In the Bayesian approach to inference, the probability density function (pdf) associated with a variable characterizes the uncertainty one has in that variable. This interpretation of probability is in contrast to viewing its role only in terms of characterizing the variation in a random (or stochastic) variable. True, a single measurement of a random variable represents a fixed value and therefore may seem to be deterministic. However, by associating a probability distribution with the measurement, one is admitting that the measurement has uncertainty, possibly because it is a sample from a stochastic process, but more fundamentally because there is certain amount of doubt about the measurement.

In this way of thinking, because all measurements are subject to some uncertainty, they have a pdf with a finite width. For example, when x is measured and the result is stated as r +/- e, where e is the rms "error," the pdf for x is often assumed to be a Gaussian with a central value of r and a standard deviation of e. Error in this context really means an estimated uncertainty in the measured value.

Bayesian inference builds on this fundamental viewpoint to consider probability distributions on any derived result, let's say a tomographic reconstruction of a 2D image based on numerous measured projections. If the reconstruction image has n x n pixels, the pdf for the reconstruction is a function of n^2 variables.

Using a multivariate Gaussian approximation, this pdf is characterized by its mean value x^hat and an n^2 by n^2 correlation matrix. For this example, the uncertainties in any pixel are correlated with those of every other pixel; there are no zero elements in the correlation matrix. Such a pdf is not easy to deal with, although there are ways to visualize it. Numerical methods (Monte Carlo, for example) allow one to sample the covariance matrix or to infer its consequences on derived quantities.

While this construct can not be contained in a computer, it is useful as a way to think about inference. The trick is to find meaningful ways to summarize the uncertainty in the reconstruction. The rms uncertainty for each pixel is clearly a minimal characterization of the uncertainty in the reconstruction.

Bayesian analysis can become quite involved, but for me it is worth the trouble because it offers a complete and coherent approach to honestly dealing with uncertainties. I wrote an "Introduction to Bayesian image analysis" about a decade ago, which might help you get started in understanding this approach http://http://kmh-lanl.hansonhub.com/publications/medim93.pdf

To my understand, when it comes to probability in an image, we are talking about information entropy. We can use entropy to measure the similarity of specific regions in an image.

If the entropy of one region is high, the similarity of this region is high, which means this region may be background in this image. If the entropy of one region is low, the similarity of this region is low, which means this region may be some edges in this image.

Hope it helps.

Maps of probabilies are commonly used to model the expected apperance of an image/presence of an object in a given reference space. They quantify the likelyhood of a specific pixel/voxel showing (part of) a specific object/tissue. These maps are then often used for segmentation purposes e.g. in medical brain imaging.

They are created from many other pre-segmented images of the same scene/organ are act as a population average.

so from my understanding the concept of probability only makes sense for a set of images, not a single one.

Thank you Zhang for your response. The idea of entropy( a difficult concept in itself) itself assumes probability but the idea of assuming an image conveys information

seems a nice way of beginning, that is... removing the uncertainty before the image is captured. I was however seeking an even more basic connection .along the lines

of explanation given by Weiss ..

Thank you Weiss for your response which tallies with my own perception.however

I would welcome your views on the following. If we assume a grey image say

of 20* 20 pixels and each pixel has a possible range 0-255 then the outcomes of the

experiment would be (20*20)^255 and then we further assume a probability for

specific outcomes would this be a meaningful interpretation? I think this is then

analogous to saying the English alphabet conveys about 1.6 bits of information. I look forward to your views.

Cheers

please note Correction in above reply .... it should read 256^(20*20) instead of (20*20)^255 ...

Narasim that is true it could be more complicated to be N*N*M where M is the number of bands in this case the problem becomes more complex. One can say for example what is the probability of having K pixels with a grey level value G within an image of N*N*M ?

In general In many image processing tasks, uncertainty plays a fundamental role. In segmentation and classification the future inputs are not known perfectly at the time of system design, but instead must be characterized only in terms of their "typical," or "likely" behavior, by means of some probabilistic model.

