# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

All Issues

## Inverse Problems & Imaging

May 2010 , Volume 4 , Issue 2

Special Issue
on Inverse Problem and Imaging in Medical Image Analysis

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2010, 4(2): i-iii doi: 10.3934/ipi.2010.4.2i +[Abstract](2182) +[PDF](37.7KB)
Abstract:
Life expectancy in the developed and developing countries is constantly increasing. Medicine has benefited from novel biomarkers for screening and diagnosis. At least for a number of diseases, biomedical imaging is one of the most promising means of early diagnosis. Medical hardware manufacturer's progress has led to a new generation of measurements to understand the human anatomical and functional states. These measurements go beyond simple means of anatomical visualization (e.g. X-ray images) and therefore their interpretation becomes a scientific challenge for humans mostly because of the volume and flow of information as well as their nature. Computer-aided diagnosis develops mathematical models and their computational solutions to assist data interpretation in a clinical setting. In simple words, one would like to be able to provide a formal answer to a clinical question using the available measurements. The development of mathematical models for automatic clinical interpretation of multi-modalities is a great challenge.

2010, 4(2): 211-222 doi: 10.3934/ipi.2010.4.211 +[Abstract](2187) +[PDF](994.1KB)
Abstract:
In this paper we will propose a novel variational framework for speckle removal in ultrasound images. Our method combines efficiently a fidelity to data term adapted to the Rayleigh distribution of the speckle and a novel spatio- temporal smoothness constraint. The regularization relies on a non parametric image model that describes the observed image structure and express inter-dependencies between pixels in space and time. The interaction between pixels is determined through the definition of new measure of similarity between them to better reflect image content. To compute this similarity measure, we take into consideration the spatial aspect as well as the temporal one. Experiments were carried on both synthetic and real data and the results show the potential of our method.
2010, 4(2): 223-240 doi: 10.3934/ipi.2010.4.223 +[Abstract](3287) +[PDF](555.8KB)
Abstract:
The aim of this work is to improve the accuracy, robustness and efficiency of the compressed sensing reconstruction technique in magnetic resonance imaging. We propose a novel variational model that enforces the sparsity of the underlying image in terms of its spatial finite differences and representation with respect to a dictionary. The dictionary is trained using prior information to improve accuracy in reconstruction. In the meantime the proposed model enforces the consistency of the underlying image with acquired data by using the maximum likelihood estimator of the reconstruction error in partial $k$-space to improve the robustness to parameter selection. Moreover, a simple and fast numerical scheme is provided to solve this model. The experimental results on both synthetic and in vivo data indicate the improvement of the proposed model in preservation of fine structures, flexibility of parameter decision, and reduction of computational cost.
2010, 4(2): 241-255 doi: 10.3934/ipi.2010.4.241 +[Abstract](2312) +[PDF](4325.0KB)
Abstract:
Brain aneurysm rupture has been reported to be closely related to aneurysm size. The current method used to determine aneurysm size is to measure the dimension of the aneurysm dome and the width of the aneurysm neck. Since aneurysms usually have complicated shapes, using just the size of the aneurysm dome and neck may not be accurate and may overlook important geometrical information. In this paper we present a level set based surface capturing algorithm to first capture the aneurysms from the vascular tree. Since aneurysms are described by level set functions, volumes, curvatures and other geometric quantities of the aneurysm surface can easily be computed for medical studies. Experiments and comparisons with models used for capturing illusory contours in 2D images are performed. Applications to medical images are also presented to show the accuracy, consistency and robustness of our method in capturing brain aneurysms and volume quantification.
2010, 4(2): 257-271 doi: 10.3934/ipi.2010.4.257 +[Abstract](2825) +[PDF](748.7KB)
Abstract:
A novel three-dimensional dental X-ray imaging method is introduced, based on hybrid data collected with a dental panoramic device. Such a device uses geometric movement of the X-ray source and detector around the head of a patient to produce a panoramic image, where all teeth are in sharp focus and details at a distance from the dental arc are blurred. A digital panoramic device is reprogrammed to collect two-dimensional projection radiographs. Two complementary types of data are measured from a region of interest: projection data with a limited angle of view, and a panoramic image. Tikhonov regularization is applied to these data in order to produce three-dimensional reconstructions. The algorithm is tested with simulated data and real-world in vitro measurements from a dry mandible. Reconstructions from limited-angle projection data alone do provide the dentist with three-dimensional information useful for dental implant planning. Furthermore, adding panoramic data to the process improves the reconstruction precision in the direction of the dental arc. The presented imaging modality can be seen as a cost-effective alternative to a full-angle CT scanner.
2010, 4(2): 273-310 doi: 10.3934/ipi.2010.4.273 +[Abstract](2643) +[PDF](5122.1KB)
Abstract:
In this work, we wish to denoise HARDI (High Angular Resolution Diffusion Imaging) data arising in medical brain imaging. Diffusion imaging is a relatively new and powerful method to measure the three-dimensional profile of water diffusion at each point in the brain. These images can be used to reconstruct fiber directions and pathways in the living brain, providing detailed maps of fiber integrity and connectivity. HARDI data is a powerful new extension of diffusion imaging, which goes beyond the diffusion tensor imaging (DTI) model: mathematically, intensity data is given at every voxel and at any direction on the sphere. Unfortunately, HARDI data is usually highly contaminated with noise, depending on the b-value which is a tuning parameter pre-selected to collect the data. Larger b-values help to collect more accurate information in terms of measuring diffusivity, but more noise is generated by many factors as well. So large b-values are preferred, if we can satisfactorily reduce the noise without losing the data structure. Here we propose two variational methods to denoise HARDI data. The first one directly denoises the collected data $S$, while the second one denoises the so-called sADC (spherical Apparent Diffusion Coefficient), a field of radial functions derived from the data. These two quantities are related by an equation of the form $S = S_0\exp(-b\cdot sADC)$ (in the noise-free case). By applying these two different models, we will be able to determine which quantity will most accurately preserve data structure after denoising. The theoretical analysis of the proposed models is presented, together with experimental results and comparisons for denoising synthetic and real HARDI data.
2010, 4(2): 311-333 doi: 10.3934/ipi.2010.4.311 +[Abstract](2377) +[PDF](2087.2KB)
Abstract:
We address the problem of detecting deformities on elastic surfaces. This is of great importance for shape analysis, with applications such as detecting abnormalities in biological shapes (e.g., brain structures). We propose an effective algorithm to detect abnormal deformations by generating quasi-conformal maps between the original and deformed surfaces. We firstly flatten the 3D surfaces conformally onto 2D rectangles using the discrete Yamabe flow and use them to compute a quasi-conformal map that matches internal features lying within the surfaces. The deformities on the elastic surface are formulated as non-conformal deformations, whereas normal deformations that preserve local geometry are formulated as conformal deformations. We then detect abnormalities by computing the Beltrami coefficient associated uniquely with the quasi-conformal map. The Beltrami coefficient is a complex-valued function defined on the surface. It describes the deviation of the deformation from conformality at each point. By considering the norm of the Beltrami coefficient, we can effectively segment the regions of abnormal changes, which are invariant under normal (non-rigid) deformations that preserve local geometry. Furthermore, by considering the argument of the Beltrami coefficient, we can capture abnormalities induced by local rotational changes. We tested the algorithm by detecting abnormalities on synthetic surfaces, 3D human face data and MRI-derived brain surfaces. Experimental results show that our algorithm can effectively detect abnormalities and capture local rotational alterations. Our method is also more effective than other existing methods, such as the isometric indicator, for locating abnormalities.

2020 Impact Factor: 1.639
5 Year Impact Factor: 1.720
2020 CiteScore: 2.6