp filter. A consequence of the non-uniform shape of the noise power spectrum is that statistical noise in CT reconstructions is correlated from point to point. Because of this correlation when the reconstructed CT values are averaged over some region, the uncertainty of the average depends on the shape of the region as well as its area. This dependence is confirmed by computer simulations.     

A method for modifying the image quality parameters of digital radiographic images

A new computer simulation approach is presented that is capable of modeling several varieties of digital radiographic systems by their image quality characteristics. In this approach, the resolution and noise characteristics of ideal supersampled input images are modified according to input modulation transfer functions (MTFs) and noise power spectra (NPS). The modification process is separated into two routines-one for modification of the resolution and another for modification of the noise characteristics of the input image. The resolution modification routine blurs the input image by applying a frequency filter described by the input MTF. The resulting blurred image is then reduced to its final size to account for the sampling process of the digital system. The noise modification routine creates colored noise by filtering the frequency components of a white noise spectrum according to the input noise power. This noise is then applied to the image by a moving region of interest to account for variations in noise due to differences in attenuation. In order to evaluate the efficacy of the modification routines, additional routines were developed to assess the resolution and noise of digital images. The MTFs measured from the output images of the resolution modification routine were within 3% of the input MTF The NPS measured from the output images of the noise modification routine were within 2% of the input NPS. The findings indicate that the developed modification routines provide a good means of simulating the resolution and noise characteristics of digital radiographic systems for optimization or processing purposes.     

Investigation of basic imaging properties in digital radiography. 3. Effect of pixel size on SNR and threshold contrast

The effect of pixel size on the signal-to-noise ratio (SNR) and threshold detection of low-contrast radiologic patterns was investigated theoretically for digital radiographic systems. The SNR based on the perceived statistical decision theory model, together with the internal noise of the human eye-brain system, was calculated by using two-dimensional displayed digital signal spectra and noise Wiener spectra. Threshold contrasts were predicted from the calculated SNR for various combinations of object size and shape, pixel size, resolution, and noise. Predicted threshold contrasts agreed well with those determined experimentally in an observer performance study. The threshold contrast of small objects increased substantially as the pixel size increased beyond 0.2 mm. For pixel sizes of 0.1 and 0.2 mm, however, the threshold contrasts were similar. Since a digital system is not shift invariant, a range of threshold contrast results for a small object and a large pixel, depending on the alignment of the object position relative to the sampling coordinates.     

Correction functions and statistical noises in transverse section picture reconstruction

An analysis of image reconstruction with one- and two-dimensional convolution methods is presented in an outline of the relationship among the correction functions, the point spread function and the statistical noise. The correction functions are derived which maximize the ratio of one-dimensional signal power to noise power for a given r.m.s. resolution width in a uniform image. The texture of the image noise is expressed by the autocovariance function.

For emission images, the variance of noise is expressed by the convolution of the source distribution with the “error kernel” which is determined from the one-dimensional correction function.     

Noise power spectra of images from digital mammography detectors.

Noise characterization through estimation of the noise power spectrum (NPS) is a central component of the evaluation of digital x-ray systems. We begin with a brief review of the fundamentals of NPS theory and measurement, derive explicit expressions for calculation of the one- and two-dimensional (1D and 2D) NPS, and discuss some of the considerations and tradeoffs when these concepts are applied to digital systems. Measurements of the NPS of two detectors for digital mammography are presented to illustrate some of the implications of the choices available. For both systems, two-dimensional noise power spectra obtained over a range of input fluence exhibit pronounced asymmetry between the orthogonal frequency dimensions. The 2D spectra of both systems also demonstrate dominant structures both on and off the primary frequency axes indicative of periodic noise components. Although the two systems share many common noise characteristics, there are significant differences, including markedly different dark-noise magnitudes, differences in NPS shape as a function of both spatial frequency and exposure, and differences in the natures of the residual fixed pattern noise following flat fielding corrections. For low x-ray exposures, quantum noise-limited operation may be possible only at low spatial frequency. Depending on the method of obtaining the 1D NPS (i.e., synthetic slit scanning or slice extraction from the 2D NPS), on-axis periodic structures can be misleadingly smoothed or missed entirely. Our measurements indicate that for these systems, 1D spectra useful for the purpose of detective quantum efficiency calculation may be obtained from thin cuts through the central portion of the calculated 2D NPS. On the other hand, low-frequency spectral values do not converge to an asymptotic value with increasing slit length when 1D spectra are generated using the scanned synthetic slit method. Aliasing can contribute significantly to the digital NPS, especially near the Nyquist frequency. Calculation of the theoretical presampling NPS and explicit inclusion of aliased noise power shows good agreement with measured values.     

