Ramp Filter Research Articles

Detailed investigation of transmission and emission data smoothing protocols and their effects on emission images

Measured attenuation correction in PET is routinely performed using transmission scans. Acquisition time and noise considerations necessitate low-pass filtering of the transmission data before generating the attenuation correction matrix. This smoothing operation reduces noise propagation from transmission into emission data but also introduces image artifacts that are mostly pronounced around areas of strongly varying attenuation coefficients. The source of these artifacts, which lies in the mismatch of the spatial resolutions of emission and transmission data, was investigated in this study. The effect of different transmission and emission sinogram smoothing protocols on the emission images was also investigated. A method is proposed that addresses the problem, in conjunction with the filtering step during reconstruction. Instead of the standard in-plane low-pass filtering of the emission data during reconstruction, emission and transmission sinograms can be volumetrically filtered to the desired image resolution prior to attenuation correction while reconstruction is performed with no smoothing (Ramp filter). This operation reduces or eliminates resolution mismatch and consequent image artifacts, especially in cardiac studies. The proposed method improves the accuracy of the activity distribution in emission images with minimal computational and signal-to-noise ratio (SNR) cost.

A wavelet-based method for multiscale tomographic reconstruction

The authors represent the standard ramp filter operator of the filtered-back-projection (FBP) reconstruction in different bases composed of Haar and Daubechies compactly supported wavelets. The resulting multiscale representation of the ramp-filter matrix operator is approximately diagonal. The accuracy of this diagonal approximation becomes better as wavelets with larger numbers of vanishing moments are used. This wavelet-based representation enables the authors to formulate a multiscale tomographic reconstruction technique in which the object is reconstructed at multiple scales or resolutions. A complete reconstruction is obtained by combining the reconstructions at different scales. The authors' multiscale reconstruction technique has the same computational complexity as the FBP reconstruction method. It differs from other multiscale reconstruction techniques in that (1) the object is defined through a one-dimensional multiscale transformation of the projection domain, and (2) the authors explicitly account for noise in the projection data by calculating maximum a posteriori probability (MAP) multiscale reconstruction estimates based on a chosen fractal prior on the multiscale object coefficients. The computational complexity of this maximum a posteriori probability (MAP) solution is also the same as that of the FBP reconstruction. This result is in contrast to commonly used methods of statistical regularization, which result in computationally intensive optimization algorithms.

A Study of Errors due to Reconstruction Filters in Computed Tomography

Abstract In the summation convolution backprojection method of image reconstruction in computed tomography, the final image accuracy depends on the convolution filter used. Filters are designed to attenuate high spatial frequencies when noisy projection data are used. This paper explores the differences between the images reconstructed using a range of filters, and compares the results with the case of the ramp filter that provides the “best” image for ideal, noise-free, projection data. It is shown that systematic errors between these images and the best image exist, and that these errors are related to the second differential of the reconstruction filter with respect to spatial frequency. This error determination may be used to correct computed tomography images that have been reconstructed using inappropriate filters, and this theory is tested using noise-free projection data from two computer simulated images. It is shown that the corrected images are far closer to the original images.

Comparison of 180/spl deg/ and 360/spl deg/ iterative reconstruction with non-uniform attenuation compensation for Tl-201 SPECT

This study compares Tl-201 SPECT images reconstructed from 180/spl deg/ and 360/spl deg/ projection data, using iterative reconstruction with non-uniform attenuation compensation. Using a mathematical phantom of the human thorax, Monte Carlo simulation is used to produce both 180/spl deg/ (45/spl deg/ RAO to 45/spl deg/ LPO) and 360/spl deg/ Tl-201 SPECT projection data. Included in the simulation are the effects of non-uniform attenuation, detector response, scatter and statistical noise. The 180/spl deg/ and 360/spl deg/ data were taken over 64 and 128 angles, respectively, assuming the same total acquisition time for both. The simulated projection data were reconstructed with nonuniform attenuation compensation using the ML-EM algorithm for 50 iterations and, for comparison, were also reconstructed without attenuation compensation using the filtered backprojection method with a ramp filter. Transaxial slices of the reconstructed images, image profiles and Bull's-eye plots all show that images reconstructed with attenuation compensation, unlike those reconstructed without attenuation compensation, are essentially the same within the region of the myocardium for the 180/spl deg/ and 360/spl deg/ data, with few artifacts and similar image contrast. Furthermore, for the same total imaging time, the images reconstructed from the 180/spl deg/ data have slightly less noise in the myocardial region.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Backprojection filtering for variable orbit fan-beam tomography

