A Reconstruction and Optimization Problem in Discrete Tomography

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A Reconstruction and Optimization Problem in
Discrete Tomography
Prof. G-W.Weber, Öznur Yasar
ESI 2004, Ankara
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Problem Definition

• Given
• a domain discrete or continuous
• a function whose range is discrete
• Aim
• reconstruct from weighted sums

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An Illustration
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An Inverse Problem

• Reconstruct a lattice
• set from its X-rays

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Outline of the talk
introduction notation
basic problems of DT complexity results
motivation Optimization in VLSI Chip
Design coding Theory optimal
Experimental Design
open problems future work
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History

• The name is due to Larry Shepp 1994

• A consistency result by Ryser 1957

• When can a planar convex body be
• uniquely determined? Hammer 1961

• DT workshops
• Germany 1994
• Hungary 1997
• France 1999

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Connections of DT to other fields of Mathematics

• Discrete Mathematics
• Combinatorics
• Functional Analysis
• Geometry
• Coding Theory
• Graph Theory

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Applications

• Medical applications
• Image processing
• Electron microscopy
• Scheduling
• Statistical data security
• Game theory
• Material sciences

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The Principle Motivation

• Reconstruct
• tissue or christaline structures from their
  images obtained by
• high resolution transmission
  electron microscopy

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Linear programming
find
s.t.
where
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An example A lattice set
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Projection on two lattice lines
in the directions (1,0) and (0,1)
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The vector x the lattice points who has a
positive projection
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Linear Programming Problem
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Extend the feasibility problem
Maximize
subject to and
There are polynomial time interior point methods
by Schwander and Hettich.
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Notation

• Lattice sets (discrete sets)
• finite subsets of integers
• Lattice directions
• nonzero vectors of
• Set of distinct lattice directions
• Lattice line
• parallel to a vector
• and is non empty.

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A Lattice
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A Lattice Set
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A Lattice Direction
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Notation

• Set of all lattice lines that are parallel
  to
• Class of finite sets in
• A set of directions in
• The
  collection of the set of lattice lines determined
  by

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Projection of a lattice set

• in direction is
  s.t.
• 
  

where is the characteristic function of
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Tomographically Equivalent
and
are tomographically
equivalent w.r.t. the directions if
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Consistency (e, L)

• Given For a function
• with finite support.

Question Does there exist an such
that for ?
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Uniqueness (e, L)

• Given An

Question Does there exist a different
such that and are
tomographically equivalent with respect to the
directions of ?
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Reconstruction (e, L)

• Given For a function
• with finite support.

Task Construct a finite set such that
for .
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Complexity results for lattice lines

• Consistency
• q2, n2 Polynomial time algortihm
• q3 NP Complete
• Uniqueness
• q2, n2 Polynomial time algortihm
• q3 NP Complete
• Reconstruction
• q2, n2 Polynomial time algortihm
• q3 NP Hard

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Complexity results in general

• Not developed yet for r-dimensional X-rays when r
  is greater than 1.
• Only a few results are known.
• For instance An open problem
• r2, n3 and L consists of the 3 coordinate
  planes

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Higher Dimensions
y
H three planes perpendicular to the axes
(1,1,0)
(1,1,1)
(0,0,0)
x
(0,0,1)
S
z
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Motivation Optimization in VLSI Chip Design

• integrated circuit (IC) - a tiny semiconductor
  chip
• the technique of fabricating an IC - creating a
  photograph with a negative.
• SSI (small scale integration) MSI, LSI, VLSI
  (very large scale integration)

SSI
VLSI
modelling, simulation, error analysis, logic
design
i.e. Optimization
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Role of DT

• Moore's Law the maximum number of transistors
  on a chip approximately doubles every eighteen
  months.
• Here DT deals with reconstruction of an atom
  cluster - faced with in the quality control of a
  silicon layer on a microchip
• The problem of reconstruction is NP-hard -- look
  for approximative algorithms based on a refined
  utilization of optimization.
• Note that other important areas of VLSI
  technology are
• Packing problems -- occurs when providing the
  connections between the circuit and the outside.
• Another point where optimization may be used
  related to VLSI chip design, is finding the
  optimal achievable pipelining period of a given
  wavefront array.

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Coding Theory

• consider the distribution along a ray as a word
  being close to one or another element of a linear
  code C
• the code - any preinformation which we have about
  the possible location of the atom cluster
• firstly only a vague initial guess about the
  atom distribution
• try to find the codeword which is closest to our
  guess by decoding
• to decode, i.e., to reconstruct, we can use the
  Optimal Decoding Rule or Maximum Likelihood
  Decoding Rule

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a training set and a test set

• the training set -- we build up a code where we
  think that the atom cluster should be an element
  in.
• the test set -- we try to validate or falsify
  and improve
• iteratively reduce the dimension or complexity
  of the code to approximate the real atom cluster

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Cyclic Codes

• a linear code - a vector space over some finite
  field
• use coding theory to find some properties of our
  atom cluster in a step by step process of
  learning
• in a word 1 means existence and 0 means
  nonexistence of an atom at a lattice point
• lattice dimension n
• to measure the difference between the iterate and
  the closest codewords, we use Hamming distance