Probability enters from the definition of the process of image formation. The most common model is Y=H*X+N where Y is the given image, X is the "true" image, H is

a blurring kernel, the star sign stands for convolution and N is noise that is presented at the sensor level (termal and quantum noises). This model assumes space invariant blur and additive noise. Usually one assumes that the noise is a random variable taken from a normal distribution. In order to recover the true image X from the given noisy and blurred image Y one uses the MAP method (Maximum a-posteriory probability). This is done via the definition P(Y|X)= P(N) = CEXP{-||Y-H*X||^2} where C is a normalisation factor. The map estimation is X_MAP=MAX_X P(X|Y) = C MAX_X P(Y|X)P(X) where P(X) is a prior over the set of all images. Usually P(X) involves smoothness assumptions or sparsity assumption wrt a given dictionary. In specific applications one can be much more precise in the prior over the specific class of images

Sochen.. Thank you for your nice explanation ..it helps me to clarify my own thoughts more precisely. especially your observation "Usually P(X) involves smoothness assumptions or sparsity assumption wrt a given dictionary. In specific applications one can be much more precise in the prior over the specific class of images " and for the moment let us ignore noise. It would be a help if you can suggest how we may look at

this wrt to a very general picture from my previous comment " If we assume a grey image say of 20* 20 pixels and each pixel has a possible range 0-255 then the total outcomes of the experiment would be 256^(20*20)" and now the question is

what how can we derive P(X) from this set ? Obviously we have to bring

cognitive interpretation to help us. It is this aspect that is the intent of the question ..

and if this can be expressed mathematically it would indeed be of great interest.

as i think subjectivity would be removed.

Call me Nir. The prior P(X) is a function that tells you how probable that a given configuration (one of your 256^(20x20) is actually an image (taken or could be taken

in the real world). Finding good P(X) is equivalent to characterise completely the set of all images. This is the holy grail of image processing. All algorithms in IP assumes (implicitly or explicitly) some form of P(X). One example is the assumption of piecewise smoothness such as in the Mumford-Shah functional or in the Beltrami framework.

For calculating the entropy where the ideal image has entropy of 8.

Nir Thanks very much. . It was indeed most satisfying to interact with you .

Adil Thank you I could not follow your answer . Would you care to amplify..

I would like to exand on the answers of Nir and Marcel.

Most tasks of image (signal) processing implicitly or explicitly consider the given image as a sample out of a distribution of possible images (signals): probability enters. These distributions can occur in quite different ways, as is nicely explained in the marvellous book "Pattern Theory. A Stochastic Analysis of Real-World Signals." by David Mumford and Agnès Desolneux. I just want to mention two examples:

1. Nir's example considers an image as a sample of a family of blurred and noisy images, which allows to estimate the original image.

I would like to give another example in the same vein, one of 'image analysis'. Suppose we want to "find a ball in an image". We have some idea of what a ball looks like under different lightings, and what typical backgrounds look like. We can try to describe this as a random generating procedure for images. We chose a prior on the possible positions of the balls, and a distribution of images given the position of the ball. This enables us to estimate the position of the ball in the given image (the Bayesian way, or via a MAP-estimate as described by Nir.) The priors and conditional probabilities encode our knowledge about the structures we are looking for.

2. Probability enters in a different way in 'a contrario methods'.

Suppose we have some algorithm that detects edges or corners in an image (which we want to use to enhance these features). Suppose we have some notion of a random image, pure noise, given by a distribution of such random images. Our algorithm is bound to find useless edges and corners in such random images. We now use the approach of statistical testing. The null-hypothesis "this is not an edge but just noise" is rejected (and the edge accepted as 'significant'), if, under the null hypothesis, the p-value of the given image is below a certain level (this is a rough sketch, for details see p. 114ff in Mumford/Desolneux, the p value has to be chosen according to the Bonferroni correction.) We end up with the interesting part of the output of our algorithm.

Is there an interesting image processing task that does not (at least implicitly) consider the image as a sample from the family of all images ? I don't think so. Of course one can apply all kinds of functions to an image , i.e. a vector \in \R^{m\times n}. But an interesting operation will relate to images as visual objects, as samples of the true distribution of images. Many of the standard operations in image processing have not been motivated by this probablistic approach, but it would enter the scene once we tried to explain why this operation is useful.

Artefacts in stereo vision occur when intrepretation of texture is inconsistent in the two images. Statistical detection theory can be applied wit success on this problem. See: Blas, M. R. and M. Blanke: Stereovision with texture learning for fault-tolerant automatic baling. Computers and Electronics in Agriculture. Vol 75 (2011) pp. 159-168. Doi. 10.1016/j.compag.2010.10.012.

Thanks to all the contributors Yu Zhang ,·Nir Sochen ,Adil AL-Rammahi , Stefan Born ,M. Awad ·and Mogens Blanke . . It has helped me to formulate a little more

satisfactorily (at least to me ) the following brief conclusions and I would welcome modifications and rectifications of the same.

