[Trilinos-Users] Model-fitting applications in trilinos

Ali - saveez at hotmail.com
Fri Feb 1 09:34:51 MST 2008 

> 
> On Monday 28 January 2008 18:01, Ali - wrote:
> > Kevin,
> >
> > While the images of interest are formed by fluorescent micro-particles or
> > bio-fluid cells, the outcome is very similar to images of galaxy clusters:
> > due to the same point-spread function, the spherical objects, being of
> > micro or macro size, are projected to 2D Airy function intensities in image
> > plane which can be well approximated by Gaussian. A typical cell or
> > micro-particle concentration does generate overlapping of images. Moreover,
> > in the case of optical imperfections such as refractive index mismatch, it
> > is well possible to have ellipsoidal Gaussians too -- it sounds like a
> > challanging problem.
> 
> Very interesting. You might look at what astronomers are doing for model-based 
> image matching. They have been working on the problem for decades. I'm sure 
> some automated method is used to identify galaxies in the Hubble Deep Field 
> images. 

While it may look like a basic problem with an obvious standard solution, locating the centre of particles with sub-pixel accuracy or, more generally, fitting a Gaussian model to particle-images has not been properly paid attention yet. Normally, the particle images are segmented first and then processed individually. The processings are simply:

- Centroid or weighted average calculation, very low accuracy when it comes to noisy data

- Least square fitting, moderate accuracy

- Levenberg-Marquardt fitting with some better accuracy than the previous ones. It seems the first person who used the LM algorithm only had access to the Numerical Recipes code, which is not an efficient implementation. For some reasons unknown to me, the next authors started following the same implementation and keep complaining about the poor fitting perfomance. No one has even thought of using a Newton-like fitting algorithm.

The point is that, neither astronomers not microscopists have found a good way of solving this problem. I introduced the problem to this mailing with the hope that there exist a 'PDE-CO inverse problem' approach for data fitting or parameter estimation. There are some active research going on about fitting deformable models to data using finite elements [1]. So, if you assume the data of interest like a simple 2D matrix filled with image intensity values, it is possible to fit 'a surface' to this data using FEM, however, this surface does not have any information about the model of interest. The question is, if we can solve the PDE in the image space, can we also solve it in the model parameters space?

[1] http://citeseer.ist.psu.edu/cache/papers/cs/22464/ftp:zSzzSzftp.cs.toronto.eduzSzpubzSztimzSzvi93.pdf/mcinerney92finite.pdf
> 
> Do you need to track the objects over time? 

Yes, and I use Sundance to track the objects :)

