Decomposition of images and objects into measurement and null components

Future trends for patient-specific dosimetry methodology in molecular radiotherapy

Molecular radiotherapy is rapidly expanding, and new radiotherapeutics are emerging. The majority of treatments is still performed using empirical fixed activities and not tailored for individual patients. Molecular radiotherapy dosimetry is often seen as a promising candidate that would allow personalisation of treatments as outcome should ultimately depend on the absorbed doses delivered and not the activities administered. The field of molecular radiotherapy dosimetry has made considerable progress towards the feasibility of routine clinical dosimetry with reasonably accurate absorbed-dose estimates for a range of molecular radiotherapy dosimetry applications. A range of challenges remain with respect to the accurate quantification, assessment of time-integrated activity and absorbed dose estimation. In this review, we summarise a range of technological and methodological advancements, mainly focussed on beta-emitting molecular radiotherapeutics, that aim to improve molecular radiotherapy dosimetry to achieve accurate, reproducible, and streamlined dosimetry. We describe how these new technologies can potentially improve the often time-consuming considered process of dosimetry and provide suggestions as to what further developments might be required.

Effect on null spaces of list-mode imaging systems due to increasing the number of attributes

There are two types of uncertainty in image reconstructions from list-mode data: statistical and deterministic. One source of statistical uncertainty is the finite number of attributes of the detected particles, which are sampled from a probability distribution on the attribute space. A deterministic source of uncertainty is the effect that null functions of the imaging operator have on reconstructed pixel or voxel values. Quantifying the reduction in this deterministic source of uncertainty when more attributes are measured for each detected particle is the subject of this work. Specifically, upper bounds on an error metric are derived to quantify the error introduced in the reconstruction by the presence of null functions, and these upper bounds are shown to be reduced when the number of attributes is increased. These bounds are illustrated with an example of a two-dimensional single photon emission computed tomography (SPECT) system where the depth of interaction in the scintillation crystal is added to the attribute vector.

Computing a projection operator onto the null space of a linear imaging operator: tutorial

Many imaging systems can be approximately described by a linear operator that maps an object property to a collection of discrete measurements. However, even in the absence of measurement noise, such operators are generally "blind" to certain components of the object, and hence information is lost in the imaging process. Mathematically, this is explained by the fact that the imaging operator can possess a null space. All objects in the null space, by definition, are mapped to a collection of identically zero measurements and are hence invisible to the imaging system. As such, characterizing the null space of an imaging operator is of fundamental importance when comparing and/or designing imaging systems. A characterization of the null space can also facilitate the design of regularization strategies for image reconstruction methods. Characterizing the null space via an associated projection operator is, in general, a computationally demanding task. In this tutorial, computational procedures for establishing projection operators that map an object to the null space of a discrete-to-discrete imaging operator are surveyed. A new machine-learning-based approach that employs a linear autoencoder is also presented. The procedures are demonstrated by use of biomedical imaging examples, and their computational complexities and memory requirements are compared.

On Hallucinations in Tomographic Image Reconstruction

Tomographic image reconstruction is generally an ill-posed linear inverse problem. Such ill-posed inverse problems are typically regularized using prior knowledge of the sought-after object property. Recently, deep neural networks have been actively investigated for regularizing image reconstruction problems by learning a prior for the object properties from training images. However, an analysis of the prior information learned by these deep networks and their ability to generalize to data that may lie outside the training distribution is still being explored. An inaccurate prior might lead to false structures being hallucinated in the reconstructed image and that is a cause for serious concern in medical imaging. In this work, we propose to illustrate the effect of the prior imposed by a reconstruction method by decomposing the image estimate into generalized measurement and null components. The concept of a hallucination map is introduced for the general purpose of understanding the effect of the prior in regularized reconstruction methods. Numerical studies are conducted corresponding to a stylized tomographic imaging modality. The behavior of different reconstruction methods under the proposed formalism is discussed with the help of the numerical studies.

Effect on null spaces of list-mode imaging systems due to increasing the size of attribute space

An upper bound is derived for a figure of merit that quantifies the error in reconstructed pixel or voxel values induced by the presence of null functions for any list-mode system. It is shown that this upper bound decreases as the region in attribute space occupied by the allowable attribute vectors expands. This upper bound allows quantification of the reduction in this error when this type of expansion is implemented. Of course, reconstruction error is also caused by system noise in the data, which has to be treated statistically, but we will not be addressing that problem here. This method is not restricted to pixelized or voxelized reconstructions and can in fact be applied to any region of interest. The upper bound for pixelized reconstructions is demonstrated on a list-mode 2D Radon transform example. The expansion in the attribute space is implemented by doubling the number of views. The results show how the pixel size and number of views both affect the upper bound on reconstruction error from null functions. This reconstruction error can be averaged over all pixels to give a single number or can be plotted as a function on the pixel grid. Both approaches are demonstrated for the example system. In conclusion, this method can be applied to any list-mode system for which the system operator is known and could be used in the design of the systems and reconstruction algorithms.

