I have personally come across this famour Heisenberg’s Uncertainty Principle many times in my lifetime – when I was a college student, when I started to learn about what is short-time FT, when I prepared my courses on wavelet. I must admit it is a difficult concept to grasp and I often adopt the strategy [...]
Archive for the ‘wavelets’ Category
Heisenberg’s Uncertainty Principle
Posted in wavelets on April 27, 2009 | Leave a Comment »
Research Advices to PhD Students
Posted in wavelets on November 24, 2008 | Leave a Comment »
I am writing this essay for first-year or second-year PhD students who are still early in their research training. Since students taking my wavelet course have diverse background, I try to provide general advices regardless of the technical field (of course your PhD advisor should be a better resource on research methodology in your own [...]
A few last words on the theory part of wavelets
Posted in wavelets on November 14, 2008 | Leave a Comment »
I hope my effort on connecting two-channel filter banks in the discrete space to scaling/wavelet functions in the continuous space at least makes some sense to you. This part is not the meat of this course (you certainly don’t need to worry about it for exams :->). But the underlying relationship between two seemingly different [...]
The idea of multiresolution
Posted in wavelets on November 9, 2008 | Leave a Comment »
It is not easy to find out when and where exactly this idea of multiresolution came about. Apparently wikipedia attributes multiresolution analysis (MRA) to S. Mallat and Y. Meyer for their pioneering work on wavelet-based MRA. However, the idea of MRA appeared much earlier than the born of wavelets. The pyramid-based image representation was suggested [...]
What we have learned about wavelets from signal denoising
Posted in wavelets on November 4, 2008 | Leave a Comment »
Restoration of a signal contaminated by additive white Gaussian noise is a classical problem in signal processing which dated back as early as Wiener’s filter solution. Despite the optimality of Wiener filtering for stationary Gaussian processes, most signals in real worlds are characterized by nonstationarity and non-Gaussianity. That is why signal modeling plays an essential [...]
Interaction between science and engineering
Posted in wavelets on October 28, 2008 | Leave a Comment »
So far, we have heard seven Friday session talks covering a wide range of topics from empirical mode decomposition and diffusion tensor MRI to image watermarking and biometrics. I believe it is fair to say that the diversity of the topics (arising from diverse background of students taking this course) is consistent with the diverse [...]
What is good and bad about exams?
Posted in wavelets on October 27, 2008 | Leave a Comment »
As midterm exam approaches, I start to sense the anxiety among my students. It reminds me how I felt about exams when I was a student. As a student coming from China, I can’t even remeber how many exams I have been through in my school years (in the worst time – say middle school, [...]
Algebraic vs. geometric aspect of wavelets
Posted in wavelets on October 21, 2008 | Leave a Comment »
As we gradually transit from first-generation to second-generation wavelets, the style of thinking varies. In Daubechie’s construction, we heavely used algebraic tricks and motivations such as the polynomial of some special property.Despite the elegance of such construction, it turns out that there are alternative approaches which rely on the language of geometry. Geometry is usually [...]
Importance of phase in signal processing and science
Posted in wavelets on October 1, 2008 | Leave a Comment »
As early as in 1981, the importance of phase was recognized by signal processing researchers (Here is the link to the article http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1456290). Note that that was before the infancy of wavelets – therefore the phase in that article solely refers to the use in Fourier transforms. Twenty years after the invention of wavelet transforms, [...]
From First-Generation to Second-Generation Wavelets
Posted in wavelets on September 25, 2008 | Leave a Comment »
Today we finished the construction of Daubechies’ maxflat filters. For those who are anxious to know what is wavelet – infinite repetition of LP/HP filter pair H0 and H1would lead to scaling and wavelet functions in continuous-space wavelet theory. So conceptually discrete filters H0/H1 and continuous scaling/wavelet functions carry exactly identical information because one is [...]
