To
reconstruct the electron wave exiting the object plane (object wave) from a
hologram, two steps are necessary [1]. First, the complex image wave has to be
regenerated from the hologram and then the microscope errors can be corrected
by deconvolution. This paper deals with the first step.
We
describe a reconstruction process in real space using an artificial neural
network. This is done because streaking artefacts known in Fourier space can be
avoided. Neural nets are establishing as a powerful tool in image processing.
In our case, a neural net has been trained to perform a pointwise calculation
if the image wave from the fringe patterns in small hologram areas
(superpixels)
The Reconstruction Procedure
The
neural net gets the intensity information from a superpixel containing 7 by 7
pixels as input data. This area contains about 1.5 hologram fringes.
The
task of the net is to calculate the image wave in the center of the superpixel.
Real and imaginary part of the image wave are represented by the two outputs.
To
reconstruct the whole image, the superpixel is shifted over the hologram and
the image wave is calculated pixel by pixel. This takes about 2 minutes on a
DEC workstation for a 1024 by 1024 pixel hologram.
The Neural Net
We
used a feedforward neural network and obtained the best results with two
hidden layers. This network was simulated and trained using the Stuttgart
Neural Net Simulator (SNNS) [3].
Generating the Training
Patterns.
The
net can only generate proper results when it is trained with patterns similar
to real world holograms. The training patterns were generated by simulating
hologram patterns of superpixel size using the interference equation.
In
order to have realistic patterns two disturbing effects are taken into account.
First,
the image wave is not exactly constant
over the superpixel area. This was simulated by allowing to have a certain, constant slope choosen
at random for each training pattern.
Second,
the inevitable noise was simulated by disturbing the calculated intensities
with poisson noise.
Neural Nets are an attempt to imitate
the flexible and massively parallel information processing in biological nerve
system in a computer. The net consists of simple units, called neurons, which are connected by links
with variable weights.
The
net can be implemented as parallel hardware, but is usually just simulated on a
conventional computer.
A
single Neuron just sums the data
from ist inputs, multiplied with weights and then applies a nonlinear sigmoid
function to the result.
In FeedForward Neural Nets, the links
form no feedback loops i.e. information is passed from the layer of input units
through some layers of hidden units to the output units. See the picture at the
top of this poster.
FFNets
can be trained to react to input signals with certain output signals and are
particulary suited for image processing tasks.
In
nature, similar structures can be found. For example, visual information is
preprocessed in a similar way in the retina.
Weights are numbers which determin
the strengths of the links between
neurons. Weights can be positve or negative,
corresponding
to exciting or inhibiting synapses
as in biology.
The
weights contain all the information the net has learned.
Training Patterns are input vectors with
known outputs. From these examples, the net learns how to react on the input
data. When trained properly, the net generalizes and gives sensible results for
input patterns not contained in the training set. As the generalisation is a kind of interpolation, such behaviour is only
possible when the training patterns are distributed over the whole space of
possible input vectors.
Backpropagation is the most popular learning algorithm for feedforward
nets. First, the weights are initialized to random values. Then, the input
vectors of the trainigs patterns are subsequently applied to the net and the
computed output is compared with the desired output. The weights are changed to
decrease the error. This is done by calculating the gradient of the error as a function of the weights.
The
algorithm was independently invented
several times but began to be widely used only after the publication of [4].
How to deal with Noise?
Real
holograms always contain a certain amount of noise. As the training patterns
should cover the whole space of possible input vectors, the training patterns
should also contain noise. On the other hand, the net should also be trained
with noiseless data to avoid disturbance of the outputs by both input and
training noise.
The
adjecent plots show the learning behaviour of the neural net.
After
each training cycle, the net is tested with a validating patterns set and the
mean square error of the net output is calculated.
The
upper plot shows that a net which is trained with noise makes big errors when
tested with noiseless data. When testing with noisy data however, the net
trained without noise produces big errors, as shown in the lower plot.
The
best way to train a net in order to deal with both cases is to start the
training with very noisy patterns and then gradually decrease the noise to
zero.
Where does the Amplitude
Information come from?
_{} 
The
amplitude of the image wave appears twice in the equation for the intensities:
The
first, nonlinear part is not
modulated by the carrier and therefore appears in the centerband, while the
second term shows its effect in
the sideband.
When and are treated as independent variables,
the
influence of both terms on the reconstructed amplitude
can
be seen. The plots show that the neural net tends to use more sideband
information than a leastsquaresfitting approach [5].
Result of the Real Space Reconstruction in Fourier Space

Feed forward neural nets can be trained to extract the complex image wave from
a hologram.

The reconstruction process is carried out in real space.

No farreaching artefacts are produced by the borders of the image.

Both center and sideband information are used for amplitude
reconstruction.

The computation time needed to reconstruct a hologram is smaller than for
leastsquares fitting with numerical minimum search.

Fourier space reconstruction and optimized analytical leastsquares fitting [5]
are even faster.
References: [1] Tonomura, A.: Electon
Holography NorthHolland 1995
[2]
Wassermann, P. D.:Neural Computing:
Theory and practice ISBN 0442207433
[3]
Zell, A.: Simulation Neuronaler Netze.
Nachdr. 1996. ISBN: 3893195548 ADDISONWESLEY,
[4]
Rumelhart, D. Hinton, G. E. and Williams, R. J.: Learning internal representations by error propagation
in Parallel
distributed processing vol. 1 pp. 318362 MIT Press, Cambridge MA
[5]
Meyer, R. and Heindl, E.: Optimized
Reconstruction of Electron Holograms in Real Space by Least Squares
Fitting Poster T1411on this
Conference.