Singular value decomposition for photon-processing nuclear imaging systems and applications for reconstruction and computing null functions

Recent advances in technology are enabling a new class of nuclear imaging systems consisting of detectors that use real-time maximum-likelihood (ML) methods to estimate the interaction position, deposited energy, and other attributes of each photon-interaction event and store these attributes in a list format. This class of systems, which we refer to as photon-processing (PP) nuclear imaging systems, can be described by a fundamentally different mathematical imaging operator that allows processing of the continuous-valued photon attributes on a per-photon basis. Unlike conventional photon-counting (PC) systems that bin the data into images, PP systems do not have any binning-related information loss. Mathematically, while PC systems have an infinite-dimensional null space due to dimensionality considerations, PP systems do not necessarily suffer from this issue. Therefore, PP systems have the potential to provide improved performance in comparison to PC systems. To study these advantages, we propose a framework to perform the singular-value decomposition (SVD) of the PP imaging operator. We use this framework to perform the SVD of operators that describe a general two-dimensional (2D) planar linear shift-invariant (LSIV) PP system and a hypothetical continuously rotating 2D single-photon emission computed tomography (SPECT) PP system. We then discuss two applications of the SVD framework. The first application is to decompose the object being imaged by the PP imaging system into measurement and null components. We compare these components to the measurement and null components obtained with PC systems. In the process, we also present a procedure to compute the null functions for a PC system. The second application is designing analytical reconstruction algorithms for PP systems. The proposed analytical approach exploits the fact that PP systems acquire data in a continuous domain to estimate a continuous object function. The approach is parallelizable and implemented for graphics processing units (GPUs). Further, this approach leverages another important advantage of PP systems, namely the possibility to perform photon-by-photon real-time reconstruction. We demonstrate the application of the approach to perform reconstruction in a simulated 2D SPECT system. The results help to validate and demonstrate the utility of the proposed method and show that PP systems can help overcome the aliasing artifacts that are otherwise intrinsically present in PC systems.

Image Science with Photon-Processing Detectors

We introduce and discuss photon-processing detectors and we compare them with photon-counting detectors. By estimating a relatively small number of attributes for each collected photon, photon-processing detectors may help understand and solve a fundamental theoretical problem of any imaging system based on photon-counting detectors, namely null functions. We argue that photon-processing detectors can improve task performance by estimating position, energy, and time of arrival for each collected photon. We consider a continuous-to-continuous linear operator to relate the object being imaged to the collected data, and discuss how this operator can be analyzed to derive properties of the imaging system. Finally, we derive an expression for the characteristic functional of an imaging system that produces list-mode data.

Scanning linear estimation: improvements over region of interest (ROI) methods

In tomographic medical imaging, a signal activity is typically estimated by summing voxels from a reconstructed image. We introduce an alternative estimation scheme that operates on the raw projection data and offers a substantial improvement, as measured by the ensemble mean-square error (EMSE), when compared to using voxel values from a maximum-likelihood expectation-maximization (MLEM) reconstruction. The scanning-linear (SL) estimator operates on the raw projection data and is derived as a special case of maximum-likelihood estimation with a series of approximations to make the calculation tractable. The approximated likelihood accounts for background randomness, measurement noise and variability in the parameters to be estimated. When signal size and location are known, the SL estimate of signal activity is unbiased, i.e. the average estimate equals the true value. By contrast, unpredictable bias arising from the null functions of the imaging system affect standard algorithms that operate on reconstructed data. The SL method is demonstrated for two different tasks: (1) simultaneously estimating a signal's size, location and activity; (2) for a fixed signal size and location, estimating activity. Noisy projection data are realistically simulated using measured calibration data from the multi-module multi-resolution small-animal SPECT imaging system. For both tasks, the same set of images is reconstructed using the MLEM algorithm (80 iterations), and the average and maximum values within the region of interest (ROI) are calculated for comparison. This comparison shows dramatic improvements in EMSE for the SL estimates. To show that the bias in ROI estimates affects not only absolute values but also relative differences, such as those used to monitor the response to therapy, the activity estimation task is repeated for three different signal sizes.