Backprojection filtering algorithms are presented for three variable orbit fan-beam geometries. Expressions for the fan-beam projection and backprojection operators are given for a flat detector fan-beam geometry with fixed focal length, with variable focal length, and with fixed focal length and off-center focusing. Backprojection operators are derived for each geometry using transformation of coordinates to transform from a parallel geometry backprojector to a fan-beam backprojector for the appropriate geometry. The backprojection operator includes a factor which is a function of the coordinates of the projection ray and the coordinates of the pixel in the backprojected image. The backprojection filtering algorithm first backprojects the variable orbit fan-beam projection data using the appropriately derived backprojector to obtain a 1/r blurring of the original image, then takes the two-dimensional (2D) Fast Fourier Transform (FFT) of the backprojected image, then multiples the transformed image by the 2D ramp filter function, and finally takes the inverse 2D FFT to obtain the reconstructed image. Computer simulations verify that backprojectors with appropriate weighting give artifact free reconstructions of simulated line integral projections. Also, it is shown that it is not necessary to assume a projection model of line integrals, but the projector and backprojector can be defined to model the physics of the imaging detection process. A backprojector for variable orbit fan-beam tomography with fixed focal length is derived which includes an additional factor which is a function of the flux density along the flat detector. It is shown that the impulse response for the composite of the projection and backprojection operations is equal to 1/r.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Pinhole collimation for ultra-high-resolution, small-field-of-view SPECT

The authors evaluated small-field-of-view ultra-high-resolution pinhole collimation for a rotating-camera SPECT system that could be used to image small laboratory animals. Pinhole collimation offers distinct advantages over conventional parallel-hole collimation when used to image small objects. Since geometric sensitivity increases markedly for points close to the pinhole, small-diameter and high-magnification pinhole geometries may be useful for selected imaging tasks when used with large-field-of-view scintillation cameras. The use of large magnifications can minimize the loss of system resolution caused by the intrinsic resolution of the scintillation camera. A pinhole collimator has been designed and built that can be mounted on one of the scintillation cameras of a triple-head SPECT system. 3 Pinhole inserts with approximate aperture diameters of 0.6, 1.2 and 2.0 mm have been built and can be mounted individually on the collimator housing. When a ramp filter is used with a 3D filtered backprojection (FBP) algorithm, the 3 apertures have in-plane SPECT spatial resolutions (FWHM) at 4 cm of 1.5, 1.9 and 2.8 mm, respectively. In-air point source sensitivities at 4 cm from the apertures are 0.9, 2.6 and 5.7 counts s-1 mu Ci-1 (24, 70 and 154 counts s-1 MBq-1) for the 0.6, 1.2 and 2.0 mm apertures, respectively. In vitro image quality was evaluated with a micro-cold-rod phantom and a micro-Defrise phantom using both the 3D FBP algorithm and a 3D maximum likelihood-expectation maximization algorithm. In vivo image quality was evaluated using 2 (315 and 325 g) rats. Ultra-high-resolution pinhole SPECT is an inexpensive and simple approach for imaging small animals that can be used with existing rotating-camera SPECT systems.

TECHNICAL ASPECTS OF BONE SCINTIGRAPHY

Optimal bone scintigraphy is obtained by using a current generation gamma camera with a high-resolution collimator, minimizing the patient-to-collimator distance, using scatter reduction techniques where possible, and obtaining a 500,000 to 1 million count image for 40-cm field of view camera. Hard copy images from an analog or digital formatter should be optimized to display all intensities either on the same images or, when necessary, to display the low count information on one image and the high count information on another. Additional images using different collimators, such as converging or pinhole collimators, and oblique and lateral views should be obtained when necessary to demonstrate or define the pathologic area. To optimize SPECT imaging, the following parameters should be used: a high-resolution collimator, a 128 x 128 acquisition matrix, and minimum separation between the patient and the collimator, which may require the use of an elliptic orbit. Between 64 and 128 views should be obtained, and depending on preference, the planar data should be prefiltered with a Butterworth, order 8-12 and a cutoff at 0.5 Nyquist. The data should then be reconstructed using a simple ramp filter. This method provides a good technique when one is first beginning to perform bone SPECT. Attenuation correction is not generally beneficial for SPECT bone studies, although sometimes weighted backprojection may improve image contrast and resolution. Finally, the use of volume rendering may help clarify the location of suspect lesions.

Time-frequency distribution inversion of the Radon transform (image reconstruction)

In using filtered backprojection to compute the inverse Radon transform, the ramp filter amplifies noise. Spatially invariant noise filters reduce resolution. It is desirable to filter noise where projections have no local high-frequency components. Using the short-time Fourier transform, the authors apply a time-frequency mask filter that zeroes out projections where local signal energy is below a threshold. Results show improvement over reconstructions using spatially invariant smoothing filters.