1. Every time an image is captured the discrete image pixels determine a large but finite universal set of outcomes. ( for the moment excluding noise).

2. Out of this set we are able to cognitively identify interesting subsets as useful

based on subjectivity or common understanding. Here I encounter a philosophical

problem viz what is the true image? eg how does one deal with optical illusion?

or an artistic interpretation? A ball for instance becomes smaller with distance

so we need to identify a portion (subset ) as it gets morphed but these

morphed objects are within the universal set..there is a connection here with

topology I suppose.

3 Out of the above, among many other ideas that seem to emerge from the

comments ,is the interesting possibility that a general structure for systems of

cognition could be generated independent of any particular image.

3. So it seems possible that in nature as Fibonacci series or Fractal patterns

are seen to occur there could be a general formulation of some Probability

Structure which could be used to draw a huge amount of information from any Image

than is normally derived. and we do see Fractal analysis , and Iterated function systems being used in Image processing and analysis.

I used sub-sampling technique (with a random distribution of pixels in the sampled set) to help reducing noise effect and also to reduce calculation time for plane and circle Hough transform from LRF image (very noisy).

Thank you Gabriel your method seems very interesting and seems ideal for images with a lot of segmentation. . I understand that generalized Hough transform is quite powerful combined with either IFS or fractals for shape identification .To me what is fascinating about fractals is the infinite length of a boundary while still enclosing a finite area (as I have understood it..) Is it possible

to share some simple examples of your work or notes that could be experimented using Scilab 5.3.3. and which would interest graduate students.?

In the Bayesian approach to inference, the probability density function (pdf) associated with a variable characterizes the uncertainty one has in that variable. This interpretation of probability is in contrast to viewing its role only in terms of characterizing the variation in a random (or stochastic) variable. True, a single measurement of a random variable represents a fixed value and therefore may seem to be deterministic. However, by associating a probability distribution with the measurement, one is admitting that the measurement has uncertainty, possibly because it is a sample from a stochastic process, but more fundamentally because there is certain amount of doubt about the measurement.

In this way of thinking, because all measurements are subject to some uncertainty, they have a pdf with a finite width. For example, when x is measured and the result is stated as r +/- e, where e is the rms "error," the pdf for x is often assumed to be a Gaussian with a central value of r and a standard deviation of e. Error in this context really means an estimated uncertainty in the measured value.

Bayesian inference builds on this fundamental viewpoint to consider probability distributions on any derived result, let's say a tomographic reconstruction of a 2D image based on numerous measured projections. If the reconstruction image has n x n pixels, the pdf for the reconstruction is a function of n^2 variables.

Using a multivariate Gaussian approximation, this pdf is characterized by its mean value x^hat and an n^2 by n^2 correlation matrix. For this example, the uncertainties in any pixel are correlated with those of every other pixel; there are no zero elements in the correlation matrix. Such a pdf is not easy to deal with, although there are ways to visualize it. Numerical methods (Monte Carlo, for example) allow one to sample the covariance matrix or to infer its consequences on derived quantities.

While this construct can not be contained in a computer, it is useful as a way to think about inference. The trick is to find meaningful ways to summarize the uncertainty in the reconstruction. The rms uncertainty for each pixel is clearly a minimal characterization of the uncertainty in the reconstruction.

Bayesian analysis can become quite involved, but for me it is worth the trouble because it offers a complete and coherent approach to honestly dealing with uncertainties. I wrote an "Introduction to Bayesian image analysis" about a decade ago, which might help you get started in understanding this approach http://http://kmh-lanl.hansonhub.com/publications/medim93.pdf

Thank you Kenneth . Your article mentioned above was indeed very illuminating and I do treasure it..and of course I need to pore over it considerably. It has indeed expanded my vision . The explanation above relies on probability entering the scene

through uncertainty in measurements ( is it connected with Heisenberg uncertainty principle which requires two variables ?) or is it a modeling of a real measurement

device. Is it reasonable to conceptualize an ideal measurement with no error

and allow probability as a subjective function of the observer? I suppose

there is a fundamental question here of whether "reality" is subjective or objective.

It would be most exhilarating to know your views on this. May i thank you once again

for a very satisfying addition to the answers above.