Null-space function estimation for the interior problem

In single-photon emission computed tomography (SPECT), projection data can be truncated when the camera's field of view is smaller than the object to be imaged. Using truncated projections to reconstruct a region of interest (ROI) is a reality we must face if small detectors are used. The truncated data result in an underdetermined system of imaging equations, which may lead to non-unique solutions. Data sampling and photon attenuation may also affect the solution uniqueness and stability. The uniqueness of the solutions in the ROI can be investigated by studying the null-space functions in the ROI. This paper uses an iterative algorithm to estimate the null-space image, to determine the sampling conditions under which a stable ROI reconstruction is possible with truncated data and to investigate whether attenuation can influence the ROI reconstruction bias. This iterative algorithm is validated by the singular value decomposition method. We show that if the ROI is sufficiently sampled, the null-space image is close to zero inside the ROI, and any almost-zero offset is insignificant in SPECT, because the noise is a much more dominating degradation factor.

Singular value decomposition analysis of a photoacoustic imaging system and 3D imaging at 0.7 FPS

Photoacoustic imaging is a non-ionizing imaging modality that provides contrast consistent with optical imaging techniques while the resolution and penetration depth is similar to ultrasound techniques. In a previous publication [Opt. Express 18, 11406 (2010)], a technique was introduced to experimentally acquire the imaging operator for a photoacoustic imaging system. While this was an important foundation for future work, we have recently improved the experimental procedure allowing for a more densely populated imaging operator to be acquired. Subsets of the imaging operator were produced by varying the transducer count as well as the measurement space temporal sampling rate. Examination of the matrix rank and the effect of contributing object space singular vectors to image reconstruction were performed. For a PAI system collecting only limited data projections, matrix rank increased linearly with transducer count and measurement space temporal sampling rate. Image reconstruction using a regularized pseudoinverse of the imaging operator was performed on photoacoustic signals from a point source, line source, and an array of point sources derived from the imaging operator. As expected, image quality increased for each object with increasing transducer count and measurement space temporal sampling rate. Using the same approach, but on experimentally sampled photoacoustic signals from a moving point-like source, acquisition, data transfer, reconstruction and image display took 1.4 s using one laser pulse per 3D frame. With relatively simple hardware improvements to data transfer and computation speed, our current imaging results imply that acquisition and display of 3D photoacoustic images at laser repetition rates of 10Hz is easily achieved.

Analysis of a photoacoustic imaging system by the crosstalk matrix and singular value decomposition

Photoacoustic imaging is a hybrid imaging modality capable of producing contrast similar to optical imaging techniques but with increased penetration depth and resolution in turbid media by encoding the information as acoustic waves. In general, it is important to characterize the performance of a photoacoustic imaging system by parameters such as sensitivity, resolution, and contrast. However, system characterization can extend beyond these metrics by implementing advanced analysis via the crosstalk matrix and singular value decomposition. A method was developed to experimentally measure a matrix that represented the imaging operator for a photoacoustic imaging system. Computations to produce the crosstalk matrix were completed to provide insight into the spatially dependent sensitivity and aliasing for the photoacoustic imaging system. Further analysis of the imaging operator was done via singular value decomposition to estimate the capability of the imaging system to reconstruct objects and the inherent sensitivity to those objects. The results provided by singular value decomposition were compared to SVD results from a de-noised imaging operator to estimate the number of measurable singular vectors for the system. These characterization techniques can be broadly applied to any photoacoustic system and, with regards to the studied system, could be used as a basis for improvements to future iterations.

Range Condition and ML-EM Checkerboard Artifacts

The expectation maximization (EM) algorithm for the maximum likelihood (ML) image reconstruction criterion generates severe checkerboard artifacts in the presence of noise. A classical remedy is to impose an a priori constraint for a penalized ML or maximum a posteriori probability solution. The penalty reduces the checkerboard artifacts and also introduces uncertainty because a priori information is usually unknown in clinic. Recent theoretical investigation reveals that the noise can be divided into two components: one is called null-space noise and the other is range-space noise. The null-space noise can be numerically estimated using filtered backprojection (FBP) algorithm. By the FBP algorithm, the null-space noise annihilates in the reconstruction while the range-space noise propagates into the reconstructed image. The aim of this work is to investigate the relation between the null-space noise and the checkerboard artifacts in the ML-EM reconstruction from noisy projection data. Our study suggests that removing the null-space noise from the projection data could improve the signal-to-noise ratio of the projection data and, therefore, reduce the checkerboard artifacts in the ML-EM reconstructed images. This study reveals an in-depth understanding of the different noise propagations in analytical and iterative image reconstructions, which may be useful to single photon emission computed tomography, where the noise has been a major factor for image degradation. The reduction of the ML-EM checkerboard artifacts by removing the null-space noise avoids the uncertainty of using a priori penalty.