Re: Photo Image out of focus

[Ludwik Kowalski <kowalskiL@MAIL.MONTCLAIR.EDU>]

Let us consider a simple case, a source (object) is a set of four
equidistant points along a line segment. Or, to be more realistic,
it is a set of four segments. Under proper focusing the image
would consist of 20 distinct small circles forming a grid. But
the lens was not focused and each circle is very large. The 20
circles overlap very significantly and the intensity  distribution
is a rectangular blab with round corners.

An observer wants to determine the distribution of sources in
the image plane. Under what condition would s/he be able to
determine that the number of point-like sources was 20 and
the distances between them were 3 cm, for example? Let
assume that there are no aberrations, that the thin lens focal
distance is known and that the thin lens formula is applicable.

To deconvolute the blab (to say what the intensity distribution
would be if the lens were positioned properly) one must know
the "rule of the convolution", for example,

1) A point object produces a circle of 3 mm on the film.
2) A point object produces a circle of 5 mm on the film.
2) A point object produces a circle of 7 mm on the film.

Each of these rules would produce a different answer
about the true distribution of sources in the object plane.

The rule of the convolution would be easy to establish
if the distance from the image plane to the lens (s) and
the distance from the lens to the film (p) were known.
Without this information the unique deconvolution would
be impossible, in my opinion.

Suppose that a tree, located further away, happened to
be photographed correctly. The photographer knows
from where the picture was taken, where the tree is and
where the students were. This information should be
sufficient to establish the rule of the convolution, at least
in principle. Once this rule is known the deconvolution
should be possible, provided some additional information
is available.

For example, is the even distribution of darkness in the
center of the blab (if it is established by using a
densitometer) due to the fact that all light bulbs were
identical and equidistant? Maybe yes and maybe no.

A periodic structure in the distribution of darkness would
help a lot but this would mean that the defocusing was not
very strong. Under strong defocusing, and in the absence
of additional information, the distribution of light bulbs in
the object plane (and how do we know they were in the
same object plane) seems to be impossible to me.

In my initial reply additional information was labeled as
subjective. An objective deconvolution should depend
only on what is measured, or on what is stated in a
particular problem. I am not sure the objective-subjective
terminology was appropriate; it would be appropriate
only if additional information was based on guessing.
Ludwik Kowalski


"John S. Denker" wrote:

> Jeff Weitz asked:
> > >  Is there an optical solution to the focus problem?
>
> Then at 06:10 AM 6/20/01 -0400, Larry Cartwright wrote:
>
> > The short answer is:  no.
>
> > I think we all start out with a not-very-well-thought-through faith in
> > symmetry which has us convinced that anything which can be done can
> > somehow be undone by reversing the process.
>
> > It doesn't take too much experimenting to shatter the focus delusion.
> > You quickly find that the enlarger can make the image even more *out* of
> > focus, but there is no purely optical way to make the image more *in*
> > focus than it is on the negative.  Not even with all the king's horses
> > and all the king's men :-)
>
> I agree that it can't be done with an enlarger lens.  But let's be careful;
> the king's men might have more resources than you think.
>
> As an illustration, let's consider a 1-dimensional image.  Model the
> blurring as a convolution, with a simple triangle-shaped kernel, such as:
>          K(i,j) =   0.25 delta(i, j+1)
>                   + 0.50 delta(i, j)
>                   + 0.25 delta(i, j-1)
>
> 1) It turns out that such blurring is a nonsingular transformation.  You
> can verify this by writing out an NxN matrix with 0.5 everywhere on the
> diagonal (N cells) and 0.25 everywhere just above the diagonal (N-1 cells)
> and just below the diagonal (another N-1 cells), and zeroes
> elsewhere.  It's invertible.  The inverse is a mess, but it exists.
>
> 1a) Under favorable conditions, it is straightforward to implement the
> inverse transformation using digital filtering techniques.  Such filters
> are built in to standard image-manipulation programs.
>
> 1b) What is perhaps more surprising is that it is possible to do it using
> all-optical techniques.  We are talking about some pretty tricky Fourier
> optics here.  (Digital image processing would be a lot easier.)
>    http://www.nobel.se/physics/laureates/1971/gabor-autobio.html
>    http://www.sira.co.uk/courses/course2.htm   (see lecture 11)
>
> 2) On the other hand, the blurring transformation is "close" to being
> singular, in the sense that inverse has a lot of large coefficients.  It's
> also rather nonlocal (unlike the blurring transformation, which is quite
> local).  Therefore the de-blurring transformation is a nasty _noise
> amplifier_.  When we account for the graininess of the film, the
> nonlinearity of the film, and other non-idealities, there will be limits on
> how much de-blurring is possible.
>
> Homework:  Fire up a spreadsheet, and type in a large matrix (at least 6x6,
> preferably larger) that represents blurring of a D=1 image as described
> above.  Calculate its inverse.  Observe what a mess it is.  Observe how
> applying it requires lots of accuracy, because of repeated small
> differences between large numbers.
>
> Hint:  In excel, remember to use control-shift-enter so that the
> minverse(...) expression gets entered in all cells of the destination array.