Radon Projections Research Articles

Asymptotic behavior of interpolation polynomials of harmonic functions based on radon projections

We study the asymptotic behavior of harmonic interpolation of harmonic functions based on Radon projections when the chords coalesce to some points, a chord and a point. We show that the limit is the Lagrange or Taylor-type interpolation at coalescing points or chords.

An Alternative Model to Radon Transform for Gamma Ray Emission Tomography

The Radon transform fits badly Single Photon Emission Tomography (SPECT). However, Thin Holes Collimator (THC) and Radon model are widely used. The CACAO project has been proposed to enhance the quality of SPECT images. CACAO is a short hand notation for computer aided collimation tomography. The main idea of this project is to use collimators with much larger holes to increase the sensitivity, and slightly longer holes to increase the spatial resolution. The acquisition sequence includes a translation. The Radon projection is replaced by a 2D sum. A dedicated reconstruction algorithm has been developed. If the physical advantage of the project in terms of sensitivity and spatial resolution is generally admitted, a question remains unanswered: Would the ill-posedness of the inverse problem ruin the quality of the reconstructed images? In this article, a representation of the 2D direct problem matrix is derived. This allows us to compare the two inverse problems (CACAO versus THC). The condition number was used for this comparison. We studied the variation of these condition numbers with several parameters. For a proper set of parameters, the CACAO inverse problem may appear easier to solve and more accurately than the THC one.

Concepts and Tradeoffs in Velocity Estimation With Plane-Wave Contrast-Enhanced Doppler.

While long Doppler ensembles are, in principle, beneficial for velocity estimates, short acoustic pulses must be used in microbubble contrast-enhanced (CE) Doppler to mitigate microbubble destruction. This introduces inherent tradeoffs in velocity estimates with autocorrelators, which are studied here. A model of the autocorrelation function adapted to the microbubble Doppler signal accounting for transit time, the echo frequency uncertainty, and contrast-agent destruction is derived and validated in vitro. It is further demonstrated that a local measurement of the center frequency of the microbubble echo is essential in order to avoid significant bias in velocity estimates arising from the linear and nonlinear frequency-dependent scattering of microbubbles and compensate for the inherent speckle nature of the received echo frequency. For these reasons, broadband Doppler estimators (2-D autocorrelator and Radon projection) are better suited than simpler narrow-band estimators (1-D autocorrelator and 1-D Fourier transform) for CE flow assessment. A case study of perfusion in a VX-2 carcinoma using CE plane-wave Doppler is also shown. We demonstrate that even when considering all uncertainties associated with microbubble-related decorrelation (destruction, pulse bandwidth, transit time, and flow gradient) and the need for real-time imaging, a coefficient of variation of 4% on the axial velocity is achievable with plane-wave imaging.

On the Szegő–Radon projection of monogenic functions

We define a version of the Radon transform for monogenic functions which is based on Szegő kernels. The corresponding Szegő–Radon projection is abstractly defined as the orthogonal projection of a Hilbert module of left monogenic functions onto a suitable closed submodule of functions depending only on two variables. We also establish the inversion formula based on the dual transform.

Inversion of the spherical means transform in corner-like domains by reduction to the classical Radon transform

We consider an inverse problem arising in thermo-/photo- acoustic tomography that amounts to reconstructing a function f from its circular or spherical means with the centers lying on a given measurement surface. (Equivalently, these means can be expressed through the solution of the wave equation with the initial pressure equal to f.) An explicit solution of this inverse problem is obtained in 3D for the surface that is the boundary of an open octet, and in 2D for the case when the centers of integration circles lie on two rays starting at the origin and intersecting at the angle equal to , . Our formulas reconstruct the Radon projections of a function closely related to f, from the values of on the measurement surface. Then, function f can be found by inverting the Radon transform.

New results on regularity and errors of harmonic interpolation using Radon projections

We study interpolation of harmonic functions in the unit disk with a finite number of values of the Radon projection along prescribed chords as the input data. We seek the interpolant in the space of harmonic polynomials in such a way that it matches the given projection values exactly. In this setting, we investigate schemes where all chords are divided into two sets of parallel chords. We give necessary and sufficient conditions for a scheme of this type to result in a uniquely solvable interpolation problem. As a second new result, we generalize the previously known error estimates for schemes with equispaced chord angles, both to allow for a larger class of chord choices and to obtain new error estimates in fractional Sobolev norms.

