On Friday, 10 November 2017 05:29:09 Keller, Jacob wrote:
> >>62500 is < 40^3, so ±20 indices on each axis.
> 50Å / 20 = 2.5Å, so not quite 2.5Å resolution
>
> Nice--thanks for calculating that. Couldn't remember how to do it off-hand,
> and I guess my over-estimate comes from most protein crystals having some
> symmetry. I don't really think it affects the question though--do you?
>
> >>All that proves is that assigning each 1x1x1 voxel a separate density value
> >>is a very inefficient use of information. Adjacent voxels are not
> >>independent, and no possible assignment of values will get around the
> >>inherent blockiness of the representation.
>
> Not sure what this means--what is the precise definition or measure of
> "efficient use of information?" Like a compression algorithm?
If it helps you to think of it that way, fine.
Suppose it is possible to compress a data set losslessly.
The information content is unchanged, but the compressed representation
is smaller than the original, so the information content per unit of size
is higher - a better use of space - hence "more efficient".
> Are diffraction data sets like compressed data?
Not the diffraction data, no.
But it is true that a truncated Fourier series is one way of compressing data.
Because of the truncation. it is a lossy, rather than lossless, compression.
An infinite series could give infinite resolution, but a truncated series is
limited by the resolution of terms that are kept after truncation.
For example the compression used in JPEG is a truncated discrete cosine
transform (DCT), making JPEG files smaller than the original pixel-by-pixel
image.
I'll throw a brain-teaser back at you.
As just noted, encoding the continuous electron density distribution in a
unit cell as a truncated Fourier series is essentially creating a JPEG image of
the original. It is lossy, but as we know from experience JPEG images are
pretty good at retaining the "feel" of the origin even with fairly severe
truncation.
But newer compression algorithms like JPEG2000 don't use DCTs,
instead they use wavelets. I won't get sidetracked by trying to describe
wavelets, but the point is that by switching from a series of cosines to
a series of wavelets you can get higher compression. They are
more efficient in representing the original data at a selected resolution.
So here's the brain-teaser:
Why does Nature use Fourier transforms rather than Wavelet transforms?
Or does she?
Have we crystallographers been fooled into describing our experiments
in terms of Fourier transforms when we could do better by using wavelets
or some other transform entirely?
Ethan
> Also, the "blockiness" of representation is totally ancillary--you can do all
> of the smoothing you want, I think, and the voxel map will still be basically
> lousy. No?
> >>I know! Let's instead of assigning a magnitude per voxel, let's assign a
> >>magnitude per something-resolution-sensitive, like a sin wave. Then for
> >>each hkl measurement we get one sin wave term. Add up all the sine waves
> >>and what do you get? Ta da. A nice map.
>
> It was good of proto-crystallographers to invent diffraction as a way to
> apply Fourier Series. I don't know--it seems funny to me that somehow
> diffraction is able to harness "efficient information use," whereas the voxel
> map is not. I am looking for more insight into this.
>
> >>Aren't Fourier series marvelous?
>
> Well, I have always liked FTs, but your explanations are not particularly
> enlightening to me yet.
>
> I will re-iterate that the reason I brought this up is that the imaging world
> might learn a lot from crystallography's incredible extraction of all
> possible information through the use of priors and modelling.
>
> Also, I hope you noticed that all of the parameters about the
> crystallographic data set were extremely optimistic, and in reality the
> information content would be far less.
>
> One could compare the information content of the derived structure to that of
> the measurements to get a metric for "information extraction," perhaps, and
> this could be applied across many types of experiments in different fields. I
> nominate crystallography for the best ratio.
>
> JPK
>
>
>
>
> > Assuming that it is apt, however: is this a possible way to see the power
> > of all of our Bayesian modelling? Could one use our modelling tools on such
> > a grainy picture and arrive at similar results?
> >
> > Are our data sets really this poor in information, and we just model the
> > heck out of them, as perhaps evidenced by our scarily low data:parameters
> > ratios?
> >
> > My underlying motivation in this thought experiment is to illustrate the
> > richness in information (and poorness of modelling) that one achieves in
> > fluorescence microscopic imaging. If crystallography is any measure of the
> > power of modelling, one could really go to town on some of these terabyte
> > 5D functional data sets we see around here at Janelia (and on YouTube).
> >
> > What do you think?
> >
> > Jacob Keller
> >
> > +++++++++++++++++++++++++++++++++++++++++++++++++
> > Jacob Pearson Keller
> > Research Scientist / Looger Lab
> > HHMI Janelia Research Campus
> > 19700 Helix Dr, Ashburn, VA 20147
> > (571)209-4000 x3159
> > +++++++++++++++++++++++++++++++++++++++++++++++++
> >
>
> --
> Ethan A Merritt, Dept of Biochemistry
> Biomolecular Structure Center, K-428 Health Sciences Bldg
> MS 357742, University of Washington, Seattle 98195-7742
--
Ethan A Merritt, Dept of Biochemistry
Biomolecular Structure Center, K-428 Health Sciences Bldg
MS 357742, University of Washington, Seattle 98195-7742