Not directly related to Heisenberg principle, but because that has to do with uncertainty, a pdf would be used to characterize it. I does have to do with real-world uncertainty and so imagining an ideal measurement with no uncertainty is fanciful.

Thanks. .. but i suppose that if we are doomed to " real world uncertainty " there is no way of being certain of what the "true " or "exact " value of the variable and as you stated we must live with a best estimate using pdf..

some doubts on further thought..

In Bayesian approach I think there is a fundamental assumption that the

prior event is random and we are only trying to improve on this with additional data.

In other words an inherent uncertainty before we even begin to process a better

conjecture with further observations ..

.... but as also stated the moment an image is captured it is definitely deterministic. and therefore the "true" reality and all we do in

image processing is to better the measurement process..is this a correct

view..? (open to corrections).... because otherwise if we do not postulate an objective

deterministic reality... it is possible that we might end up

with a processed image which is accepted (based on our perception) rather

than the correct one .. .I certainly hope that my comments are clear and I am not tying myself up in confusion..

see eg http://mnikolova.perso.math.cnrs.fr/MAP.pdf and http://mnikolova.perso.math.cnrs.fr/Bounds.pdf

Using probability can lead to very bad choices, but this is a tool for some heuristics

Thanks for a very incisive , elegant and bringing out little known contradictions in your analysis . it certainly cleared up so many uncertainties.. .much indebted to you for your correspondence. thanks once again.

some doubts on further thought..

In Bayesian approach I think there is a fundamental assumption that the prior event is random and we are only trying to improve on this with additional data.

> ironically, you use the phrase "prior event", ironic because a fundamental concept in Bayesian analysis is that one must state a prior pdf on the variables about which you wish to make an inference. Bayes law is p(x|d) ~ p(d|x) p(x). p(x) is the prior on x, the variable to be infered and p(d|x) is the likelihood (of the data d, given x). I won't go into detail, for that see the paper I referred to, but the basic idea is that the original knowledge about x, encapsulated by p(x), is updated by the measurements to arrive at the state of knowledge after the measurement is incorporated, called the posterior.

In other words an inherent uncertainty before we even begin to process a better

conjecture with further observations ..

.... but as also stated the moment an image is captured it is definitely deterministic and therefore the "true" reality and all we do in image processing is to better the measurement process.

> I think deterministic is the wrong term. Perhaps, fixed is better. Don't know what is meant by "true" reality. As for improving the measurement process, I would rather say making the most of the existing measurements.

is this a correct view..? (open to corrections).... because otherwise if we do not postulate an objective deterministic reality... it is possible that we might end up with a processed image which is accepted (based on our perception) rather than the correct one .. .I certainly hope that my comments are clear and I am not tying myself up in confusion.

> See, phrases like "correct one" and " objective deterministic reality" are misleading. Data analysis involves obtaining the best possible estimate of reality, given data and what else we know, not uncovering the true reality. That is unattainable.

I tried to read between your statements to determine what you were getting at. I hope my comments addressed your concerns.

> As for Mila's comments: Nice papers - I can't follow all the math but let me observe that the choice of the estimate of x always has to be considered carefully, especially when the posterior pdf is not Gaussian like. My preference is to think of the posterior itself to be the most important result of Bayesian analysis. Having obtained that, then one can consider how to present a representative estimate, along with its uncertainties. Admittedly, that task is sometimes difficult, and at times, perhaps impossible.

Kenneth thank you.. for your considered and valuable views.it addresses the issues raised .. My original doubts are contained in earlier comments reproduced far below . my concern based on the inputs received till now is.. should we always add cautionary qualifying remarks to Image processing results.based on probability.." I believe

Mila's work justifies my doubts .

.please see below .IMHO .my views

" It has helped me to formulate a little more

satisfactorily (at least to me ) the following brief conclusions and I would welcome modifications and rectifications of the same.

1. Every time an image is captured the discrete image pixels determine a large but finite universal set of outcomes. ( for the moment excluding noise).

2. Out of this set we are able to cognitively identify interesting subsets as useful

based on subjectivity or common understanding. Here I encounter a philosophical

problem viz what is the true image? eg how does one deal with optical illusion?

or an artistic interpretation?

"

thanks to everyone for their valuable inputs..

should we always add cautionary qualifying remarks to Image processing results based on probability..

kh> certainly: we learn in out physics training that a measurement without and uncertainty estimate is useless. Actually, the same goes for any inference. However, expressing an uncertainty in an estimate image is very difficult, which is why almost no one does it.

