2024 Faculty Courses School of Engineering Undergraduate major in Information and Communications Engineering
Functional Analysis and Inverse Problems
- Academic unit or major
- Undergraduate major in Information and Communications Engineering
- Instructor(s)
- Isao Yamada / Takashi Obi
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Mon / 5-6 Thu
- Class
- -
- Course Code
- ICT.S302
- Number of credits
- 200
- Course offered
- 2024
- Offered quarter
- 1Q
- Syllabus updated
- Mar 14, 2025
- Language
- Japanese
Syllabus
Course overview and goals
To establish solution strategies for various inverse problems in modern data sciences such as signal processing, image processing, pattern recognition, optimization and machine learning, the unified mathematical perspective built through Functional Analysis will certainly serve as helpful guide. Starting from definitions of convergence of real number sequence and vector space which serve as prerequisites of Functional Analysis, this lecture surveys its central ideas, e.g., in Metric space, Normed space, Inner product space, Banach space and Hilbert space, together with their applications to typical inverse problems.
Course description and aims
Through the lectures, the students will be able to:
1) understand mathematical meanings of spaces, convergences and operators and apply these to real world problems.
2) build mathematical perspectives to grasp many real world inverse problems in unified ways.
Keywords
Metric space, Complete metric space, Open set, Closed set, Contraction mapping theorem, Normed space, Bounded linear operator, Inner product space, Parallelogram law, Banach space, Hilbert space, Projection theorem, Orthogonal projection onto linear variety, Normal equation, Generalized inverse, Singular value decomposition, regularization, Iterative image reconstruction, Constraint, L1 norm minimization, sparse modeling, incomplete data
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Two lectures are given in every week.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Role of functional analysis in engineering and review of first-year mathematics | Explain about the reasons for studying functional analysis the role of functional analysis in engineering |
| Class 2 | Metric space and Complete metric space | Explain about metric space and complete metric space |
| Class 3 | Open set and closed set | Explain definitions and properties of open and closed sets. |
| Class 4 | Contraction mapping theorem with applications to functional equations (Note: Lecture will be given via Zoom from overseas) | Explain about the contraction mapping theorem and its applications to functional equations. |
| Class 5 | Normed space and Bounded linear operator | Explain about normed space and bounded linear operator. |
| Class 6 | Inner product space and Parallelogram law, Banach space and Hilbert space | Explain about inner product space, parallelogram law, Banach space and Hilbert space. |
| Class 7 | Projection theorems in Hilbert space | Explain about the projection theorems in Hilbert space. |
| Class 8 | Orthogonal projection, Generalized inverse, Fourier series expansion -revisited | Explain about Orthogonal projection, Generalized inverse, Fourier series expansion. |
| Class 9 | Singular value decomposition and Image processing | Explain about Singular value decomposition, the image processing and image compression using the SVD. |
| Class 10 | Noise and Regularization | Explain the relationship between signal containing noise and regularization method. |
| Class 11 | Iterative image reconstruction | Explain the image reconstruction method by the iterative reconstruction method such as ART. |
| Class 12 | Image Reconstruction Problem with Constraints | Explain the image reconstruction method with constraints. |
| Class 13 | Norm minimization and Sparce modeling | Explain the relationship between norm minimization and sparse modeling. |
| Class 14 | Medical image reconstruction from incomlete data set | Explain the medical image reconstruction method from incomplete observation data set. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
I. Yamada, Kougaku no tameno Kansuu kaiseki, Saiensu co ltd, 2009.
Reference books, course materials, etc.
D.G.Luenberger, Optimization by Vector Space Mathods, Wiley, 1997.
C.W. Groetsche, Inverse Problems in the Mathematical Sciences, Springer, 1993.
Evaluation methods and criteria
Grading is made based on the final exam., etc. including the contents of the first half (basic idea on functional analysis) and the second half (basic idea on inverse problems).
Related courses
- LAS.M102 ： Linear Algebra I / Recitation
- LAS.M101 ： Calculus I / Recitation
- ICT.S206 ： Signal and System Analysis
- ICT.S210 ： Digital Signal Processing
- ICT.M316 ： Numerical Analysis (ICT)
- ICT.C201 ： Introduction to Information and Communications Engineering
- ICT.H504 ： Medical Image Processing
- ICT.H421 ： Medical Imaging Systems
- ICT.S414 ： Advanced Signal Processing (ICT)
Prerequisites
As a general rule, we accept only applications from students in the department of Information and communications Engineering.