Tomografia discreta: problemi, modelli e algoritmi

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Tomografia discretaproblemi, modelli e
algoritmi
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• Elena Barcucci
• Dipartimento di Sistemi e Informatica
• Università degli Studi di Firenze

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Discrete Tomography

• It can be thought of as a special kind of
  Computerized Tomography.
• It needs its own theory for answering questions
  related to consistency, reconstruction and
  uniqueness of discrete unknown structures.
• This theory is mostly based on notions of
  discrete mathematics, combinatorics and geometry.
• Information about the structure are usually
  retrieved from the projections along a finite set
  of discrete directions. Sometimes different kind
  of a priori information are also taken into
  consideration, i.e. connectivity, convexity, and
  similarity with other structures.
• Characteristics
• The examined structure is composed by one or two
  different kinds of atoms/molecules
• The number of directions of the projections
  varies from two to four.

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Applications

• Computerized tomography for non-destructive
  tests on homogeneous structures. The
  reconstructed image contains only two values 1
  for the presence, and 0 for the absence of the
  material.
• Data compression, data coding, and image
  processing projections can be considered as an
  encoding of the object. The difficulties of the
  reconstruction process may assure some security
  of the data, and results about uniqueness may
  assure their perfect coding.
• Medical applications as an example, for the
  reconstruction, and so the visualization of a
  cardiac ventricle the points 1s and 0s depends
  on the presence or absence of the dye. Usually
  only two projections are collected for each
  planar section of the structure, and then they
  are glued together for a three dimensional
  visualization.
• Electron microscopy images produced by
  transmission microscopes can be considered as
  projections of the studied object. This technique
  has been applied to atomic structures in
  crystals in order to obtain information for
  quality tests.

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Discrete Tomography
H.Ryser, Combinatorial properties of matrices of
zeros and ones, Canad. J. Math., 9 (1957),
371-377. G.T. Herman, Image Reconstruction from
Projections The Foundamentals of Computerized
Tomography, Academic Press, New York (1980). R.J.
Gardner and P. Gritzmann, Discrete tomography
determination of finite sets by X-rays, Trans.
Amer. Math. Soc., 349 (1997), 2271-2295. R. J.
Gardner, P. Gritzmann and D. Prangenberg, On the
computational complexity of reconstructing
lattice sets from their X-rays, Disc. Math., 202
(1999), 45-71. G. T. Herman and A. Kuba (eds.),
Discrete Tomography Foundations, Algorithms and
Applications, Birkhauser Boston, Cambridge, MA
(1999). G. T. Herman and A. Kuba (eds.), Advances
in Discrete Tomography and its Applications,
Birkhauser Boston, Cambridge, MA (2007).
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Discrete Tomography
C. Kiesielolowski, P. Schwander, F. H. Baumann,
M. Seibt, Y. Kim, A. Ourmazd, An approach to
quantitative hight-resolution transmission
electron microscopy of crystalline materials,
Ultramicroscopy, 58 (1995) 131-155. E. Barcucci,
A. Del Lungo, M. Nivat and R. Pinzani,
Reconstructing convex polyominoes from their
horizontal and vertical projections, Theoret.
Comput. Sci., 155 (1996), 321-347. M. Chrobak, C.
Dürr, Reconstructing hv-Convex Polyominoes from
Orthogonal Projections, Inform. Process. Lett.,
69 (1999), 283-289. E. Barcucci, A. Del Lungo, M.
Nivat, R. Pinzani, X-rays characterizing some
classes of discrete sets, Linear Algebra and its
Applications, 339 (2001), 3-21. G. J. Woeginger,
The reconstruction of polyominoes from their
orthogonal projections, Inform. Process. Lett.,
77 (2001), 225-229.
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The model
Single component of the physical structure
Point x in Z3
Finite subset S of Z3
Physical structure
Direction of projection
Vector v of Z3
Projection of S along the line l (v)
having direction v
PS (l (v)) S ? l (v)
When dealing with crystalline structures, we
consider

