没有合适的资源?快使用搜索试试~ 我知道了~
Mathematical methods in biomedical imaging
需积分: 10 9 下载量 40 浏览量
2015-04-22
20:36:14
上传
评论
收藏 498KB PDF 举报
温馨提示
试读
30页
Mathematical methods in biomedical imaging
资源推荐
资源详情
资源评论
GAMM-Mitt. 37, No. 2, 154 – 183 (2014) / DOI 10.1002/gamm.201410008
Mathematical methods in biomedical imaging
Martin Burger
1∗
, Jan Modersitzki
2,3
, and Daniel Tenbrinck
4
1
Institute for Computational and Applied Mathematics, University of M¨unster , M¨unster
2
Institute of Mathematics and Image Computing, University of L¨ubeck, L¨ubeck
3
Fraunhofer MEVIS Project group image registration, L¨ubeck
4
GREYC, UMR 6072 CNRS,
´
Ecole Nationale Sup´erieure d’Ing´enieurs de Caen, Caen
Receiv ed 19 August 2014
Published online 19 November 2014
Key words Bayesian modeling, biomedical imaging, expectation maximization, maximum-
a-posteriori estimation, motion correction, motion estimation, reconstruction, registration,
regularization, segmentation, singular energies, total variation.
Dedicated to the memory of Professor Bernd Fischer, 1957–2013.
Biomedical imaging is an important and exponentially growing field in life sciences and clini-
cal practice, which strongly depends on the advances in mathematical image processing.
Biomedical data presents a number of particularities such as non-standard acquisition tech-
niques. Thus, biomedical imaging may be considered as an own field of research. Typical
biomedical imaging tasks, as outlined in this paper, demand for innovative data models and
efficient and robust approaches to produce solutions to challenging problems both in basic
research as well as daily clinical routine.
This paper discusses typical specifications and challenges of reconstruction and denois-
ing, segmentation, and image registration of biomedical data. Furthermore, it provides an
overview of current concepts to tackle the typically ill-posed problems and presents a unified
framework that captures the different tasks mathematically.
c
2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction
Biomedical imaging has become of enormous importance in research and clinical practice in
the last decades. In Germany alone around ten million patients are examined each year with a
variety of different imaging devices. The number of techniques is rapidly growing and modal-
ities range from established ones such as Computed Tomography (CT), Magnetic Resonance
Imaging (MR), Ultrasound, Single Photon Emission Computed Tomography (SPECT), and
Positron Emission Tomography (PET), to novel ones such as Diffusion Tensor Imaging (DTI)
Elastography, Magnetic Particle Imaging, EE/MEG, Optical Microscopy and Tomography,
Photoacoustic Tomography, Electron Tomography, and Atomic Force Microscopy.
This development is accompanied by a number of mathematical challenges. First of all,
image formation is typically not only based on measurements but also requires so-called re-
construction methods; see also Sec. 2. As to be expected, these methods rely on a proper
mathematical model of the physics of the imaging device and a decent handling of noise and
∗
Corresponding author E-mail: burger@wwu.de, Phone: +49 251 83 33793, Fax: +49 251 83 32729
c
2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
GAMM-Mitt. 37, No. 2 (2014) 155
artifacts. Image reconstructio n is ill-posed and therefore issues like regularization, bias, exis-
tence and uniqueness of solution s, condition, stability, and robustness immediately enter in to
play. As the amount of acquired data is typically relatively large (a single 1.024
3
CT image
yields 16 GB), memory requirements and computation times are also important.
Interpretation and analysis of biomedical images is not trivial. It is for good reason that
the education of radiolo gists requires typically seven years until a trained expert is ready to
“read” an image. Although current technology can produce complex images of dimensions
three (volumetric) or four (space and time), even with vector (color) or tensor (DTI) valued
output, human perception and output devices are in general only two dimensional. A fun-
damental task is the identification of structures-of-interest, also called segmentation; see also
Sec 3. Today, a slice-by-slice manual segmentation of 3D d ata may still be considered as state-
of-the-art. Limited resources for expensive experts in combination with the fast growing use
of imaging techniques for diagnosis make automatization inevitable. Another important chal-
lenge is image registration, the spatial normalization or alignment of image data for various
tasks suc h as motion correction or data fusion; see also Sec. 4.
In contrast to industrial imaging, where high quality images can almost always get acquired
by simply increasing dose or measurement time, biomedical imaging has limitations on dose
prescription. Thus, acquired data typically realize a trade-off between image quality and
wellfare of the imaged subjects and hence image formation always aims for less measurement
time, less dose o f radiation or less laser light, and higher resolution leading to more image
details.
It is tempting to subordinate biomedical imaging as a special application of image process-
ing, computer vision, and inverse problems. But there are several reasons for a self-contained
treatment of mathematics in b iomedical imaging. Some of theses are briefly outlined in this
exposition.
Data Ty pe Biomedical images are visualizations of physical measurements and can thus be
very different to natural images. Indeed, many tomographical techniques capture a density
of some physical object (body density in CT, weighted proton densities in MR, densities of
radioactive tracers in emission tomography). The measurements are visualized as images only
in order to allow manual inspection. In an automated an alysis pipeline there is not even a need
to form an image in a classical sense. For example, cardiologists are used to analyze so-called
bullseye plots (a collection of average densities in divided subregions of the heart) rather than
images in the original sp a tial co ntext.