GPU-accelerated 3D mipmap for real-time visualization of ultrasound volume data

Ultrasound volume rendering is an efficient method for visualizing the shape of fetuses in obstetrics and gynecology. However, in order to obtain high-quality ultrasound volume rendering, noise removal and coordinates conversion are essential prerequisites. Ultrasound data needs to undergo a noise filtering process; otherwise, artifacts and speckle noise cause quality degradation in the final images. Several two-dimensional (2D) noise filtering methods have been used to reduce this noise. However, these 2D filtering methods ignore relevant information in-between adjacent 2D-scanned images. Although three-dimensional (3D) noise filtering methods are used, they require more processing time than 2D-based methods. In addition, the sampling position in the ultrasonic volume rendering process has to be transformed between conical ultrasound coordinates and Cartesian coordinates. We propose a 3D-mipmap-based noise reduction method that uses graphics hardware, as a typical 3D mipmap requires less time to be generated and less storage capacity. In our method, we compare the density values of the corresponding points on consecutive mipmap levels and find the noise area using the difference in the density values. We also provide a noise detector for adaptively selecting the mipmap level using the difference of two mipmap levels. Our method can visualize 3D ultrasound data in real time with 3D noise filtering.     

Reduction of aliasing in modulation transfer function measurements

The images formed by many radiological systems are difficult to sample at spatial intervals small enough to avoid aliasing in the calculation of the system's modulation transfer function. However, if a system's response can be assumed to be symmetrical, this assumption can be used to effectively double the sampling density and to double the frequency limit before aliasing occurs. To accomplish this, a more complex algorithm is required. In this work, the formula for the calculation of the modulation transfer function from a symmetrical, one-dimensional line spread function is derived and a similar result for a symmetrical, two-dimensional point spread function is presented. The effect of noisy data and errors in the estimation of the offset of the center of the line spread function from a sampling point are investigated by simulation studies. For low noise (relative standard deviation of 1%) and an offset error of no more than 2% or 3% of a sampling interval, reasonable precision is obtained. These conditions appear to be achievable, especially when the noise is Poisson distributed.     

Wiener noise power spectra of radiological television systems using a digital oscilloscope

Measurement of spatial noise power spectra from television based radiographic and fluoroscopic systems is essential to the understanding of their operation and optimization. However, conventional methods require acquisition and processing of large numbers of complete images, thus confining such measurements to special applications where accessible frame buffers already exist or elaborately equipped laboratories. We have developed a method which only requires storage of single TV lines or point scans. A digital oscilloscope captures these point scans and a laboratory microcomputer facilitates manipulation of the data to separate out different components of the noise power spectra. The x-ray dependent component of the noise power spectrum so produced is not the ordinary Wiener spectrum. However, it is shown that reconstruction of the full Wiener spectrum from this is possible subject only to the requirement that the x-ray noise spectrum at the output of the imaging system is circularly symmetric.     

Development and validation of a hybrid simulation technique for cone beam CT: application to an oral imaging system

This paper proposes a hybrid technique to simulate the complete chain of an oral cone beam computed tomography (CBCT) system for the study of both radiation dose and image quality. The model was developed around a 3D Accuitomo 170 unit (J Morita, Japan) with a tube potential range of 60-90 kV. The Monte Carlo technique was adopted to simulate the x-ray generation, filtration and collimation. Exact dimensions of the bow-tie filter were estimated iteratively using experimentally acquired flood images. Non-flat radiation fields for different exposure settings were mediated via 'phase spaces'. Primary projection images were obtained by ray tracing at discrete energies and were fused according to the two-dimensional energy modulation templates derived from the phase space. Coarse Monte Carlo simulations were performed for scatter projections and the resulting noisy images were smoothed by Richardson-Lucy fitting. Resolution and noise characteristics of the flat panel detector were included using the measured modulation transfer function (MTF) and the noise power spectrum (NPS), respectively. The Monte Carlo dose calculation was calibrated in terms of kerma free-in-air about the isocenter, using an ionization chamber, and was subsequently validated by comparison against the measured air kerma in water at various positions of a cylindrical water phantom. The resulting dose discrepancies were found <10% for most cases. Intensity profiles of the experimentally acquired and simulated projection images of the water phantom showed comparable fractional increase over the common area as changing from a small to a large field of view, suggesting that the scatter was accurately accounted. Image validation was conducted using two small phantoms and the built-in quality assurance protocol of the system. The reconstructed simulated images showed high resemblance on contrast resolution, noise appearance and artifact pattern in comparison to experimentally acquired images, with <5% difference for voxel values of the aluminum and air insert regions and <3% difference for voxel uniformity across the homogeneous PMMA region. The detector simulation by use of the MTF and NPS data exhibited a big influence on noise and the sharpness of the resulting images. The hybrid simulation technique is flexible and has wide applicability to CBCT systems.     

