Interactive Segmentation of 3D Medical Images with Implicit Surfaces

To cope with a variety of clinical applications, research in medical image processing has led to a large spectrum of segmentation techniques that extract anatomical structures from volumetric data acquired with 3D imaging modalities. Despite continuing advances in mathematical models for automatic segmentation, many medical practitioners still rely on 2D manual delineation, due to the lack of intuitive semi-automatic tools in 3D. In this thesis, we propose a methodology and associated numerical schemes enabling the development of 3D image segmentation tools that are reliable, fast and interactive. These properties are key factors for clinical acceptance. Our approach derives from the framework of variational methods: segmentation is obtained by solving an optimization problem that translates the expected properties of target objects in mathematical terms. Such variational methods involve three essential components that constitute our main research axes: an objective criterion, a shape representation and an optional set of constraints. As objective criterion, we propose a unified formulation that extends existing homogeneity measures in order to model the spatial variations of statistical properties that are frequently encountered in medical images, without compromising efficiency. Within this formulation, we explore several shape representations based on implicit surfaces with the objective to cover a broad range of typical anatomical structures. Firstly, to model tubular shapes in vascular imaging, we introduce convolution surfaces in the variational context of image segmentation. Secondly, compact shapes such as lesions are described with a new representation that generalizes Radial Basis Functions with non-Euclidean distances, which enables the design of basis functions that naturally align with salient image features. Finally, we estimate geometric non-rigid deformations of prior templates to recover structures that have a predictable shape such as whole organs. Interactivity is ensured by restricting admissible solutions with additional constraints. Translating user input into constraints on the sign of the implicit representation at prescribed points in the image leads us to consider inequality-constrained optimization.

Segmentation of Gated Tl-SPECT Images and Computation of Ejection Fraction: A Different Approach

Michaël Unser

Background. We describe a set of image processing algorithms and mathematical models that can be advantageously used in schemes for the segmentation of thallium-201-single photon emission computed tomography (SPECT) images and for computation of left ventricular ejection fraction (EF). Methods. The system consists of two independent blocs for image segmentation and computation of function. The former is based on a multiresolution elliptical coordinate transformation and dynamic contour tracking. Computation of EF is formulated on the basis of both the endocardial and epicardial contours, and we compare this formulation with that using only the endocardial border for images with low signal-to-noise ratios. The accuracy of border detection was validated against manual border tracing on FDG-PET images, simulated Tl-201-SPECT images where the true underlying borders were known, and actual Tl-201-SPECT images. Finally, we compared EFs computed for FDG-PET, technetium-99m-SPECT and Tl-201-SPECT with those obtained from planar gated blood pool imaging. Results. The automatically obtained results always were within the manual uncertainty range. Agreement between myocardial volumes from positron emission tomography and automatically obtained values from the simulated Tl-201-SPECT images was excellent (r = 0.95, n = 32). Agreement between EFs from planar gated blood pool imaging and the other image modalities was good (FDG-PET: y = 5.89 + 1.21x, r = 0.92, see = 6.24, n = 19, Tc-99m-SPECT: y = -3.86 + 1.06x, r = 0.88, see = 7.78, n = 9, Tl-201-SPECT: y = 17.8 + 0.81x, r = 0.77, see = 7.44, n = 26). For noisy input data the combined use of information from epicardial and endocardial contours gives more accurate EF values than the traditional formula on the basis of the endocardial contour only. Conclusions. Alternate approaches for segmentation and computation of function have been presented and validated. They might also be advantageously incorporated into other existing techniques.

Journal of Nuclear Cardiology1999

An Efficient Algorithm for Level Set Method Preserving Distance Function

Xavier Bresson, Virginia Estellers Casas, Jean-Philippe Thiran, Dominique Zosso

The level set method is a popular technique for tracking moving interfaces in several disciplines including computer vision and fluid dynamics. However, despite its high flexibility, the original level set method is limited by two important numerical issues. Firstly, the level set method does not implicitly preserve the level set function as a distance function, which is necessary to estimate accurately geometric features s.a. the curvature or the contour normal. Secondly, the level set algorithm is slow because the time step is limited by the standard CFL condition, which is also essential to the numerical stability of the iterative scheme. Recent advances with graph cut methods and continuous convex relaxation methods provide powerful alternatives to the level set method for image processing problems because they are fast, accurate and guaranteed to find the global minimizer independently to the initialization. These recent techniques use binary functions to represent the contour rather than distance functions, which are usually considered for the level set method. However, the binary function cannot provide the distance information, which can be essential for some applications s.a. the surface reconstruction problem from scattered points and the cortex segmentation problem in medical imaging. In this paper, we propose a fast algorithm to preserve distance functions in level set methods. Our algorithm is inspired by recent efficient

