数学方法在医学图像处理中的应用

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"MATHEMATICAL METHODS IN MEDICAL IMAGE PROCESSING.pdf" 这本PDF书籍"MATHEMATICAL METHODS IN MEDICAL IMAGE PROCESSING"是医学图像处理领域的一份基础教材，特别适合初学者。由SIGURD ANGENENT, ERIC PICHON, 和 ALLEN TANNENBAUM三位专家合著，内容涵盖了医学成像中的关键数学问题。随着硬件和软件的不断进步，医学图像处理领域经历了快速发展。书中强调了软件开发中新颖的方法，这些方法结合了几何偏微分方程、传统的信号/图像处理技术以及计算机图形学，以促进人机交互。在这一研究领域，研究人员致力于将生物医学工程原理建立在严格的数学基础上，以发展出可集成到完整治疗输送系统中的软件方法。这些系统支持了放射疗法、活组织检查以及微创手术等许多图像引导程序的有效实施。作者们通过数学方法探讨了该领域的主要问题，包括图像增强、配准以及分割。图像增强是医学图像处理的一个重要环节，旨在提高图像的视觉质量和诊断信息。通过数学算法，可以调整图像的对比度、亮度和噪声水平，使医生更容易识别病灶或异常区域。例如，利用傅立叶变换进行频域分析，可以有效去除图像噪声并突出特定频率成分。配准是将多幅图像对齐的过程，常用于比较不同时间点的图像（如疾病进展监测）或来自不同成像模态的图像（如CT和MRI）。数学方法在这里起着关键作用，例如采用优化算法寻找最佳配准参数，确保图像间对应结构的一致性。图像分割则是将图像划分为多个具有特定意义的区域，例如区分肿瘤组织和正常组织。数学模型如水平集、活动轮廓模型和基于概率的分割方法被广泛应用于解决这个问题。这些方法需要解决复杂的优化问题，以找到最能反映图像特征的分割边界。此外，书中可能还涉及其他主题，如形态学操作、偏微分方程在图像平滑和边缘检测中的应用，以及机器学习和深度学习在图像分析中的角色。数学在医学图像处理中的应用不仅提高了诊断准确性，也促进了医疗技术的创新和临床实践的进步。通过深入理解和应用这些数学方法，医学图像处理领域的专业人士能够更好地理解和解决实际问题，从而改善患者护理和治疗效果。

6 SIGURD ANGENENT, ERIC PICHON, AND ALLEN TANNENBAUM

(a) Ultrasound. (foetus) (b) Computed Tomography (brain, 2D

axial slice).

(brain, 2D axial slice).

(d) Positron Emission Tomography

(brain, 2D axial slice).

Figure 3.2. Examples of diﬀerent image modalities.

in various anatomical structur e s (as a measure of the injected positron emitter)

can be visualized and relatively quantiﬁed. Since the patient has to be injected

with radioactive mater ial, PET is relatively invasive. T he radiation dose however

is similar to a CT sc an. Image resolution may b e p oor and major pr eprocessing

may be necessary. See Figure 3.2(d).

4. Mathematics and Imaging

Medical imaging needs highly trained technicians and clinicians to determine

the details of ima ge ac quisition (e.g. choice of modality, of patient position, of

an optional contrast agent, etc.), a s well as to analyz e the results. The dramatic

MATHEMATICAL METHODS IN MEDICAL IMAGE PROCESSING 7

increase in availability, diversity, and res olution of medical imag ing devices over the

last 50 years threatens to overwhelm these human experts.

For image analysis, modern image process ing techniques have therefore become

indispensable. Artiﬁcial systems must be designed to analyze medical datasets

either in a partially or even a fully automatic manner. This is a challenging ap-

plication of the ﬁeld known as artiﬁcial vision (see Section 4.1). Such algorithms

are based on mathematical models (see Section 4.2). In medical image analysis,

as in many practical mathematical applications, numerical simulations should be

regarded as the e nd product. The purpose of the mathematical analysis is to guar-

antee that the constructed algorithms will behave as desired.

4.1. Artiﬁcial Vision. Artiﬁcial Intelligence (AI) was initiated as a ﬁeld in the

1950’s with the ambitious (and so-far unrealized) goal of creating artiﬁcial systems

with human-like intelligence

. Whereas classical AI had been mostly concerned with

symbolic representation and reasoning, new subﬁelds were cr e ated a s r e searchers

embraced the complexity of the goal and realized the importance of sub-symbolic

information and perception. In particular, artiﬁcial vision [32, 44, 39, 92] emerged

in the 1970’s with the more limited goal to mimic human vision with man-made

systems (in practice, computers).

Vision is such a basic aspect of human cognition that it may superﬁcially ap-

pear somewha t trivial, but after decades of rese arch the scientiﬁc understanding

of biological vision remains extremely fra gmentary. To date, artiﬁcial vision has

produced important applications in medical imaging [18] as well as in other ﬁelds

such as Earth observation, industrial automation, a nd robotics [92].

The human e ye-brain system evolved over tens of millions of years and at this

point no artiﬁcial system is as versatile and powerful for everyday tasks. In the

same way that a chess-playing program is not directly modelled after a human

player, many mathematical techniques a re e mployed in artiﬁcial vision that do not

pretend to simulate biological vision. Artiﬁcial vision systems will therefore not

be set within the natural limits of human per c e ptio n. For example, human vision

is inherently two dimensional

. To accommodate this limitation, radiologists must

resort to visualizing only 2D planar slices of 3D medical images. An a rtiﬁcial system

is free of that limitation a nd can “see” the image in its entirety. Other advantages

are that artiﬁcial systems can work on very large image datasets, are fast, do not

suﬀer from fatigue, and produce repeatable results. Because artiﬁcial vision system

designers have so far bee n unsuccessful in incorpora ting high level understanding

of real-life applications, artiﬁcial systems typically complement rather than replace

of human exp erts.

4.2. Algorithms and PDEs. Many mathematical approaches have been investi-

gated for applications in artiﬁcial vision (e.g., fr actals and self-similarity, wavelets,

pattern theory, stochastic point process, ra ndom graph theory; see [42]). In partic-

ular, methods based on partial diﬀerential equations (PDEs) have been extremely

popular in the pa st few years [2 0, 35]. Here we brieﬂy outline the major concepts

involved in us ing PDEs for ima ge processing.

The deﬁnition of “intelligence” is still very problematic.

Stereoscopic vision does not allow us to see inside objects. It is sometimes described as “2.1

dimensional perception.”

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