> 
> 
> >
> > Perhaps one way of making the solutions convex is to assign each object
> > image a 'property' which labels it as an individual. For example, it is
> > very fast and easy to find the centriod of each object image; assigning
> > each object a unique centroid should avoid the problem of object exchange
> > which creates a new solution, this may be used as an initial guess for the
> > Gaussian centre parameters while providing the total number of objects
> > which is also unknown. In this case, which numerical technique (which is
> > very likely to be already implemented in DAKOTA) can solve the problem of
> > multiple peak detection? 
> 
> 
> 
> > What are the keywords for this 'one-shot bunch 
> > fitting' approach?
> 
> I'm not sure. I'm basing my knowledge of the method on a couple of talks I saw 
> a few years back. However, I would guess that people have also tried the "one 
> shot bunch fitting" approach in automating spectroscopic analysis. 
> 
> - k
> 
> >
> >
> > -Ali
> >
> > > From: kevin.long at ttu.edu
> > > To: trilinos-users at software.sandia.gov
> > > Subject: Re: [Trilinos-Users] Model-fitting applications in trilinos
> > > Date: Mon, 28 Jan 2008 17:15:36 -0600
> > > CC: saveez at hotmail.com; rabartl at sandia.gov
> > >
> > >
> > > Ali,
> > >
> > > Trying to fit a bunch of Gaussians in one shot is a non-convex problem
> > > (because you can obviously exchange the positions of any two Gaussians to
> > > produce a new solution). I've seen talks by people trying "brute force"
> > > fitting of a large number of peaks, and they've found it's rather tricky
> > > to pull off in practice.  You'd certainly want a good initial guess.
> > >
> > > How isolated are your objects? If they are widely spaced (i.e.,
> > > object-to-object distance more than a few times sigma) you could look at
> > > smaller segments of the picture to get rough fits for a few peaks in each
> > > segment, then use those fits as initial guesses in a final fit including
> > > all objects. This could of course be done recursively.
> > >
> > > Can you have ellipsoidal Gaussians? (i.e., different sigmas along two
> > > different principal axes, oriented arbitrarily). That makes the problem
> > > even more fun. Astronomers have to do this when deducing models of
> > > individual models from clusters of galaxies; the objects can overlap
> > > quite a bit, and worse, can be significantly distorted from ellipsoids
> > > (due to tidal effects, and even in undisturbed galaxies there are
> > > distortions due to projection of a 3D triaxial body onto 2D).
> > >
> > > - kevin
> > >
> > > On Monday 28 January 2008 16:17, Ali - wrote:
> > > > Ross,
> > > >
> > > > Thanks for refering me to DAKOTA. I am trying to carry out model-based
> > > > segmentation image processing using optimisation approches. As an
> > > > example, assume an image involving randomly distributed Gaussian
> > > > intensities, or something like this:
> > > >
> > > > http://piv.vsj.or.jp/piv/data/01/piv01_1.bmp .
> > > >
> > > > The aim is to detect all the Gaussian objects in the image by fitting
> > > > them to a 2D Gaussian model which has 4 parameters: (x, y, sigma, I)
> > > > with x and y being the centre, sigma being the deviation and I being
> > > > the peak intensity. Typically each image includes a few hundereds of
> > > > these little Gaussian objects, multiplied by the 4 parameters, we are
> > > > dealing with over 1e+3 parameters in total.
> > > >
> > > > Normally people solve this problem by segmenting the individual object
> > > > (eg using watershed algorithm) and then fit them one by one using
> > > > Levenberg-Marquadt method. But I was wondering, while the Trilinos
> > > > solvers can handle millions of unknown in a typical PDE problem solved
> > > > by FEM, why shouldn't it be possible to fit 'all' of the objects at
> > > > once to a model which is simply a linear superposition of the
> > > > individual models?
> > > >
> > > >
> > > > -Ali
> > > >
> > > >
> > > > Subject: RE: [Trilinos-Users] Model-fitting applications in trilinos
> > > > Date: Mon, 28 Jan 2008 13:35:27 -0700
> > > > From: rabartl at sandia.gov
> > > > To: saveez at hotmail.com; trilinos-users at software.sandia.gov
> > > >
> > > >
> > > >
> > > >
> > > >
> > > >
> > > >
> > > >
> > > >
> > > >
> > > > Ali,
> > > >
> > > > Depending on what you need there are a few possibilities.
> > > > Without knowing the details, I would recommend that you first look at a
> > > > related project called Dakota which as parameter estimation and
> > > > least-squares fitting methods.  You can find more information at:
> > > >
> > > >     http://www.cs.sandia.gov/DAKOTA/software.html
> > > >
> > > > If you only need to fit a smaller
> > > > number of parameters, I would definitely look at Dakota first.  Fitting
> > > > a large number of parameters (i.e. 1e+3 and up) is a lot of work to set
> > > > up and requires some serious optimization algorithms.
> > > >
> > > > Cheers,
> > > >
> > > > Ross
> > > >
> > > >
> > > >
> > > >
> > > > From: trilinos-users-bounces at software.sandia.gov
> > > > [mailto:trilinos-users-bounces at software.sandia.gov] On Behalf Of Ali
> > > > -
> > > > Sent: Monday, January 28, 2008 12:19 PM
> > > > To:
> > > > trilinos-users at software.sandia.gov
> > > > Subject: [Trilinos-Users]
> > > > Model-fitting applications in trilinos
> > > >
> > > >
> > > >
> > > > Hi,
> > > >
> > > > Is there any packages in Trilinos (or
> > > > other Sandia software) specifically designed for model-fitting
> > > > (curve-fitting)?
> > > >
> > > >
> > > > -Ali
> > > >
> > > >
> > > >
> > > > Messenger on the move. Text MSN to 63463 now!
> > > >
> > > > _________________________________________________________________
> > > > Telly addicts unite!
> > > > http://www.searchgamesbox.com/tvtown.shtml
> > >
> > > --
> > > Dr. Kevin Long
> > > Department of Mathematics and Statistics
> > > Texas Tech University
> > > Lubbock TX 79409
> > > kevin.long at ttu.edu
> >
> > _________________________________________________________________
> > Share what Santa brought you
> > https://www.mycooluncool.com
> 
> -- 
> Dr. Kevin Long
> Department of Mathematics and Statistics
> Texas Tech University
> Lubbock TX 79409
> kevin.long at ttu.edu

_________________________________________________________________
Telly addicts unite!
http://www.searchgamesbox.com/tvtown.shtml
-------------- next part --------------
An HTML attachment was scrubbed...
URL: http://software.sandia.gov/mailman/private/trilinos-users/attachments/20080201/8b670368/attachment-0001.html

More information about the Trilinos-Users mailing list