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• Hamming distance of two words of a binary linear
  code is defined as the number of places where the
  words differ. Ihringer
• Look for cyclicity which brings many
  simplifications with it
• A cyclic code is a cyclic subspace of the vector
  space where it is defined in.
• Let us enumerate the position of the word and
  define a correspondence.
• Working on Z_qn is equivalent to working on
  Fx/f(x), the set of polynomials of degree less
  than the degree n of some given polynomial f(x)
• g(x)h(x) mod (f(x)) iff g-h is divisible by f(x)
  
• If f(x) has degree n, then Fx/f(x)qn.
• Define the generator polynomial of a code
• Ca(x)g(x) a(x) in Fx/f(x)
• Define generator matrix G for C

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• For multiplication with such as special matrix G
  we can use shift registers.
• Using the generator polynomial g(x) instead of
  general generator matrices G, i.e., polynomial
  instead of matrix algebra and computation, is
  much more convenient. Herewith, we have reduced
  the problem complexity to a great extent.
• At this point, by a suitable decoding algorithm
  one can find out what the actual words, i.e.,
  atom locations, are.

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Optimal Exprimental Design

• The basic idea is to look for symmetry in the set
  of approximative data, i.e., of possible atom
  clusters. Herewith, we continue our reflections
  on symmetry and equivariance started in the
  context of coding theory. This will simplify the
  problem and reduce the dimension. The symmetry
  may be detected by trial and error. With these
  information at hand one can observe some
  properties of the system, such as maximum
  likelihood or minimum variance. Invariance and
  equivariance information about the atom
  distribution could be extracted by using optimal
  experimental design from statistics (Bertram et
  al.).

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experimental design

• Let us formalize our experimental design and
  explain how our discrete tomography problem can
  be approached by design theory.
• Define a linear regression model on the
  experimental region with compact range and an
  unknown parameter vector.
• By an approximate design we mean a discrete
  probability measure with finite support, i.e., it
  assigns a nonnegative weight to finitely many
  points and these weights sum to 1.
• The moment matrix, which reflects the statistical
  properties of the design is a positive
  semidefinite matrix.
• The components are the covariances.

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Optimal Design

• In order to decide about the optimality of a
  design developed in this way, we need a
  real-valued convex criterion on the convex cone
  of symmetric matrices.
• This approach is similar to nonlinear interior
  point methods. Our feasible set, being a convex
  cone, can be viewed as the infinite intersection
  of hyperplanes. This makes the problem also be a
  semi-infinite optimization problem Hettich.
• Define an optimal design
• There are various optimality criteria the most
  common ones being Kiefer's Phi-Criteria or
  integrated variance criterion.
• By such an approximate criterion or objective
  function, we aim at highest probability of
  reconstruction of the atom cluster, at smallest
  standard deviation of measurement error or noise,
  etc..
• A drawback of this approach occurs when we
  increase the dimension. We can make use of the
  special structures of the model in order to
  reduce the dimension. To this end we will focus
  on invariant designs.

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invariant designs

• Consider moment matrices lying in some convex
  cone with semidefinite elements and the given
  measurement space
• f is some suitable criterion on this cone to be
  minimized
• F stands for the measurements of atom
  distribution
• Q is a compact group of regular matrices
• The concerted interactions of these functions and
  the matrices, called invariance and equivariance,
  leads to dimensional reduction of our
  measurements.
• Gaffke et al. have already used this for
  optimizing elasticity of crystals
• Finally, semidefinite and nonlinear integer
  programming problems have to be resolved. The
  semidefinite ones with their feasibility being
  defined by the cone are closely related to
  semi-infinite problems.
• Adapt and apply iterative algorithms suitably for
  DT problems.
• To achieve a better convergence rate quadratic
  approximation is constructed.

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• An illustration of the outcome of the algorithm

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About the figure

• Gaffke and Heiligers, Sloane
• All of the points constitute a (symmetrical)
  exact design with rational weights. The dark
  points constitute an approximate design with
  real-valued weights. Since it is the usual case
  that the set to be reconstructed is nonsymmetric,
  one can simply apply our orbits g, e.g., make
  rotations, perturbations and sign changes, and
  also include the lattice points which are not
  included giving them zero weights. Namely, we can
  add some part of a discrete set to get more
  symmetries so that we can optimize more easily,
  later on we can delete these new points.

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open problems

• It has been pointed out by Fishburn and Shepp
  that for a discrete Radon base with at least two
  elements it is difficult to construct sets S
  which are unique but not additive. It stimulated
  the idea that the number of non-similar sets
  which are unique but not additive tends to zero
  as the cardinality of S increases.
• Another open problem considers a grid with a
  known number of objects of the same size to be
  put on each row or column. The aim is to cover
  the whole area with the objects. The problem is
  solvable in polynomial time if there is only one
  object which was considered in detail in Section
  3. The computational complexity of the two object
  case is open. For three or more objects the
  problem is NP-complete. It is also known that it
  is NP hard for six or more objects.

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