You keep coming back to statements like "what is the true image?" Give it up and you will be free.

Thanks Alex .. perhaps that is one way out....on reflection... I suppose it does sound like religion :-)..however following rankles my mind...would like your views on same

if it is indeed not too ..tiresome..

1 as you pointed out . measurement is usually with indication of +- error = ("true"- measured) which I think is different from uncertainty (which implies no knowledge

of " correct"? value). Uncertainty to my understanding ( Heisenberg) comes with two variables eg either we know the velocity(momentum) accurately or the position .

2 The classic y =1/ (1-x) = 1+x+ x^2 ...x^n (n to infinity ) RHS which from GP

is valid only for mod(x)1..

so it is necessary on RHS to fix it with convergent series constraint.In case of Image

processing with probability are we on LHS or RHS?

Also have a look at

http://www.123opticalillusions.com/pages/albert-einstein-marilyn-monroe.php

It is rather difficult to Give it up :- )....this is the difficulty..

Ironically, Einstein reportedly said: The definition of insanity is doing the same thing over and over again and expecting different results.

but Ironically,this is not the case with probability..

Thanks Kenneth for the patient and kind response . I do appreciate it. and following your advice . I hope this is a closure on this topic. thanks once again.

Closures on scientific topics are extremely rare events. See some books on epistemology eg the classical Kuhn. Paradigms play an important role. It appears that breaking a paradigm is a real scientific achievement.

As to Bayes, see some papers on sampling the posterior by Darbon & Sigelle, also Yuille & Papandreou.

Other refs Gribonval:

http://hal.inria.fr/docs/00/57/60/41/PDF/MAPvsOptim-revision-HAL.pdf

among others

Mila I briefly went through the introduction and will definitely spend time on it in detail ..I am extremely indebted to you . and for your interpretation of scientific achievements.. It is exactly what I had in mind and your reference addresses my concerns with signal processing using probability. in a very satisfying manner.

probability to my mind is definitely useful but we have to be careful with it and its

application (i think it cannot be a replacement for more deterministic knowledge however) i would refer to the following :

http://www.dailymail.co.uk/news/article-2241854/What-odds-New-study-shows-guessing-heads-tails-isnt-really-50-50-game.html

. Thank you once again for a very

encouraging and valuable response. . actually i was fearing Intellectual exhaustion

on this.

If I may add my two cents the debate here is not about probability in image processing but on the status of "truth" or "objective reality". Strange as it may sound I don't think that these concepts are useful in science and sometimes they can be misleading and harmful. Science is not about finding "the truth". It is about finding the best explanation for data. Science has criteria to decide between competing explanations. These criteria are, inter alia, "simplicity", coherent explanation, conformity with other explanations, scope of application and power of predictability. In deciding between explanations we don not refer at all to some transcendental truth.

Regarding MAP I have a slightly different or maybe complementary view to that of Kenneth. I view it as a way to encode our knowledge and not a way to encode uncertainty. The Bayes rule helps us to separate our knowledge about the noise (expressed usually in the likelihood function) from our knowledge about the image/signal/process expressed in the prior.

Take for example the problem of interpolation. Here there is no noise at all just a finite number of points and the (exact) value of a function at these points. Can we infer from that knowledge the value of the function at other points ? Clearly this is impossible without providing knowledge about the type of function that we expect (continuous, polynomial, etc) We may also demand the the "explanation", i.e. the function, wil be the "simplest" one that explains the data so we choose the polynomial with the smallest degree that passes through the points and so on.

Similarly in other cases we may choose to reduce the space of possible solutions to be in a certain subspace (L_2 functions, bounded variation fns, etc). In my eyes the prior reflects the narrowing of the space of possibilities not enlarging it to accomodate for uncertainties.

Thanks Nir. I find your definition of science appropriate. and in the context of image processing I have meant "objective reality " as what was the "original object"

when we capture an image of something which has not been seen except as

the image..I would also refer to optical illusion referred earlier where even after seeing the image

it becomes difficult to say which object it is?. what is of concern here is whether after processing we get this object? but your observation(narrowing of the space of possibilities not enlarging it to accomodate for uncertainties.) is indeed very appealing to me

It seems all this is supposed to be contained in "fixing" the prior approximately

and then "cleaning up" the data with MAP to produce the final image. On what basis is this fixing to be done then? I hope I have used all the semantics correctly and am open to correction always..Surely I

have benefited immensely from all the answers. I am indebted to you . Kenneth ,

and Mila for so much knowledge and a definite expansion of my understanding

of probability especially wrt to image and signal processing. I have no intention

however of belabouring an obvious issue if it is so deemed .