Some noise properties of 2DFT MR images from asymmetrically sampled data.

This report describes noise statistics in 2DFT MR images, expanding the earlier work of Henkelman and others to include variably asymmetric sampling and conjugate synthesis reconstruction. The effects of low-order polynomial and Fourier phase correction used with conjugate synthesis are also explicitly considered. This analysis shows that complex images obtained by conjugate synthesis have an elliptical noise distribution, with the smaller axis corresponding to the imaginary image channel. Derivations and simulations predict a ratio of mean to standard deviation in the background of magnitude images varying from the known value of square root of pi/(4 - pi) (approximately 1.91) for full symmetry to square root of 2/(pi - 2) (approximately 1.32) at fully asymmetric or half-echo sampling; these predictions are validated over a range of asymmetry by experimental measurements. These results are important for predicting and interpreting image noise when using asymmetric sampling.     

An FDK-like cone-beam SPECT reconstruction algorithm for non-uniform attenuated projections acquired using a circular trajectory

In this paper, Novikov's inversion formula of the attenuated two-dimensional (2D) Radon transform is applied to the reconstruction of attenuated fan-beam projections acquired with equal detector spacing and of attenuated cone-beam projections acquired with a flat planar detector and circular trajectory. The derivation of the fan-beam algorithm is obtained by transformation from parallel-beam coordinates to fan-beam coordinates. The cone-beam reconstruction algorithm is an extension of the fan-beam reconstruction algorithm using Feldkamp-Davis-Kress's (FDK) method. Computer simulations indicate that the algorithm is efficient and is accurate in reconstructing slices close to the central slice of the cone-beam orbit plane. When the attenuation map is set to zero the implementation is equivalent to the FDK method. Reconstructed images are also shown for noise corrupted projections.     

A Technique for Simulating the Effect of Dose Reduction on Image Quality in Digital Chest Radiography

Purpose: The purpose of this study is to provide a pragmatic tool for studying the relationship between dose and image quality in clinical chest images. To achieve this, we developed a technique for simulating the effect of dose reduction on image quality of digital chest images. Materials and Methods: The technique was developed for a digital charge-coupled-device (CCD) chest unit with slot-scan acquisition. Raw pixel values were scaled to a lower dose level, and a random number representing noise to each specific pixel value was added. After adding noise, raw images were post processed in the standard way. Validation was performed by comparing pixel standard deviation, as a measure of noise, in simulated images with images acquired at actual lower doses. To achieve this, a uniform test object and an anthropomorphic phantom were used. Additionally, noise power spectra of simulated and actual images were compared. Also, detectability of simulated lesions was investigated using a model observer. Results: The mean difference in noise values between simulated and real lower-dose phantom images was smaller than 5% for relevant clinical settings. Noise power spectra appeared to be comparable on average but simulated images showed slightly higher noise levels for higher spatial frequencies and slightly lower noise levels for lower spatial frequencies. Comparable detection performance was shown in simulated and actual images with slightly worse detectability for simulated lower dose images. Conclusion: We have developed and validated a method for simulating dose reduction. Our method seems an acceptable pragmatic tool for studying the relationship between dose and image quality.     

Variation of the count-dependent Metz filter with imaging system modulation transfer function

A systematic investigation was conducted of how a number of parameters which alter the system modulation transfer function (MTF) influence the count-dependent Metz filter. Since restoration filters are most effective at those frequencies where the object power spectrum dominates that of the noise, it was observed that parameters which significantly degrade the MTF at low spatial frequencies strongly influence the formation of the Metz filter. Thus the radionuclide imaged and the depth of the source in a scattering medium had the most influence. This is because they alter the relative amount of scattered radiation being imaged. For low-energy photon emitters, the collimator employed and the distance from the collimator were found to have less of an influence but still to be significant. These cause alterations in the MTF which are more gradual, and hence are most pronounced at mid to high spatial frequencies. As long as adequate spatial sampling is employed, the Metz filter was determined to be independent of the exact size of the sampling bin width, to a first approximation. For planar and single photon emission computed tomographic (SPECT) imaging, it is shown that two-dimensional filtering with the Metz filter optimized for the imaging conditions is able to deconvolve scatter and other causes of spatial resolution loss while diminishing noise, all in a balanced manner.     

