Medical imaging techniques have evolved to expand our ability to visualize new contrast information of electrical, optical, and mechanical properties of tissues in the human body using noninvasive measurement methods. In particular, electrical tissue property imaging techniques have received considerable attention for the last few decades since electrical properties of biological tissues and organs change with their physiological functions and pathological states. We can express the electrical tissue properties as the frequency-dependent admittivity, which can be measured in a macroscopic scale by assessing the relation between the time-harmonic electric field and current density. The main issue is to reconstruct spectroscopic admittivity images from 10 Hz to 1 MHz, for example, with reasonably high spatial and temporal resolutions. It requires a solution of a nonlinear inverse problem involving Maxwell's equations. To solve the inverse problem with practical significance, we need deep knowledge on its mathematical formulation of underlying physical phenomena, implementation of image reconstruction algorithms, and practical limitations associated with the measurement sensitivity, specificity, noise, and data acquisition time. This paper discusses a number of issues in electrical tissue property imaging modalities and their future directions.

The goal of this note is to describe the asymptotic behaviour of problems set in cylinders when the size of them is becoming infinite. This leads to consider problems in unbounded domains as well as new singular perturbations issues.

We propose a variational segmentation model based on statistical information of intensities in an image. The model consists of both a local region-based energy and a global region-based energy in order to handle misclassification which happens in a typical statistical variational model with an assumption that an image is a mixture of two Gaussian distributions. We find local ambiguous regions where misclassification might happen due to a small difference between two Gaussian distributions. Based on statistical information restricted to the local ambiguous regions, we design a local region-based energy in order to reduce the misclassification. We suggest an algorithm to avoid the difficulty of the Euler-Lagrange equations of the proposed variational model.

This paper aims to give a quick view on denoising without comprehensive details. Denoising can be understood as removing unwanted parts in signals and images. Noise incorporates intrinsic random fluctuations in the data. Since noise is ubiquitous, denoising methods and models are diverse. Starting from what noise means, we briefly discuss a denoising model as maximum a posteriori estimation and relate it with a variational form or energy model. After that we present a few major branches in image and signal processing; filtering, shrinkage or thresholding, regularization and data adapted methods, although it may not be a general way of classifying denoising methods.

If there are metals located in the X-ray scanned object, a point outside the metals has its range of projection angle at which projections passing through the point are disturbed by the metals. Roughly speaking, this implies that attenuation information at the point is missing in the blocked projection range. So conventional projection completion MAR algorithms to use the undisturbed projection data on the boundary of the metaltrace is less efficient in reconstructing the attenuation coefficient in detailed parts, in particular, near the metal region. In order to overcome this problem, we propose the algebraic correction technique (ACT) to utilize a pre-reconstructed interim image of the attenuation coefficient outside the metal region which is obtained by solving a linear system designed to reduce computational costs. The reconstructed interim image of the attenuation coefficient is used as prior information for MAR. Numerical simulations support that the proposed correction technique shows better performance than conventional inpainting techniques such as the total variation and the harmonic inpainting.

In this paper we consider a novel reconstruction method in electrical impedance tomography (EIT) and its application for monitoring and detecting a hydrargyrum (mercury) polluted soil near to the surface of underground. We use electrodes placed on the surface of land to collect the data which provides the relations of voltage and current map and to produce a projected image of interior conductivity distribution onto the surface of land. Here the projected image reconstruction method is used to monitor the pollution in soil underneath the ground without any destruction and any digging into a land.

In this paper, we propose a robust real-time myocardial border tracking algorithm for echocardiography. Commonly, after an initial contour of LV border is traced at one or two frame from the entire cardiac cycle, LV contour tracking is performed over the remaining frames. Among a variety of tracking techniques, optical flow method is the most widely used for motion estimation of moving objects. However, when echocardiography data is heavily corrupted in some local regions, the errors bring the tracking point out of the endocardial border, resulting in distorted LV contours. This shape distortion often occurs in practice since the data acquisition is affected by ultrasound artifacts, dropout or shadowing phenomena of cardiac walls. The proposed method deals with this shape distortion problem and reflects the motion realistic LV shape by applying global deformation modeled as affine transform partitively to the contour. We partition the tracking points on the contour into a few groups and determine each affine transform governing the motion of the partitioned contour points. To compute the coefficients of each affine transform, we use the least squares method with equality constraints that are given by the relationship between the coefficients and a few contour points showing good tracking results. Many real experiments show that the proposed method supports better performance than existing methods.

In this paper we explore the emergence of the notion of compactness within its historical beginning through rigor versus intuition modes in the treatment of Dirichlet's principle. We emphasize on the intuition in Riemann's statement on the principle criticized by Weierstrass' requirement of rigor followed by Hilbert's restatement again criticized by Hadamard, which pushed the ascension of the notion of compactness in the analysis of PDEs. A brief overview of some techniques and problems involving compactness is presented illustrating the importance of this notion. Compactness is discussed here to raise educational issues regarding rigor vs intuition in mathematical studies. The concept of compactness advanced rapidly after Weierstrass's famous criticism of Riemann's use of the Dirichlet principle. The rigor of Weierstrass contributed to establishment of the concept of compactness, but such a focus on rigor blinded mathematicians to big pictures. Fortunately, Poincar$\acute{e}$ and Hilbert defended Riemann's use of the Dirichlet principle and found a balance between rigor and intuition. There is no theorem without rigor, but we should not be a slave of rigor. Rigor (highly detailed examination with toy models) and intuition (broader view with real models) are essentially complementary to each other.