another aspect that might be of interest is for example. if there is a connection between

Chaos theory ( I understand this as as small input perturbations leading to large

scale differences in output.please correct if this concept is not valid ) and prior input. What i have in mind is can a

weather map image show possible storms with a single image if small changes in

priors produce chaotic output with a known likelihood function) . I hope again that I have not stepped on any sensitive toes..wrt to semantics or even understanding ..

but it is just a thought , Would be happy to receive gentle reactions if it is

a worthwhile area for exploration. thanks..

Nir - I agree with most of what you say.

Narasim - I think I may have an example of what you are referring to. It has to do with visualizing the variability in weather predictions. Slide 25 of this presentation: http://kmh-lanl.hansonhub.com/talks/lansce05-t1vgr.pdf , shows the weather forecast made on October 30, 2003, taken from a National Geographic article. Shown are 22 predictions of the expected path of future storms. The variability in predictions is likely driven by uncertainties in measured weather conditions on that date, but could also include uncertainties in the calculations. The nonlinear effects that you mention could also be included.

The above presentation is from a lecture series on Bayesian analysis that I gave at the Los Alamos Neutron Scattering Center (LANSCE). The second half of the next lecture http://kmh-lanl.hansonhub.com/talks/lansce05-t2vgr.pdf describes ways to sample the posterior probability distribution. These sample can be used for visualization of uncertainty, or to estimate the expected variance in a result. I am a proponent of using quasi-random samples from the posterior, which can be calculated using the centroidal Voronoi tessellation technique.

Kenneth , thanks a lot, I just skimmed through the references above.. it is eminently readable..and elegant. and must be included in Engineering probability texts... I am thankful to Mila for helping carry forward the topic..I am indebted to you for for a very

exciting guidance...

Would welcome comments/ corrections/ suggestions on the attached pdf..

NR> Is it possible to say only from detection that there is only one source?

Likely answer ..No!

KMH> Depends on number of detected quanta. . If n is large, for example n=1000, the shape of the posterior distribution for a single source should be observably different than if two sources are present and separated by more than their distance from the detector. The way to see this is as follows: as the number of quanta increases, the calculated posterior for a single source approaches a Cauchy distribution, centered on the source position x0 and with an approximate width of y0, the distance from the detector. Two sources will yield a posterior that is close to the sum of two Cauchy distributions.

NR> The simulation has been done for n=2 and 5 .

It may be noted that we do not get same result for every simulation as

might be expected..

KMH> Sure. Because the shape of the posterior depends on the position of the detected quanta and their position is a random process, the posterior for each experiment will be different. In my paper, to which you referred, I ran 1000 experiments for each situation to get some idea of the average performance of each estimator.

In that work, I didn't check to see how well the position at the maximum posterior does. However, I wrote another paper with David Wolf ( http://kmh-lanl.hansonhub.com/publications/maxent93.pdf ) on the estimators of the central position of the Cauchy distribution, in which we gave some closed formulas for the posterior mean estimator. As you surmised, in our simulations with 10^5 experiments, the posterior maximum does almost as well that the posterior mean. Interestingly, the median of the positions of the detected events does quite well. Because of its simplicity, it would be a natural choice.

Kenneth thanks a lot for your valuable views..I think a more exciting problem is the

one you worked on ..viz weather prediction.. would you suggest a simplified problem

for that case as in the detection problem but including chaos..I am a fan of Scilab

and would like to try some simple coding on that ...would be much indebted for your suggestions..

The weather-prediction example in my presentation was not my work. Although I brushed shoulders with folks who worked on hydrodynamics, I am not an expert in that field.

You are asking good questions - good luck with your research. Perhaps some of my papers will provide some guidance as to probability, sampling, etc.

Thanks Kenneth I am reaching the biblical life limit.. as I teach Probability .

at grad level..for engineers..( I am an electrical engineer..) I look for something exciting to pass on to students.. indeed

I enjoyed thoroughly ..the nice interaction in this discussion..and hope to pass on

what I gathered here to them .. thanks a lot ..Cheers..

First response links to a document http://http://kmh-lanl.hansonhub.com/publications/medim93.pdf . I found this introductory document very helpful - thanks Prof. Hanson for this.