• two-dimensional structures
• horizontal and vertical directions v1(1,0) e
  v2(0,1)

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Representation of the model
1 1 1 1 0 1 1 0 0 0 0 1 0 1 0 1 0
1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0
1
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Main problems
Let O be a class of (finite) subsets of Z2 and ?
v1,v2. We define
Consistency (O , ?) Given two vectors H and
V. Question does there exist an S? O having H
and V as projections along v1 and
v2? Reconstruction (O , ?) Given two vectors
H and V. Task Construct an element S? O
having H and V as projections along v1 and
v2. Uniqueness (O , ?) Given an element S?
O Question does there exist a different
element S? O, tomographically equivalent to S
with respect to the directions v1 e v2 ?
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Basic results
Ryser, Gale (1957) Consistency (O , ?) has
solution in polynomial time. Reconstruction (O ,
?) can be fulfilled in polynomial time.
Uniqueness (O , ?) has solution in polynomial
time.
Sometimes the number of matrices which satisfy a
given couple of projections is huge, and,
furthermore, these matrices can be very different
one from the other.
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Switching components
A switching component of a binary matrix A is a
2x2 sub-matrix of either of the following two
forms
1 0 0 1
0 1 1 0
A1
A2
A switching operation is a transformation of the
elements of A that changes a sub-matrix of type
A1 into type A2 or vice versa (and leaves all the
other elements of A unaltered).
1 0 1 1 0 1 0 1 0 0 1 0 1 0 1 1
0 1 1 0 0 1 0 1 0 1 1 0 0 1
1 0 1 1 0 1 0 0 0 0 1 1 1 0 1 1
0 1 1 1 0 1 0 0 0 1 1 0 0 1
?
A
A
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Uniqueness

• Ryser (1957)
• A binary matrix A is non-unique (with respect to
  its row and column sums) if and only if it has a
  switching component.
• If A and B are two binary matrices with same row
  and column sums, then A is transformable into B
  by a finite number of switching.
• In many cases, a lot of binary matrices have the
  same row and column sums (n! for permutation
  matrices) therefore, the horizontal and vertical
  projections do not provide sufficient information
  for unique reconstruction.

Possible strategy Increase the number of
projections
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Uniqueness
INCREASING THE NUMBER OF PROJECTIONS
Gardner, Gritzmann, Prangerberg (1999) Let ??
3, it holds Reconstruction (O , ?) is
NP-hard. Consistency (O , ?) is
NP-complete. Uniqueness (O , ?) is NP-complete.
In polynomial time one can obtain only
approximate solutions i.e. solutions having
projections close to those of the starting
object. Unfortunately, if the input data can
not uniquely determine the solution, then again
the reconstructed object can be very different
from the starting one!
Another possible strategy impose some properties
on O
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Some classes of discrete sets
(Ø)
(p)
(h)
(p,h)
(p,h,v)
(c)
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Uniqueness
USING A PRIORI INFORMATION
Reconstruction algorithms may draw advantages
from geometrical properties of the structure as
connection or convexity. These information are
useful for reducing the wideness of the class O
to whom the solution has to belong as an example
Barcucci, Del Lungo, Nivat, Pinzani (1996) If O
(p), (h), (v), (p,h), (p,v), (h,v) and ?
v1,v2, then Reconstruction (O , ?) is
NP-hard. Consistency (O , ?) is NP-complete. If
O (p,h,v), then Reconstruction (O , ?) can be
fulfilled in polynomial time. Consistency (O ,
?) can be solved in polynomial time.
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Results on the class ? (c)
A set of discrete directions ? characterizes a
class ? if two different discrete sets in ?
having the same projections along ? do not
exist Uniqueness Theorem (Gardner, Gritzmann
1997) 1. (c) cannot be characterized by any set
of 3 discrete directions 2. There is a set of 6
discrete directions not characterizing (c) 3.
(c) is characterized by any set of 7
directions 4. If ? is a set of 4 directions
such that ?(?) ? 4/3, 3/2, 2, 3, 4 then (c) is
characterized by ?
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Results on the class ? (p, h, v)
Barcucci, Del Lungo, Nivat, Pinzani (2001) 1.
(p, h, v) cannot be characterized by any set of 3
discrete directions 2. There is a set of 6
discrete directions not characterizing (p, h,
v) 3. There are some sets of 7 discrete
directions not characterizing (p, h, v) 4. If ?
(0,1), (1,0), v3, v4 and ?(?) ? 4/3, 3/2,
2, 3, 4 then (p, h, v) is not characterized
by ?
Conjecture There is a set ? of 4 directions such
that ?(?) ? 4/3, 3/2, 2, 3, 4 characterizing
(p, h, v)
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L-convex polyominoes
G. Castiglione, A. Restivo, Reconstruction of
L-convex polyominoes, Electron. Notes in Discrete
Math., 12 (2003) G. Castiglione, A. Frosini, A.
Restivo, S. Rinaldi, A tomographical
characterization of L-convex polyominoes, (DGCI
2005), Lecture Notes in Comp. Sci., 3429 (2005),
115-125.
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L-convex polyominoes
A classification of the h-v convex
polyominoes A polyomino P is kL-convex if any
pair of its cells is connected by a path composed
by horizontal and vertical steps which is
entirely contained in P and changes direction k
times at most.

