Feedback control for sparsity based regularization parameter selection in X-ray tomography

Röntgentomografia on kuvantamismenetelmä, jolla pyritään selvittämään objektin sisärakenne eri suunnista otettujen röntgenkuvien perusteella. Kun käytettävissä on kuvia vain hyvin pienestä määrästä suuntia, tomografiaongelma on äärimmäisen huonosti asetettu ja sen ratkaisu vaatii regularisointia. Regularisointi koostuu sopivan regularisoijan ja regularisointiparametrin arvon valinnasta. Tämä tutkielma käsittelee tapausta, jossa halutaan regularisoida rekonstruktion harvuutta Haarin aallokekannan suhteen. Se johtaa minimisaatio-ongelmaan, joka ratkaistaan iteratiivisella pehmeää kynnystystä käyttävällä algoritmilla (iterative soft thresholding algorithm eli ISTA). Regularisointiparametri valitaan olettamalla että käytettävissä on a priori tunnettu harvuustaso, eli luku joka kertoo kuinka suuri osuus objektia kuvaavista aallokekertoimista on erisuuria kuin nolla, ja säätämällä regularisointiparametria iteraation aikana niin, että rekonstruktion harvuus saavuttaa kyseisen tason. Tätä varten käytämme variaatioita proportional-integral-derivative-säätimestä (PID-säädin). Jotta haluttu harvuustaso saavutetaan tulee säädin virittää sopivasti. Tutkimme eri virityksien vaikutusta rekonstruktioprosessiin ja erityisesti käsittelemme kahta adaptiivista säädinvarianttia parametrin valinnassa. Vertailemme näitä kahta varianttia, adaptiivista integraalisäädintä ja neuroverkkoihin perustuva PID-säädintä, toisiinsa. Lopuksi vielä vertaamme adaptiivisella integraalisäätimellä säädettyä ISTA:a kahteen klassiseen rekonstruktioalgoritmiin: suodatettuun takaisinprojektioon (filtered back projection, FBP) ja Tikhonov regularisointiin. Kokeissa käytetään sekä aitoa että simuloitua röntgendataa sekä verrattain tiheällä että harvemmalla mittauskulmien jakaumalla. Integraalisäätö on osoitettiin tärkeäksi regularisointiparametrin valinnassa, kun taas kahta muuta termiä voidaan hyödyntää tarpeen vaatiessa. Adaptiivisista säätimistä adaptiivinen integraalisäädin osoittautui kaikin kriteerein paremmaksi. Adaptiivisella integraalisäätimellä säädetty ISTA myös päihitti molemmat klassiset menetelmät sekä suhteellisen virheen että visuaalisen arvioinnin suhteen harvan datan tapauksessa. Tulokset osoittavat että eri PID-säädinvariantit voivat toimia regularisointiparametrin valinnassa. Adaptiiviset säätimet ovat hyvin käyttäjäystävällisiä, koska ne eivät vaadi manuaalista parametrien säätöä. Lisäksi säätimet ovat verrattain yksinkertaisia, joten niiden soveltaminen eri tilanteissa on helppoa. PID-säätimet mahdollistavat regularisointiparametrin valinnan algoritmin suorituksen aikana, tehden näin koko rekonstruktioprosessista verrattain nopean.X-ray computed tomography is an imaging method where the inner structure of an object is reconstructed from X-ray images taken from multiple directions around the object. When measurements from only a few measurement directions are available, the problem becomes severely ill-posed and requires regularization. This involves choosing a regularizer with desirable properties, as well as a value for the regularization parameter. In this thesis, sparsity promoting regularization with respect to the Haar wavelet basis is considered. The resulting minimization problem is solved using the iterative soft thresholding algorithm (ISTA). For the selection of the regularization parameter, it is assumed that an a priori known level of sparsity is available. The regularization parameter is then varied on each iteration of the algorithm so that the resulting reconstruction has the desired level of sparsity. This is achieved using variants of proportional-integral-derivative (PID) controllers. PID controllers require tuning to guarantee that the desired sparsity level is achieved. We study how different tunings affect the reconstruction process, and experiment with two adaptive variants of PID controllers: an adaptive integral controller, and a neural network based PID controller. The two adaptive methods are compared to each other, and additionally the adaptive integral controlled ISTA is compared to two classical reconstruction methods: filtered back projection and Tikhonov regularization. Computations are performed using both real and simulated X-ray data, with varying amounts of available measurement directions. The integral control is shown to be crucial for the regularization parameter selection while the proportional and derivative terms can be of use if additional control is required. Of the two adaptive variants, the adaptive integral control performs better with respect to all measured figures of merit. The adaptive integral controlled ISTA also outperforms the classical reconstruction methods both in terms of relative error and visual inspection when only a few measurement directions are available. The results indicate that variants of the PID controllers are effective for sparsity based regularization parameter selection. Adaptive variants are very end user friendly, avoiding the manual tuning of parameters. This makes it easier to use sparsity promoting regularization in real life applications. The PID control allows the regularization parameter to be selected during the iteration, thus making the overall reconstruction process relatively fast