Characterization of color CRT display systems for monochrome applications

Soft-copy presentation of medical images is becoming more and more important as medical imaging is strongly moving toward digital technology, and health care facilities are converting to filmless hospital and radiological information management. Although most medical images are monochrome, frequently they are displayed on color CRTs, particularly if general-purpose workstations or PCs are used for medical viewing. In the present report, general measurement and modeling procedures for the characterization of color CRT monitors for monochrome presentation are introduced. The contributions from the three color channels (red, green, and blue) are weighted according to the spectral sensitivity of the human eye for photopic viewing. The luminance behavior and the resolution capabilities of color CRT monitors are analyzed with the help of photometer and charge-coupled device (CCD) camera measurements. For the evaluation of spatial resolution, a two-dimensional Fourier analysis of special test images containing white noise (broadband response) is employed. A stage model for a color CRT monitor is developed to discuss the effects of scanning and dot sampling. Furthermore, display intrinsic veiling glare and reflectivity of typical color CRT monitors are measured and compared with those of monochrome CRT monitors. The developed methods and models allow one to describe the image quality aspects of color monitors if they are applied for medical monochrome image presentation. Particularly, because of the reduced luminance and dynamic range of color monitors, the calibration and control of their luminance curves is a very important task. For present color CRT monitors, 1,280 x 1,024 turns out to be an intrinsic limit for the displayable matrix of medical images.     

Characterization of a fluoroscopic imaging system for kV and MV radiography

An on-line kilovoltage (kV) imaging system has been implemented on a medical linear accelerator to verify radiotherapy field placement. A kV x-ray tube is mounted on the accelerator at 90 degrees to the megavoltage (MV) source and shares the same isocenter. Nearly identical CCD-based fluoroscopic imagers are mounted opposite the two x-ray sources. These systems are being used in a clinical study of patient setup error that examines the advantage of kV imaging for on-line localization. In the investigation reported here, the imaging performance of the kV and MV systems are characterized to provide support to the conclusions of the studies of setup error. A spatial-frequency-dependent linear systems model is used to predict the detective quantum efficiencies (DQEs) of the two systems. Each is divided into a series of gain and spreading stages. The parameters of each stage are either measured or obtained from the literature. The model predicts the system gain to within 7% of the measured gain for the MV system and to within 10% for the kV system. The systems' noise power spectra (NPSs) and modulation transfer functions (MTFs) are measured to construct the measured DQEs. X-ray fluences are calculated using modeled polyenergetic spectra. Measured DQEs agree well with those predicted by the model. The model reveals that the MV system is well optimized, and is x-ray quantum noise limited at low spatial frequencies. The kV system is suboptimal, but for purposes of patient positioning yields images superior to those produced by the MV system. This is attributed to the kV system's higher DQE and to the inherently higher contrasts present at kV energies.     

Fast and exact 2D image reconstruction by means of Chebyshev decomposition and backprojection

A new algorithm for the reconstruction of two-dimensional (2D) images from projections is described. The algorithm is based on the decomposition of the projections into Chebyshev polynomials of the second kind, which are the ideal basis functions for this application. The Chebyshev decomposition is done via the fast discrete sine transform. A discrete reconstruction filter is applied that corresponds to the ramp filter used in standard filtered backprojection (FBP) reconstruction. In contrast to FBP, the filter is applied to the Chebyshev coefficients and not to the Fourier coefficients of the projections. Then the reconstructed image is simply obtained by means of backprojection. Consequently, the method can be considered as a Chebyshev-domain filtered backprojection (CD-FBP). The total calculation time is dominated by the backprojection step only and is comparable to FBP. The merits of CD-FBP as compared with standard FBP are that: (a) The result is exact if the 2D function to be reconstructed can be decomposed into polynomials of finite degree, and if the sampling is adequate. Otherwise a polynomial approximation results. (b) The algorithm is inherently discrete. (c) It is particularly well suited for reconstructions from projections with non-equidistant samples that occur for instance in 2D PET (positron emission tomography) imaging and in a special form of fan beam scanning. Examples of applications comprise reconstructions of the Shepp and Logan head phantom in various sampling geometries, and a real PET test object. In the PET example an increased resolution is observed in comparison with standard FBP.     