Interpolating solutions of the Poisson equation in the disk based on Radon projections

We consider an algebraic method for reconstruction of a function satisfying the Poisson equation with a polynomial right-hand side in the unit disk. The given data, besides the right-hand side, is assumed to be in the form of a finite number of values of Radon projections of the unknown function. We first homogenize the problem by finding a polynomial which satisfies the given Poisson equation. This leads to an interpolation problem for a harmonic function, which we solve in the space of harmonic polynomials using a previously established method. For the special case where the Radon projections are taken along chords that form a regular convex polygon, we extend the error estimates from the harmonic case to this Poisson problem. Finally we give some numerical examples.

Projection-based medical image compression for telemedicine applications.

Recent years have seen great development in the field of medical imaging and telemedicine. Despite the developments in storage and communication technologies, compression of medical data remains challenging. This paper proposes an efficient medical image compression method for telemedicine. The proposed method takes advantage of Radon transform whose basis functions are effective in representing the directional information. The periodic re-ordering of the elements of Radon projections requires minimal interpolation and preserves all of the original image pixel intensities. The dimension-reducing property allows the conversion of 2D processing task to a set of simple 1D task independently on each of the projections. The resultant Radon coefficients are then encoded using set partitioning in hierarchical trees (SPIHT) encoder. Experimental results obtained on a set of medical images demonstrate that the proposed method provides competing performance compared with conventional and state-of-the art compression methods in terms of compression ratio, peak signal-to-noise ratio (PSNR), and computational time.

Cubature rules for harmonic functions based on Radon projections

We construct a class of cubature formulae for harmonic functions on the unit disk based on line integrals over $$2n+1$$2n+1 distinct chords. These chords are assumed to have constant distance $$t$$t to the center of the disk, and their angles to be equispaced over the interval $$[0,2\pi ]$$[0,2?]. If $$t$$t is chosen properly, these formulae integrate exactly all harmonic polynomials of degree up to $$4n+1$$4n+1, which is the highest achievable degree of precision for this class of cubature formulae. For more generally distributed chords, we introduce a class of interpolatory cubature formulae which we show to coincide with the previous formulae for the equispaced case. We give an error estimate for a particular cubature rule from this class and provide numerical examples.

Interpolation of harmonic functions based on Radon projections

We consider an algebraic method for reconstruction of a harmonic function in the unit disk via a finite number of values of its Radon projections. The approach is to seek a harmonic polynomial which matches given values of Radon projections along some chords of the unit circle. We prove an analogue of the famous Marr's formula for computing the Radon projection of the basis orthogonal polynomials in our setting of harmonic polynomials. Using this result, we show unique solvability for a family of schemes where all chords are chosen at equal distance to the origin. For the special case of chords forming a regular convex polygon, we prove error estimates on the unit circle and in the unit disk. We present an efficient reconstruction algorithm which is robust with respect to noise in the input data and provide numerical examples.

Super-resolution compressive imaging with anamorphic optics

A new imaging technique that combines compressive sensing and super-resolution techniques is presented. Compressive sensing is accomplished by capturing optically a set of Radon projections. Super-resolution measurements are simply taken by introducing a slanted two-dimensional array in the optical system. The goal of the technique is to overcome resolution limitation that occurs in imaging scenarios where dense pixels sensors with large number of pixels are not available or cannot be used. With the presented imaging technique, owing to the compressive sensing approach, we were able to reconstruct images with significantly more number of pixels than measured, and owing to the super-resolution design we have been able to achieve resolution significantly beyond that limited by the sensor's pixels size.