1L-convex
2L-convex
3L-convex
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L-convex polyominoes state of the art
A polyomino is L-convex
It can be decomposed into maximal rectangles
having crossing intersection
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L-convex polyominoes state of the art
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L-convex polyominoes state of the art
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L-convex polyominoes state of the art
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L-convex polyominoes state of the art
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L-convex polyominoes state of the art
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L-convex polyominoes state of the art
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L-convex polyominoes state of the art
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L-convex polyominoes tomographical properties
If a polyomino is L-convex
it is uniquely determined by its horizontal and
vertical projections
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L-convex polyominoes tomographical properties
No way of getting a switching component!!!
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L-convex polyominoes tomographical properties
If a polyomino is L-convex
it is uniquely determined by its horizontal and
vertical projections
its horizontal and vertical projections are
unimodal
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L-convex polyominoes tomographical properties
1 1 1 1 1 1
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L-convex polyominoes tomographical
characterization
If a polyomino is L-convex
it is uniquely determined by its horizontal and
vertical projections
its horizontal and vertical projections are
unimodal
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L-convex polyominoes a quick reconstruction

• set in decreasing order the horizontal
  projections

• place the entries on each row using a
• greedy strategy (on the vertical projections)

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L-convex polyominoes a quick reconstruction

• set in decreasing order the horizontal
  projections

1 1 3 5 6 2

• place the entries on each row using a
• greedy strategy (on the vertical projections)

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L-convex polyominoes a quick reconstruction

• set in decreasing order the horizontal
  projections

1 1 3 5 6 2

• place the entries on each row using a
• greedy strategy (on the vertical projections)

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L-convex polyominoes a quick reconstruction

• set in decreasing order the horizontal
  projections

1 1 3 5 6 2

• place the entries on each row using a
• greedy strategy (on the vertical projections)

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L-convex polyominoes a quick reconstruction

• set in decreasing order the horizontal
  projections

1 1 3 5 6 2

• place the entries on each row using a
• greedy strategy (on the vertical projections)

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L-convex polyominoes a quick reconstruction

• set in decreasing order the horizontal
  projections

1 1 3 5 6 2

• place the entries on each row using a
• greedy strategy (on the vertical projections)

