Why do I Love and Hate mathematics at the same time?

I love mathematics – because it is the only “language” that humans can communicate each other without the barrier of their native language; because it is as beautiful as art and you can really enjoy the joy of mastering a mathematical topic (just like playing a musical instrument or painting a picture); because it is [...]

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Understanding the Impact of Mathematics on Image Processing

This blog is a continuation of my previous blog on the relationship between mathematics and image processing and aims at more technical virtuosity than conceptual understanding. What are the most influential works by mathematicians on image processing research in the past three decades? I would say MRF in 1980s; Wavelets and PDEs in 1990s; still-too-early-to-tell [...]

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On Bias-Variance Dilemma and Occam’s Razor

In frequentists’ framework, there is a well-known phenomenon called bias-variance dilemma: a simple model tends to have lower variance but introduce higher bias; a more complex model could reduce the bias at the price of higher variance. Therefore, it is often necessary to find a tradeoff and avoid over-complicated models. The preference of simple models [...]

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Deductive Logic vs. Inductive Reasoning

For a long period I was baffled by the relationship between mathematical and physical sciences (like Platonism vs. Aristotelian). If mathematics and physics are about the study of mentally reproducible and experimentally reproducible objects respectively, which one is more fundamental? Especially in view of the increasing influence of mathematics on science and engineering (for better [...]

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Where Does Prior Knowledge Come From?

One problem left by Laplace is the specification of prior probability. From Bayes’ formula, one can connect the posterior probability with the prior probability – but where does this prior model come from? Uniform prior or Jeffreys prior – they are surely mathematically different but do we have any principled way of choosing from them? [...]

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The Fight Between Frequentists and Bayesians

(I spent two hours writing the blog this afternoon but apparently I forgot to click the update button before heading to the P&T meeting. So I have to rewrite everything from the scratch. Given my poor short-term memory these days, I do not expect I can reproduce the original blog. I am still baffled why [...]

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Categorization of Memes

If you don’t know what is a meme? Here is the link to wiki http://en.wikipedia.org/wiki/Meme It is worth mentioning Richard Dawkins’ popular book “The Selfish Gene” which is a kind of meme itself. How do we categorize memes? Similar to genes, good memes propagate prolifically both in space and time; while bad memes cannot survive [...]

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Image Regularization/Prior

Many image processing problems I have worked on (e.g., denoising, interpolation, deblurring) can be formulated as special cases of inverse problems. Image prior (also known as regularization in a deterministic setting) plays the fundamental role in the solution to those problems. What is a prior P(x)? In fact, as long as P(x) is known, so [...]

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Lossless Image Coding Revisited

I have decided to write a series of blogs to reenforce my understanding of various image processing problems which I have worked in the past ten years. Many of those problems have common grounds because any they are all connected with two fundamental aspects: representation and optimization. However, since different problems (e.g., coding vs. non-coding) [...]

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Influential Thinkers

Philosophy: René Descartes Ludwig  Wittgenstein Immanuel Kant Arthur Schopenhauer Karl Popper Physical Sciences: I Newton A Einstein Joseph Fourier James Clerk Maxwell Ludwig Eduard Boltzmann J Gibbs Werner Heisenberg Erwin Schrödinger Richard Feynman Linus Pauling Edwin Hubble John Hopfield Lee Smolin Mathematical Sciences: Karl Gauss Leonhard Euler Joseph Louis Lagrange Pierre-Simon Laplace Henri Poincare Harold [...]

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