Spectral analysis of full field digital mammography data

The spectral content of mammograms acquired from using a full field digital mammography (FFDM) system are analyzed. Fourier methods are used to show that the FFDM image power spectra obey an inverse power law; in an average sense, the images may be considered as 1/f fields. Two data representations are analyzed and compared (1) the raw data, and (2) the logarithm of the raw data. Two methods are employed to analyze the power spectra (1) a technique based on integrating the Fourier plane with octave ring sectioning developed previously, and (2) an approach based on integrating the Fourier plane using rings of constant width developed for this work. Both methods allow theoretical modeling. Numerical analysis indicates that the effects due to the transformation influence the power spectra measurements in a statistically significant manner in the high frequency range. However, this effect has little influence on the inverse power law estimation for a given image regardless of the data representation or the theoretical analysis approach. The analysis is presented from two points of view (1) each image is treated independently with the results presented as distributions, and (2) for a given representation, the entire image collection is treated as an ensemble with the results presented as expected values. In general, the constant ring width analysis forms the foundation for a spectral comparison method for finding spectral differences, from an image distribution sense, after applying a nonlinear transformation to the data. The work also shows that power law estimation may be influenced due to the presence of noise in the higher frequency range, which is consistent with the known attributes of the detector efficiency. The spectral modeling and inverse power law determinations obtained here are in agreement with that obtained from the analysis of digitized film-screen images presented previously. The form of the power spectrum for a given image is approximately l/f2beta with beta approximately 1.4-1.5.     

Optimized motion estimation for MRE data with reduced motion encodes

Motion estimation is an essential step common to all magnetic resonance elastography (MRE) methods. For dynamic techniques, the motion is obtained from a sinusoidal fit of the image phase at multiple, uniformly spaced relative phase offsets, phi, between the motion and the motion encoding gradients (MEGs). Generally, eight values of phi sampled at the Nyquist interval pi/4 over [0, 2pi). We introduce a method, termed reduced motion encoding (RME), that reduces the number of phi required, thereby reducing the imaging time for an MRE acquisition. A frequency-domain algorithm was implemented using the discrete Fourier transform (DFT) to derive the general least-squares solution for the motion amplitude and phase given an arbitrary number of phi. A closed form representation of the condition number of the transformation matrix which is used for estimating motion was introduced to determine the sensitivity to noise for different sampling patterns of phi. Simulation results confirmed the minimum error sampling patterns suggested from the condition number maps. The minimum noise in the motion estimate is obtained when the sampled phi are essentially evenly distributed over the range [0, pi) with an interval pi/n, where n is the number of phi sampled, or alternatively with an interval 2pi/n over the range [0, 2pi) which represents the Nyquist interval. Simulations also show that the noise level decreases as n increases as expected. The decrease in noise is the largest when n is small and it becomes less significant as n increases. The algorithm also makes it possible to estimate the motion from only two values of phi, which cannot be accomplished with traditional methods because sampling at the Nyquist interval is indeterminate. Finally, noise levels in motion estimated from phantom studies and in vivo results taken with different n agreed with that predicted by simulation and condition number calculations.     

Image reconstruction with shift-variant filtration and its implication for noise and resolution properties in fan-beam computed tomography

In computed tomography (CT), the fan-beam filtered backprojection (FFBP) algorithm is used widely for image reconstruction. It is known that the FFBP algorithm can significantly amplify data noise and aliasing artifacts in situations where the focal lengths are comparable to or smaller than the size of the field of measurement (FOM). In this work, we propose an algorithm that is less susceptible to data noise, aliasing, and other data inconsistencies than is the FFBP algorithm while retaining the favorable resolution properties of the FFBP algorithm. In an attempt to evaluate the noise properties in reconstructed images, we derive analytic expressions for image variances obtained by use of the FFBP algorithm and the proposed algorithm. Computer simulation studies are conducted for quantitative evaluation of the spatial resolution and noise properties of images reconstructed by use of the algorithms. Numerical results of these studies confirm the favorable spatial resolution and noise properties of the proposed algorithm and verify the validity of the theoretically predicted image variances. The proposed algorithm and the derived analytic expressions for image variances can have practical implications for both estimation and detection/classification tasks making use of CT images, and they can readily be generalized to other fan-beam geometries.