Data Degradation The form of degradation and noise is very special in biomedical imaging.
First of all, each modality has a specific imag e formation process and detector noise that needs
to be taken into account. Even more important and specific is the fact that biomedical imaging
means to measure on living species. This yields a variety of specific artifacts in images (e.g.,
due to motion or tissue absorption), for which mathematical methods need to take care for. For
example, a single PET measurement can take several minutes, therefore not only small scale
motion such as heart beat, breathin g, or tremo r degrade imag e quality but also large scale
motion such as repositioning are to be considered. The latter is a particular problem if the
patient is not cooperative such as children, elderly persons, and animals. Other challenging
problems in medical imaging include inhomogeneous image regions, low contrast between
target and background, and fuzzy edges.
www.gamm-mitteilungen.org
c
2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
156 M. Burger, J. Modersitzki, and D. Tenbrinck: Mathematical methods in biomedical imaging
Image Analysis The focus of image analysis in biomedicine can be very different to the
analysis of natural images. While in computer vision a major focus is to make images and
videos appealing to the human eye, on e usually seeks for quantitative information to be ex-
tracted from biomedical images. The particular question to be answered by mathematical
image analysis needs to be defined in interdisciplinary collaborations. Hence, biomedical im-
age analysis is not just a possible application o f mathematical image computing techniques
but also a driving force for research and the development of novel concepts and algorithms.
Validation An often underestimated aspect is the validation of results. One of the main ob-
stacles is that there is typically no ground truth. For example, the current gold standard for
tumour segmentation is an average of experts opinion. There is generally no unique under-
standing about the delineation and classification of tissue. In addition, structures can manifest
in different ways in different imaging modalities. For example, bones are excellently visible
in CT images but not in PET images, while tumou r tissue may be seen in PET but not in CT.
Standard approaches for validation are the use of hard- and software phantoms, animal mod-
els, and clinical trials. However, the transfer of results is delicate. Hardware phantoms have
different physical properties than living tissue, software models have a bias to the underlying
forward model, animals have different physiology and scale than humans, clinical studies are
expensive and ethically critical. The situation even complicates for image registration. For
example, a clinical task would be to fuse the information of a patient’s anatomy (bones and
organs from CT) with functional data (tumours from PET). The CT imaging protocol requires
the patient to rest his arms next to him (space limitations by the CT tube), while the PET
imaging protocol requires his arms to b e lifted-up (minimization of attenuation). In addition
to a potential re-positioning, the data is acquired in different poses. Image registration aims to
align the data into a spatially unified framework. This task is complicated as no motion model
for each individual patient is available, since the deformation of internal organs depends on in-
dividual parameters, e.g., relation of fat and muscle or age. In contrast to segmentation, where
clinical experts are basically able to rate the results, in image registration little is known.
We can only provide a small glimpse into the wide field of mathematical developments in
biomedical imaging and thus focus on three different aspects, chosen due to their relevance
and the authors’ research interests. The first one is image reconstruction, which we discuss in
a unified way together with image enhancement. The second is image segmentation, which
is fundamental for biomedical image analysis and visualization of target structures. Finally,
we discuss the problem of image registration, which aims to spatially align different images,
for example for motion correction or data fusion. We restrict ourselves to variational (energy
minimization) methods, which present a flexible approach and receive by far most attention in
biomedical imaging. Since the numerical treatment of the arising problems is a huge field of
its own, overlapping with other fields in mathematical imaging, we omit its discussion here.
2 Image Reconstruction and Enhancement
Image reconstruction may be interpreted as the most fundamental task to be performed in
biomedical imaging, since most subsequent processing steps directly rely on the reconstructed
data. Furthermore, one has to take special care as the measured data are often only indirectly
related to the quantity one would like to visualize during image formation. Already the first
www.gamm-mitteilungen.org
c
2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
GAMM-Mitt. 37, No. 2 (2014) 157
modern imaging technique of X-ray computed tomography (CT, cf. [1]) owes its name to the
fact that images need to be computed from physical measurements. Especially macroscopic
imaging of living subjects is necessarily indirect, since one wants to visualize structur es-of-
interest or processes in the body non-invasively, i.e., without opening it. Also at a nanoscale
level one encounters similar circumstances, e.g., in electron tomography (cf. [2, 3]). A re-
lated problem of image reconstruction is image enhancement, which we therefore also dis-
cuss in this section. In both cases we start with an overview of challenging problems and
well-established techniques and then proceed to current and future challenges.
2.1 Reconstruction and Inverse Problems
Image reconstruction has been a major source for the development of the field of inverse
problems, since frequently one has to solve an integral equation of the first kind,
Ku = f, (1)
where f ∈Vare the given data (in general corrupted by noise) and u ∈Uis the desired
unperturbed image. Here, K : U→Vis a forward operator which in most relevant cases
is a linear integral operator. Due to its compactness the solu tion of (1) becomes an ill-posed
problem (cf. [4]), which needs to be approximated by regularization methods (cf. [4, 5]). We
will discuss one of the most frequently used approach of variational regularization in the
next section. In the following we give an overview of used operators for prominent imaging
techniques.