• set back the horizontal projections

• moving with them the entries in the
• correspondent rows

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Model with absorption
A. Kuba, M. Nivat, Reconstruction of discrete
sets with absorption, Linear Algebra Appl., 339
(2001), 171-194. E. Balogh, A. Kuba, A. Del
Lungo, M. Nivat, Reconstruction of binary
matrices from absorbed projections, Lecture Notes
in Comp. Sci., 2301 (2002), 392-403. A. Kuba, A.
Nagy, E. Balogh, Reconstruction of hv-convex
binary matrices from their absorbed projections,
Disc. Appl. Math., 139 (2004), 137-148 A. Kuba,
M. Nivat, A sufficient condition for
non-uniqueness in binary tomography with
absorption, Theoret. Comput. Sci. 346 (2005),
335-357. E. Barcucci, A. Frosini, S. Rinaldi, An
algorithm for the reconstruction of discrete sets
from two projection in presence of absorption,
Disc. Appl. Math. 151 (2005) 21-35. E. Barcucci,
A. Frosini, A. Kuba, S. Rinaldi, An efficient
algorithm for recostructing binary matrices from
horizontal and vertical absorbed projections,
Workshop on Discrete Tomography and Its
Applications, New York City, 2005.
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Model with absorption
DEFINITIONS
The previously defined model can be extended in
the spirit of modelling emitting structures, as
follows each point of the structures can be
considered both an emitting and an absorbing
point, while the surroundings can be considered
composed only by absorbing points. Let ? be the
common absorption coefficient.
? -2 ? -5
detectors
projections
Each detector detects a signal which is
proportional to the number of emitting points
situated on the row it controls and inverse
proportional to the length of the path d from an
emitting point to the detector, as stated by the
law I I0 ?? -d
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Model with absorption
DEFINITIONS
Elementary 1D switch 1 0 0 ? 0 1 1
1D switch 1 0 x 0 0 ? 0 1 x
1 1 1 0 x 0 x
0 0 ? 0 1 x 1 x 1 1
1 (0 x)k 0 0 ?
0 (1 x)k 1 1
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Model with absorption
2D SWITCHES
A switch along the horizontal and vertical
directions is defined as the composition of the
following two basic switches
1 0 0 0 1 1 Ex. E(1)E(0)E(0) 0 1
x 1 x 1 1 1 0 x 0 0 1 0 x 0
0
1 0 0 E(1) 0 1 1 0 1 1
0 1 1 E(0) 1 0 0 1 0 0

• Kuba, Nivat (2001)
• A matrix is univocally determined by H and V if
  and only if it has no switches.
• Given two matrices A and B which are equivalent
  with respect to the vectors H and V, it exists a
  sequence of switches which leads from A to B.

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Model with absorption
MAIN RESULTS

• The horizontal and vertical projections can be
  used to compute all possible starting points for
  the switches inside a matrix consistent with
  them.
• The shape of each switch inside a matrix
  consistent with H and V can be detected, and it
  is independent from the other, eventually
  present, switches.
• Two switches which intersect in one or more
  positions are coincident.
• Each switch has a fixed number of different
  possible configurations. This number is linear in
  the elementary components which form the switch
  itself.

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Model with absorption
RECONSTRUCTION STRATEGY SOME HINTS

• The solution matrix is reconstructed column by
  column, searching for ALL
• possible starting points of 2D switches and
  following ALL their possible
• evolutions (shapes).
• During the computation each switch grows
  independently from the others, so
• the number of possible evolutions of all the
  switches of the matrix is the
• maximum number between the possible
  configuration of each (hypothetical)
• switch.
• Column by column the algorithm search for
  switches which are completely
• detected. Their points will not be further
  changed by the computation.