A 4-quadrant Curvelet Transform for Denoising Digital Images

The conventional discrete wavelet transform (DWT) introduces artifacts during denoising of images containing smooth curves. Finite ridgelet transform (FRIT) solved this problem by mapping the curves in terms of small curved ridges. However, blind application of FRIT all over an image is computationally heavy. Finite curvelet transform (FCT) selectively applies FRIT only to the tiles containing small portions of a curve. In this work, a novel curvelet transform named as 4-quadrant finite curvelet transform (4QFCT) based on a new concept of 4-quadrant finite ridgelet transform (4QFRIT) has been proposed. An image is band pass filtered and the high frequency bands are divided into small non-overlapping square tiles. The 4QFRIT is applied to the tiles containing at least one curve element. Unlike FRIT, the 4QFRIT takes 4 sets of radon projections in all the 4 quadrants and then averages them in time and frequency domains after denoising. The proposed algorithm is extensively tested and benchmarked for denoising of images with Gaussian noise using mean squared error (MSE) and peak signal to noise ratio (PSNR). The results confirm that 4QFCT yields consistently better denoising performance quantitatively and visually.

TU‐C‐103‐06: A Simple Method for Simulating Reduced‐Dose Images for Evaluation of Clinical CT Protocols

Purpose: To develop a simple method using readily‐available software and equipment to simulate clinically‐realistic dose reduction levels for CT images from regular‐dose clinical images. The resulting images will be used to demonstrate the effects of dose reduction on image quality to clinicians and aid in guiding protocol revision. Methods: A reconstruction of an anthropomorphic phantom scanned under a clinical protocol was imported into MATLAB. The image was converted to attenuation coefficients before forward‐projecting using the built‐in radon function. Noise was added to the projections using a Poisson statistics model. The original radon projections were then subtracted from the simulated projections to produce noise‐only projections, which could then be reconstructed using the iradon function with a custom filter. The filter was calculated by first measuring the radial average of the two‐dimensional noise power spectrum (NPS) in a noise‐only image generated from subtracted back‐to‐back scans of a water cylinder. The filter used by the scanner is proportional to the square‐root of the radially‐averaged NPS after normalizing it to radial frequency. The final noise‐only reconstruction is then added to the original clinical image to simulate a reduced‐dose scan. Results: The noise‐only reconstructions show accurate structure and texture for both the standard and bone reconstructions of the head phantom. The final simulated images (with the noise reconstruction added to the original reconstruction) closely match the appearance of the actual reduced‐dose images. Conclusion: Previous methods for noise simulation that rely on raw data access and advanced system knowledge allow for very low dose levels to be accurately simulated. However, these ultra‐low dose levels are often too low to be of clinical significance. This work provides a simple and accurate tool for dose reduction simulations that are more clinically realistic. In addition, this method only requires measurements that can be acquired using standard procedures and phantoms.

Breast cancer detection from thermal images using bispectral invariant features

Highly sensitive infrared (IR) cameras provide high-resolution diagnostic images of the temperature and vascular changes of breasts. These images can be processed to emphasize hot spots that exhibit early and subtle changes owing to pathology. The resulting images show clusters that appear random in shape and spatial distribution but carry class dependent information in shape and texture. Automated pattern recognition techniques are challenged because of changes in location, size and orientation of these clusters. Higher order spectral invariant features provide robustness to such transformations and are suited for texture and shape dependent information extraction from noisy images. In this work, the effectiveness of bispectral invariant features in diagnostic classification of breast thermal images into malignant, benign and normal classes is evaluated and a phase-only variant of these features is proposed. High resolution IR images of breasts, captured with measuring accuracy of ±0.4% (full scale) and temperature resolution of 0.1 °C black body, depicting malignant, benign and normal pathologies are used in this study. Breast images are registered using their lower boundaries, automatically extracted using landmark points whose locations are learned during training. Boundaries are extracted using Canny edge detection and elimination of inner edges. Breast images are then segmented using fuzzy c-means clustering and the hottest regions are selected for feature extraction. Bispectral invariant features are extracted from Radon projections of these images. An Adaboost classifier is used to select and fuse the best features during training and then classify unseen test images into malignant, benign and normal classes. A data set comprising 9 malignant, 12 benign and 11 normal cases is used for evaluation of performance. Malignant cases are detected with 95% accuracy. A variant of the features using the normalized bispectrum, which discards all magnitude information, is shown to perform better for classification between benign and normal cases, with 83% accuracy compared to 66% for the original.