\ell^1

optimization techniques, which will provide an efficient and easy to implement algorithm. It is interesting to note that our algorithm is not limited by the CFL condition and it naturally preserves the level set function as a distance function during the evolution, which avoids the classical re-distancing problem in level set methods. We apply the proposed algorithm to carry out image segmentation, where our methods proves to be 5 to 6 times faster than standard distance preserving level set techniques. We also present two applications where preserving a distance function is essential. Nonetheless, our method stays generic and can be applied to any level set methods that require the distance information.

Institute of Electrical and Electronics Engineers2012

Bits from Photons

Feng Yang

The trends in the design of image sensors are to build sensors with low noise, high sensitivity, high dynamic range, and small pixel size. How can we benefit from pixels with small size and high sensitivity? In this dissertation, we study a new image sensor that is reminiscent of traditional photographic film. Each pixel in the sensor has a binary response, giving only a one-bit quantized measurement of the local light intensity. The response function of the image sensor is non-linear and similar to a logarithmic function, which makes the sensor suitable for high dynamic range imaging. We first formulate the oversampled binary sensing scheme as a parameter estimation problem based on quantized Poisson statistics. We show that, with a single-photon quantization threshold and large oversampling factors, the Cramér-Rao lower bound (CRLB) of the estimation variance approaches that of an ideal unquantized sensor, that is, as if there were no quantization in the sensor measurements. Furthermore, the CRLB is shown to be asymptotically achievable by the maximum likelihood estimator (MLE). By showing that the log-likelihood function is concave, we guarantee the global optimality of iterative algorithms in finding the MLE. We study the performance of the oversampled binary sensing scheme in presence of dark current noise. The noise model is an additive Bernoulli noise with a known parameter, and the noise only flips the binary output from "0" to "1". We show that the binary sensor is quite robust with respect to noise and its dynamic range is only slightly reduced. The binary sensor first benefits from the increasing of the oversampling factor and then suffers in term of dynamic range. We again use the MLE to estimate the light intensity. When the threshold is a single photon, we show that the log-likelihood function is still concave. Thus, the global optimality can be achieved. But for thresholds larger than "1", this property does not hold true. By proving that when the light intensity is piecewise-constant, the likelihood function is a strictly pseudoconcave function, we guarantee to find the optimal solution of the MLE using iterative algorithms for arbitrary thresholds. For the general linear light field model, the log-likelihood function is not even quasiconcave when thresholds are larger than "1". In this circumstance, we find an initial solution by approximating the light intensity field with a piecewise-constant model, and then we use Newton's method to refine the estimation using the exact model. We then examine one of the most important parameters in the binary sensor, i.e., the threshold used to generate binary values. We prove the intuitive result that large thresholds achieve better estimation performance for strong light intensities, while small thresholds work better for low light intensities. To make a binary sensor that works in a larger range of light intensities, we propose to design a threshold array containing multiple thresholds instead of a single threshold for the binary sensing. The criterion is to minimize the average CRLB which is a good approximation of the mean squared error (MSE). The performance analysis on the new binary sensor verifies the effectiveness of our design. Again, the MLE is used for reconstructing the light intensity field from the binary measurements. By showing that the log-likelihood function is concave for arbitrary threshold arrays, we ensure that the iterative algorithms can find the optimal solution of the MLE. Finally, we study the reconstruction problem for the binary image sensor under a generalized piecewise-constant light intensity field model, which is quite useful when parameters like oversampling factors are unknown. We directly estimate light exposure values, i.e., the number of photons hitting on each pixel. We assume that the light exposure values are piecewise-constant and we use the MLE for the reconstruction. This optimization problem is solved by iteratively working out two subproblems. The first one is to find the optimal light exposure value for each segment, given the optimal segmentation of the binary measurements. The second one is to find the optimal segmentation of the binary measurements given the optimal light exposure values for each segment. Several algorithms are provided for solving this optimization problem. Dynamic programming can obtain the optimal solution for 1-D signals, but the computation is quite heavy. To reduce the burden of computation, we propose a greedy algorithm and a method based on pruning of binary trees or quadtrees.

EPFL2012