Spatial resolution and count density requirements in brain SPECT imaging

A set of simulations has been performed to investigate the spatial resolution and count density requirements for brain SPECT imaging. Projections were drawn from a matrix representation of the Hoffman brain phantom. These projections were convolved with realistic point spread functions and Poisson noise was added to simulate a wide range of imaging situations normalized to a fixed imaging time. The projections were optimally smoothed with a Wiener filter and were reconstructed with a ramp filter. The quality of the reconstructed images was determined objectively from the normalized mean square between the simulated data and the true distribution. This ranking was validated against the preferences of a group of trained observers. The results from this study indicate that the optimal choice of spatial resolution (collimation) depends on the available count density.

Reconstruction-reprojection processing of transmission scans and the variance of PET images

Two methods for processing noisy transmission sinograms are compared on the basis of the sample variances of emission images reconstructed from simulated data. The first method, reconstruction-reprojection, reconstructs a transmission image, and then reprojects and exponentiates it to form a smoothed set of attenuation correction factors. The other method low-pass filters each projection in the transmission data. It showed analytically that reconstruction-reprojection significantly reduces the variance of the attenuation correction factors even when the reconstruction step uses a ramp filter. Simulations were done to determine whether this reduction emission image when compared to low-pass filtering. The results of these simulations are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A simple method for the measurement of local statistical noise levels in SPECT

A simple method that can be used to assess the magnitude and variation of local noise within a reconstructed section is described. This permits the application of general formulae for the prediction of noise levels to be applied to specific SPECT systems. It was found, however, that the use of real as opposed to simulated data changed the nature of the noise expression. Large errors can be obtained if the results from simulation studies are applied directly to real data. A variation in the RMS% noise of around 20% was measured within an individual section, the noise being highest towards the centre of the source distribution. The method was also used to compare the effects of different reconstruction filters. A soft Shepp-Logan filter was shown to reduce the noise level by a factor of around three when compared to the Ramp filter.

Optimal edge detectors for ramp edges

It is argued that the best way to model an edge is by assuming all ideal mathematical function passed through a low-pass filter and and immersed in noise. Using techniques similar to those developed by J. Canny (1983, 1986) and L.A. Spacek (1986), optimal filters are derived for ramp edges of various slopes. The optimal nonrecursive filter for ideal step edges is then derived as a limiting case of the filters for ramp edges. Because there are no step edges in images, edge detection is improved when the ramp filter is used instead of the filters developed for step edges. For practical purposes, some convolution masks are given which can be used directly for edge detection without the need to go into the details of the subject.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An artificial neural network for SPECT image reconstruction

An artificial neural network has been developed to reconstruct quantitative single photon emission computed tomographic (SPECT) images. The network is trained with an ideal projection-image pair to learn a shift-invariant weighting (filter) for the projections. Once trained, the network produces weighted projections as a hidden layer when acquired projection data are presented to its input. This hidden layer is then backprojected to form an image as the network output. The learning algorithm adjusts the weighting coefficients using a backpropagation algorithm which minimizes the mean squared error between the ideal training image and the reconstructed training image. The response of the trained network to an impulse projection resembles the ramp filter typically used with backprojection, and reconstructed images are similar to filtered backprojection images.

Characterization of the modulation transfer function of discrete filtered backprojection

A mathematical expression for the modulation transfer function (MTF) of image reconstruction by discrete filtered backprojection (DFBP) is derived. A simulation study is used to investigate the dependence of the MTF of DFBP on: (1) the number of projection views; (2) the type of ramp filter used; (3) the interpolation method used during backprojection; and (4) the position of the object. These results were compared to MTFs calculated from point-source single-photon-emission computed tomographic (SPECT) acquisitions in air. The experimentally obtained MTFs contained much of the same structure as the MTFs of DFBP obtained through simulation. It is shown that the discretization of the filtered backprojection process can cause the tomographic transfer function to be anisotropic and nonstationary. However, through proper selection of the methods used in reconstruction, a nearly isotropic and stationary MTF can be obtained.

Interactive reconstruction in single-photon tomography.

A new method is described to allow interactive selection of the reconstruction filter at the time of interpretation of images from single-photon tomography. In the filtered back projection algorithm, the only part of the reconstruction process requiring user interaction is the selection of the window function. Since the ramp and window filters have different purposes, they can be separated, placing the window at the end of the reconstruction process as a three-dimensional filter. All stages of reconstruction except the window filtering are performed before the physician begins to interpret the study. The three-dimensional filtering is performed very rapidly with use of the Chebyshev convolution algorithm. A 64 x 64 x 64 pixel cube of data is filtered in 13-33 s using filters of 3-11 lengths. Smaller volumes of image data can be filtered in less than 1 s; thus, the user can interactively choose any desired filter for a given tomographic study at the time of interpretation of the images.