X-ray Computed Tomography (CT) In CT one measures the attenuation of X-rays sent
through a subject, e.g., a human body. The forward operator K is th e Radon transfo rm of u,
i.e., th e co llection of all its line integrals [1, 6]. Depending on the way the X-rays are emitted
and measured, different variants of the Radon transform are obtained, in particular for modern
3D (spiral) scanning geometries [6–8].
Magnetic Resonance Imaging (MRI) In MRI one measures electrical signals induced by
nuclear magnetic relaxation of protons. The forward operator K can b e identified with the
Fourier transform of u and thus yields the only imaging case with a continuous inverse that
can also be computed efficiently [9]. However, the situation already changes if one wants
to reco nstruct phases or other parameters, such as coil sensitivities, as well, since then the
problem becomes ill-posed and nonlinear [10–12].
Positron-Emission Tomography (PET) In a PET detector ring one records coincidences of
photons emitted in opposite directions, which are emitted when a positron (generated during
radioactive decay) hits a nearby electron. Hence, one can infer that the decay event occurred
on the line between and since the decay intensity is proportional to the tracer density, one has a
random sampling of the Radon transform [13]. In practice one also encounters an attenuation
proportional to the line integral of body density, which means that the forward operator can be
factorized in the form K = AR,whereR is the Radon transform and A a diagonal operator
implementing the attenuation (usually determined by an supplemental CT scan).
Single-Photon Emission Computed Tomography (SPECT) The image formation process
in SPECT is relatively similar to PET discussed above with the difference that only a single
photon is measured and hence the attenuation depends on the path of the photon. The mathe-
matical consequence is that the forward operator cannot be factorized anymore as in PET, but
www.gamm-mitteilungen.org
c
2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
158 M. Burger, J. Modersitzki, and D. Tenbrinck: Mathematical methods in biomedical imaging
one is forced to use the attenuated Radon transform [14, 15]. An interesting problem is the
joint reconstruction of tracer density and attenuation [16].
Microscopy The image reconstruction step is often ignored in microscopy since the forward
operator K is simp ly a convolutio n with a kernel, which is often resembled by a Gaussian
function. Since microscopes scan an object-of-interest slice by slice one usually has a signif-
icantly higher resolution in the xy-plane compared to the z-direction. Although anisotropic
resolution does also occur in other biomedical imaging devices based on slice acquisition,
e.g., in MRI imaging, it is a particularly important point in microscopy and one has to take
special care when using an anisotropic point-spread-function during the 3D reconstruction of
microscopic images [17].
Electron Tomography In case of electron tomography the operator K can be factorized
into two parts. The main part is again a Radon transform with th e additional difficulty that
one can only sample a finite range of angles (limited angle CT) and also a limited field of
view. The second part consists of convolutions caused by the optics of the detector system.
We refer to [2, 3] for a detailed discussion.
Less regularly used image reconstruction techniques are based on the solution of diffusion,
wave, or Helmholtz equations as forward operators. Examples include ultrasound tomography
[18], optical tomography [19], and electrical impedance tomography [20]. Here, the problems
arise from the severe ill-posedness [4]. A rising technology with good potential for future use
is photoacoustic tomography, where the forward operator is determined by taking spherical
means [21]. This allows analogous developments to the Radon transform [22, 23]. Recently,
there is a strong trend of including more and more details about the physics in the modeling
of the forward operators, e.g., scattering in emission tomography and optics.
Enhancement of biomedical images, which presents another important imaging task, is
also covered by the general formulation (1). We give two examples for image enhancement in
the following.
Denoising or Sharpening The right-hand side f in (1) is assumed to present a corrupted
version of the image, hence one can choose the forward operator K eq ual to the identity, or
in a more precise functional analytic treatment as an embedding operator from an appropriate
space U for the image (usually the space BV (Ω) of functions of bounded variation) to a larger
space containing the noise [24, 25].
Inpainting or Superresolution The forward operator is a restriction of the image u to a
subset of the image domain (the complement of the inpainting region) or a downsampling
operator [24, 26].
2.2 Variational Reconstruction Methods
State-of-the art approaches for the reconstruction problem (1) are based on variational models,
i.e., one aims to find a solution of the following problem:
Find a minimizer u ∈Uof the functional
J : U→R,J(u):=D(Ku,f)+R(u),
(2)
where D is an appropriate data term an d R a regularization functional.
www.gamm-mitteilungen.org
c
2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
剩余29页未读,继续阅读
资源评论
math_learning
- 粉丝: 12
- 资源: 27
上传资源 快速赚钱
- 我的内容管理 展开
- 我的资源 快来上传第一个资源
- 我的收益 登录查看自己的收益
- 我的积分 登录查看自己的积分
- 我的C币 登录后查看C币余额
- 我的收藏
- 我的下载
- 下载帮助
安全验证
文档复制为VIP权益,开通VIP直接复制
信息提交成功