Belief-propagation reconstruction for discrete tomography

We consider the reconstruction of a two-dimensional discrete image from a set of tomographic measurements corresponding to the Radon projection. Assuming that the image has a structure where neighbouring pixels have a larger probability of taking the same value, we follow a Bayesian approach and introduce a fast message-passing reconstruction algorithm based on belief propagation. For numerical results, we specialize to the case of binary tomography. We test the algorithm on binary synthetic images with different length scales and compare our results against a more usual convex optimization approach. We investigate the reconstruction error as a function of the number of tomographic measurements, corresponding to the number of projection angles. The belief-propagation algorithm turns out to be more efficient than the convex-optimization algorithm, both in terms of recovery bounds for noise-free projections and reconstruction quality when moderate Gaussian noise is added to the projections.

Harmonic interpolation based on Radon projections along the sides of regular polygons

Abstract Given information about a harmonic function in two variables, consisting of a finite number of values of its Radon projections, i.e., integrals along some chords of the unit circle, we study the problem of interpolating these data by a harmonic polynomial. With the help of symbolic summation techniques we show that this interpolation problem has a unique solution in the case when the chords form a regular polygon. Numerical experiments for this and more general cases are presented.

Reducing frequency-domain artifacts of binary image due to coarse sampling by repeated interpolation and smoothing of Radon projections

We develop a method to calculate 2D spectrum of a binary image with better quality than that obtained via direct 2D-FFT. With FFT, jagged edges of objects due to coarse sampling introduce artifacts into the frequency domain, especially in the high-frequency area. With the proposed method, Radon projections of the binary image along lines at different viewing angles are calculated. Each projection is extended by interpolation, then smoothed and decimated. The interpolation–smoothing–decimation operation is repeated several times to reduce ruggedness and improve quality of the Radon projections considerably. One-dimensional FFT of each refined Radon projection is calculated, resulting in a set of frequency-domain samples distributed on a polar coordinate system. These samples are interpolated onto a Cartesian grid to give the required 2D spectrum of the sampled binary image. Numerical computations on several objects show that the method can provide significant improvement to the spectrum as compared with direct 2D-FFT.

Progressive compressive imaging from Radon projections

In this work we propose a unique sampling scheme of Radon Projections and a non-linear reconstruction algorithm based on compressive sensing (CS) theory to implement a progressive compressive sampling imaging system. The progressive sampling scheme offers online control of the tradeoff between the compression and the quality of reconstruction. It avoids the need of a priori knowledge of the object sparsity that is usually required for CS design. In addition, the progressive data acquisition enables straightforward application of ordered-subsets algorithms which overcome computational constraints associated with the reconstruction of very large images. We present, to the best of our knowledge for the first time, a compressive imaging implementation of megapixel size images with a compression ratio of 20:1.

Performance Analysis of Text-Independent Speaker Identification System

In this paper, the text independent speaker identification system is implemented and analyzed using the speaker specific features extracted by applying image processing techniques. The speech is converted into spectrogram, an efficient representation of the speech signal in the form of pattern. Radon Transform (RT) and Discrete Cosine Transform (DCT) are applied on the spectrogram to extract the features. Radon Transform is applied for eight orientations on the acoustic characteristics of the spectrogram. After applying RT, the two dimensional DCT has been applied on Radon projections yields low dimensional feature vectors. The performance of the algorithm has been evaluated with our own created BCS database using sound forge version 5.0. The effect of number of Radon projections and DCT coefficients are analyzed. The demonstration program is coded by MATLAB V 7.0.4. The recognition rate of this algorithm is 96% on BCS database

Performance Analysis of Text-Independent Speaker Identification System

In this paper, the text independent speaker identification system is implemented and analyzed using the speaker specific features extracted by applying image processing techniques. The speech is converted into spectrogram, an efficient representation of the speech signal in the form of pattern. Radon Transform (RT) and Discrete Cosine Transform (DCT) are applied on the spectrogram to extract the features. Radon Transform is applied for eight orientations on the acoustic characteristics of the spectrogram. After applying RT, the two dimensional DCT has been applied on Radon projections yields low dimensional feature vectors. The performance of the algorithm has been evaluated with our own created BCS database using sound forge version 5.0. The effect of number of Radon projections and DCT coefficients are analyzed. The demonstration program is coded by MATLAB V 7.0.4. The recognition rate of this algorithm is 96% on BCS database.