Compensation for collimator divergence in SPECT using inverse Monte Carlo reconstruction

The inverse Monte Carlo (IMC) reconstruction algorithm achieved compensation for two degrading collimator effects in SPECT (single-photon-emission computed tomography) imaging: position dependence and radial/tangential asymmetry in reconstructed resolution. With filtered backprojection using a ramp filter, full width at half maximum (FWHM, radial and tangential, respectively) for a line source on axis with a high-sensitivity collimator were 19.3 mm and 16.4 mm. At 12 cm off axis, FWHM were 14.8 mm and 8.2 mm, respectively. With IMC after 35 iterations, FWHM on axis were 8.4 mm and 7.7 mm and off axis were 6.0 mm and 7.4 mm, respectively. Thus, for the simple imaging task of line sources with no background activity using simulated projection data, IMC provides compensation for the collimator effects in addition to providing higher resolution. >

Inversion of the 3D Radon transform for a multidetector, point-focused SPECT brain scanner

The Radon transform is presented for unattenuated projection data acquired with a multidetector, point-focused SPECT brain scanner. The three-dimensional (3D) integral transform is shown to be exactly invertible only for the case of an ideal machine whose detectors scan away from the source to infinity in all directions. The two-dimensional (2D) ramp filter reconstruction algorithm previously used for this scanner is a limiting case of the 3D method and is exact only for sources with cylindrical symmetry (f(x,y,z)=f(x,y) for all z). Projection data from a long cylinder and a 1.5 cm thick disc of equal diameters and activity concentrations were simulated by computer for the same scan pattern used on the machine. The data were reconstructed with both the 2D and 3D analytic methods. The 2D method produced a 20% bowl-shaped dip in the centre of the disc, whereas the central slice of the 3D reconstruction was more than 97% accurate. The effective, noise-equivalent sensitivity when reconstructing the central slice of the disc with the necessary 3D method is 10.1 times lower than the sensitivity obtained for the long cylindrical source when reconstructing with the 2D method.

5.SPECTのイメージング技術(第41回総会)(I.核医学イメージング技術の標準化)

At present, a large number of SPECT systems are being widely used in Japan, hence, it is reasonable for us to know the physical and imaging characteristics of these SPECT devices, and also to recommend the optimum utility of SPECT systems. For this reason, a survey respect of characteristics of the commercialy available SPECT devices was carried out. In addition to this, various factors which have significant influence over SPECT image quality, such as, data acquisition matrix, reconstruction filter, γ-ray attenuation correction and daily quality control procedure, were also investigated. The materials used for this study are PET/SPECT phantom. Alderson liver phantom filled with Tc-99m solution, and either LFOV-E or ZLC-7500 interfaced to Scintipac 2400 minicomputer with 256 K byte of memory. Following are the results of this study. 1) The suitable data acquisition procedure was 128×128 matrix for linear sampling and approximately 64 views for angular sampling. 2) Reconstructed image using pre-processing filter with Wiener and Butterworth filters provided high quality image as compared with the Ramp filter. 3) Weighted backprojection method (WBP) proposed by Tanaka was superior to other methods, such as Sorenson method and Chang method in the object with non-uniform distribution of radionuclide. 4) It was found that uniformity correction of gamma camera and precise abjustment of the center of rotation are most important to maintain the images with a high quality.

Analysis of x-ray computed tomography images using the noise power spectrum and autocorrelation function

A discrete representation of the reconstruction process, consistent with the method of data collection, has been used to derive expressions for the noise power spectrum, autocorrelation function and noise equivalent quanta (NEQ) of a computed tomography (CT) image. These parameters have been expressed in terms of basic scanning factors such as tube current, exposure time, slice width and number of detectors. Each of these factors affects the overall magnitude of the noise power spectrum, but the spatial frequency dependence is also determined by the type of reconstruction filter used in the computer algorithm. The noise power spectrum has been calculated for scanners employing either a ramp or Hanning weighted ramp filter. Predictions made from this theoretical analysis have been compared with experimental measurements made on various CT scanners. Measurements were made of the modulation transfer function (MTF) by techniques which permitted the authors to deduce the contributions of the algorithmic and nonalgorithmic components to the overall MTF. NEQ values have been calculated for a number of CT scanners.

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 mod G(k) mod 2/k where k is the radial spatial frequency and G(k) is the one-dimensional corrective filter used in the filtered back-projection reconstruction 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.