Final Report Summary - HD-TOMO (High-Definition Tomography)
With computed tomography (CT) we can see inside objects – we send signals through an object and measure the response, from which we compute an image of the object’s interior. Medical doctors can look for cancer, physicists can study microscopic details of new materials, engineers can identify internal defects in pipes, and security personnel can inspect luggage for suspicious items.
It is of vital importance that the images are as sharp, detailed and reliable as possible, so scientists, engineers, doctors etc. can make the correct decisions. To achieve high-definition tomography – sharper images with more reliable details – we must use prior information consisting of accumulated knowledge about the object.
Overall Outcome: Insight and Framework
Previous efforts were often based on ad-hoc techniques and naive algorithms with limited applications and ill-defined results. This project focused on obtaining deeper insight and developing a rigorous framework. We carefully analyzed the underlying mathematical problems and algorithms, and we developed new theory that provides better understanding of their challenges and possibilities. This insight allowed us to develop a solid framework for precisely formulated CT algorithms that compute much more well-defined results. We laid the groundwork for the next generation of rigorously defined algorithms that will further optimize the use of prior information.
The road to this insight involved specific case studies related to the formulation and use of prior information, involving such applications as X-ray phase-contrast tomography, fusion plasma physics, and underwater pipeline inspection. Below we list the highlights of these cases.
Understanding of Sparsity for Low-Dose CT
We characterize how the prior information that an object is “simple” – in mathematical terms, sparse – allows us to compute reliable images from very limited data, and we show that the sufficient amount of CT data depends in a simple way on the sparsity. This is essential in medical and engineering CT where one must minimize the X-ray dose and shorten measurement time.
Superior Localization in Electrical Impedance Tomography
By incorporation the prior information that the details stand out from the background, we can now compute images with superior localization and contrast. Moreover we developed new theory that, for the first time, precisely describes the obtainable resolution and the optimal measurement configuration. This is essential in industrial process monitoring where measurement constraints often limit the amount of data.
Superior Use of Textural Training Images
For textural images, we developed a new mathematical and computational framework that is superior to other methods for limited-data. It is particularly suited for computing reliable segmentations of these images. To do this we use prior information in the form of training images that the computed image must resemble.
Novel Convergence Analysis of Iterative Methods
We developed novel theoretical insight into the advantages and limitations of the iterative methods that are required for 3D tomography computations. This insight guided the development of new software suited for many-core and GPU computers, as well as public-domain software with model implementations of these algorithms.
Correct Handling of Noise
We formulate correct mathematical models for the measurement noise and we develop new computational algorithms especially suited for using prior information about non-Gaussian noise. We show that these noise priors improve both the algorithms and the images, compared to the standard algorithms that are based on cruder models.
Novel Use of Prior Information about Structure
Structural prior information states that the image contains visual structures, e.g. texture along certain directions. Incorporation of this kind of information prompted us to develop new anisotropic higher-order techniques that avoid the unwanted artifacts of traditional methods (such as total variation).
It is of vital importance that the images are as sharp, detailed and reliable as possible, so scientists, engineers, doctors etc. can make the correct decisions. To achieve high-definition tomography – sharper images with more reliable details – we must use prior information consisting of accumulated knowledge about the object.
Overall Outcome: Insight and Framework
Previous efforts were often based on ad-hoc techniques and naive algorithms with limited applications and ill-defined results. This project focused on obtaining deeper insight and developing a rigorous framework. We carefully analyzed the underlying mathematical problems and algorithms, and we developed new theory that provides better understanding of their challenges and possibilities. This insight allowed us to develop a solid framework for precisely formulated CT algorithms that compute much more well-defined results. We laid the groundwork for the next generation of rigorously defined algorithms that will further optimize the use of prior information.
The road to this insight involved specific case studies related to the formulation and use of prior information, involving such applications as X-ray phase-contrast tomography, fusion plasma physics, and underwater pipeline inspection. Below we list the highlights of these cases.
Understanding of Sparsity for Low-Dose CT
We characterize how the prior information that an object is “simple” – in mathematical terms, sparse – allows us to compute reliable images from very limited data, and we show that the sufficient amount of CT data depends in a simple way on the sparsity. This is essential in medical and engineering CT where one must minimize the X-ray dose and shorten measurement time.
Superior Localization in Electrical Impedance Tomography
By incorporation the prior information that the details stand out from the background, we can now compute images with superior localization and contrast. Moreover we developed new theory that, for the first time, precisely describes the obtainable resolution and the optimal measurement configuration. This is essential in industrial process monitoring where measurement constraints often limit the amount of data.
Superior Use of Textural Training Images
For textural images, we developed a new mathematical and computational framework that is superior to other methods for limited-data. It is particularly suited for computing reliable segmentations of these images. To do this we use prior information in the form of training images that the computed image must resemble.
Novel Convergence Analysis of Iterative Methods
We developed novel theoretical insight into the advantages and limitations of the iterative methods that are required for 3D tomography computations. This insight guided the development of new software suited for many-core and GPU computers, as well as public-domain software with model implementations of these algorithms.
Correct Handling of Noise
We formulate correct mathematical models for the measurement noise and we develop new computational algorithms especially suited for using prior information about non-Gaussian noise. We show that these noise priors improve both the algorithms and the images, compared to the standard algorithms that are based on cruder models.
Novel Use of Prior Information about Structure
Structural prior information states that the image contains visual structures, e.g. texture along certain directions. Incorporation of this kind of information prompted us to develop new anisotropic higher-order techniques that avoid the unwanted artifacts of traditional methods (such as total variation).