Articles by Christian Chapman

  1. A Vector-Gaussian Quantizer using Spatial Tours

    By representing a high-dimensional random variable in terms of its time-of-occurrence in a space-filling tour, we gain a systematic way of pruning out atypical regions of the observation space. This allows for efficient source coding since it excludes encodings that correspond to bogus observations.

    This could lead to design of …

  2. Source Encoders as Channels

    It is well known that a rate-distortion-optimal source encoder's output generally doesn't match its source's distribution. This can make some analyses a pain in the neck. For example, say you want to investigate the relationship between a signal that appears in a source, and that signal's appearance in an encoding …

  3. Combinatoral Iteration in Matlab

    Often we need a to write a program that iterates over all combinations of \(k\) elements from some set (elements chosen without replacement). It isn't obvious how to efficiently traverse through all these sets, although if bitwise arithmetic is readily available one can use Gosper's Hack. This kind of problem …

  4. Some Interaction Information Examples

    Interaction information between three random variables has a much less immediate interpretation compared to other information quantities. This makes it more tricky to work with. An i.i. term \(I(X;Y;Z)\) between \(X,\ Y\) and \(Z\), could be either negative:

    If we are trying to specify \(X\) with …

  5. Point KL-Divergence is not Very Negative Very Often

    If \(X\sim P\) then for any distribution \(Q\) it is unlikely that \(Q\) ascribes much greater density to \(X\)'s outcome than \(P\) does. In fact if \(P,Q\) have PDFs \(f_P, f_Q\), then:

    \begin{align} \mathbb{P}(f_P(X)\leq c f_Q(X)) &= \int \mathbf{1}_{\{x …
  6. Tails of Probability Distributions are Nowhere Dense

    Lately I have been thinking about Kullback-Liebler divergence on probability distributions, also called the KL discriminant or relative entropy. It is often useful to think about this quantity as a distance, even though it doesn't behave much like one: it's not symmetric, it doesn't follow the triangle inequality, and even …

  7. Thoughts on Demosaicing for X-Trans Sensors

    A lot of new Fujifilm cameras use their own brand of 'X-Trans' sensors which have a non-Bayer color mosaic. Fujifilm claims that the less regular arrangement makes it more uncommon for edges in the scene to make moire patterns with the mosaic. They say this justifies not including an optical …

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