Controlled wavelet domain sparsity for x-ray tomography

Tomographic reconstruction is an ill-posed inverse problem that calls for regularization. One possibility is to require sparsity of the unknown in an orthonormal wavelet basis. This, in turn, can be achieved by variational regularization, where the penalty term is the sum of the absolute values of the wavelet coefficients. The primal-dual fixed point algorithm showed that the minimizer of the variational regularization functional can be computed iteratively using a soft-thresholding operation. Choosing the soft-thresholding parameter mu > 0 is analogous to the notoriously difficult problem of picking the optimal regularization parameter in Tikhonov regularization. Here, a novel automatic method is introduced for choosing mu, based on a control algorithm driving the sparsity of the reconstruction to an a priori known ratio of nonzero versus zero wavelet coefficients in the unknown.Peer reviewe

An Automatic Regularization Method : An Application for 3-D X-Ray Micro-CT Reconstruction Using Sparse Data

X-ray tomography is a reliable tool for determining the inner structure of 3-D object with penetrating X-rays. However, traditional reconstruction methods, such as Feldkamp-Davis-Kress (FDK), require dense angular sampling in the data acquisition phase leading to long measurement times, especially in X-ray micro-tomography to obtain high-resolution scans. Acquiring less data using greater angular steps is an obvious way for speeding up the process and avoiding the need to save huge data sets. However, computing 3-D reconstruction from such a sparsely sampled data set is difficult because the measurement data are usually contaminated by errors, and linear measurement models do not contain sufficient information to solve the problem in practice. An automatic regularization method is proposed for robust reconstruction, based on enforcing sparsity in the 3-D shearlet transform domain. The inputs of the algorithm are the projection data and a priori known expected degree of sparsity, denoted as 0 <C-prPeer reviewe

Tomographic X-ray data of a lotus root filled with attenuating objects

<p>This is an open-access dataset of tomographic X-ray data of a slice of a lotus root. The dataset consists of</p> <ul> <li>the X-ray sinogram of a single 2D slice of the lotus root with two different resolutions, and</li> <li>the corresponding measurement matrices modeling the linear operation of the X-ray transform.</li> </ul> <p>Each of these sinograms was obtained from a measured 360-projection fan-beam sinogram by down-sampling and taking logarithms. The original (measured) sinogram is also provided in its original form and resolution.</p> <p>Also MATLAB code for reconstructions using filtered back-projection, Landweber iteration, and Tikhonov regularization are provided.</p> <p>Documentation of the dataset is available at <a href="https://arxiv.org/abs/1609.07299">arxiv.org/abs/1609.07299</a>. See also <a href="https://www.fips.fi/dataset.php">www.fips.fi/dataset.php</a>.</p

An automatic regularization method:an application for 3-D X-ray micro-CT reconstruction using sparse data

Abstract X-ray tomography is a reliable tool for determining the inner structure of 3-D object with penetrating X-rays. However, traditional reconstruction methods, such as Feldkamp-Davis-Kress (FDK), require dense angular sampling in the data acquisition phase leading to long measurement times, especially in X-ray micro-tomography to obtain high-resolution scans. Acquiring less data using greater angular steps is an obvious way for speeding up the process and avoiding the need to save huge data sets. However, computing 3-D reconstruction from such a sparsely sampled data set is difficult because the measurement data are usually contaminated by errors, and linear measurement models do not contain sufficient information to solve the problem in practice. An automatic regularization method is proposed for robust reconstruction, based on enforcing sparsity in the 3-D shearlet-transform domain. The inputs of the algorithm are the projection data and a priori known expected degree of sparsity, denoted as 0 <; C pr ≤ 1. The number Cpr can be calibrated from a few dense-angle reconstructions and fixed. Human subchondral bone samples were tested, and morphometric parameters of the bone reconstructions were then analyzed using standard metrics. The proposed method is shown to outperform the baseline algorithm (FDK) in the case of sparsely collected data. The number of X-ray projections can be reduced up to 10% of the total amount 300 projections over 180° with uniform angular step while retaining the quality of the reconstruction images and of the morphometric parameters