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Model with absorption two horizontal
projections
The left and right projections of the same row
may have different values!
A ?0 representation which does not contain the
substring 011 is called ?0 expansion.
A real number can have many ?0 representations of
the same length, but only one is a ?0 expansion.
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Model with absorption two horizontal
projections
Example
0 1 0 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1 0 0
1 1 1 0 0 0 0 0 1 0 0
are ?0 representations of the same number r
?0-1 ?0-7. The last one is the ?0 expansion.
Equivalent ?0 representations can be considered
as 1D switches!
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Model with absorption two horizontal
projections
Lemma Let a1 ak and b1 bk be different ?0
representations of the same number, a1 ?0 1
ak ?0 k b1 ?0 1 bk ?0 k If a1 ?0
k ak ?0 -1 b1 ?0 k bk ?0 1
, then ai bi , for all i, 1 ? i ? k.
Theorem A row of a binary matrix is uniquely
determined by its left and right ?0 absorbed
projections.
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Model with absorption two horizontal
projections
MAIN RESULTS
1) The values rleft and rright univocally
determine a sequence of fixed length.
Each matrix is determined by its left and right
?0 -horizontal absorbed projections.
2) Let O be the class of the finite subsets of
Z2 and ? (rleft, rright).
Consistency (O , ?) has solution in polynomial
time. Uniqueness (O , ?) has solution in
polynomial time. Reconstruction (O , ?) can be
fulfilled in polynomial time.
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Model with absorption two horizontal
projections
Property 1 Let a1 ak and b1 bk be different
?0 representations of the same number. Then b1
bk can be obtained from a1 ak by a finite
number of 1D switches. Furthermore the two
representations differ for sub-sequences of the
form 1 (0 x)t 0 0 or 0 (1 x)t 1 1
with t ? 0.
Property 2 Let z a1 ak be the ?0 expansion
of r. Then any ?0 representation of r can be
obtained from z by using only 1D switches of the
type 1 (0 0)t 0 0 ? 0 (1 0)t 1 1
with 0 ? t ? ?(k-3)/2?
Example 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0
1 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 1 0 1 1 0
1 0 0 1 0 1 0 1 1 1 0
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Model with absorption two horizontal
projections
Property 3 Any switch 1 (0 0)t 0 0 ? 0 (1
0)t 1 1 can be obtained by a sequence of t1
elementary switches 1 0 0 ? 0 1 1.
Example 1 0 0 0 0 0 0 0 0 1 (0 0)3 0 0
0 1 1 0 0 0 0 0 0 0
1 0 1 1 0 0 0 0 0 1 0 1 0 1 1 0 0 0 1 0 1 0 1
0 1 1 0 (1 0 )3 1 1
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Model with absorption two horizontal
projections
Property 4 Any ?0 representation of r can be
obtained from the ?0 expansion of r by using
only 1D elementary switches of the type 1 0 0
? 0 1 1 Example 0 1 0 0 1 0 0 1 0 1 0 0 0 1
0 0 1 0 0 1 0 0 1 1 0 1 0 0 1 0 0 0 1 1 1 1 0
1 0 0 0 1 1 0 1 1 1 1 0 0 1 1 0 1 1 0 1 1 1 1
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Model with absorption two horizontal
projections
Property 5 The ?0 expansion of r has the
minimal right projection among all the ?0
representations of r. Any switch 1 (0 0)t 0 0 ?
0 (1 0)t 1 1 increases the right projection
without modifying the left one
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Model with absorption two horizontal
projections
Reconstruction algorithm Input two real numbers
rleft and rright and an integer n Output a
binary sequence z of length n having rleft and
rright as projections Step 1 let z be the ?0
expansion of rleft of length n. (by
property 4, the right projection zright of z is ?
rright) Step 2 scan z from right to left till
the first position j different from 0 and s.t.
?0-j2 ?0-j1 - ?0-j ? rright
zright perform, if possible, the
switch 1 0 0 ? 0 1 1 starting from such a
position Step 3 repeat Step 2 till rright
zright 0 Step 2 return z as output.
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Model with absorption two horizontal
projections
Example let us reconstruct the sequence z of
length 7 which satisfies r left 2?0-1 ?0-5
and r right 2?0-1 ?0-3. Let z 1 1
1 0 1 0 0 be the ?0 expansion of rleft of
length 7 j 3 is the detected starting
position for the switch 1 0 0 ? 0 1 1 Update z
1 1 1 0 0 1 1 since rright zright gt 0, we
perform a second switch starting in position
j 5 Update z 1 1 0 1 1 1 1 now rright
zright 0 and the desired sequence is
achieved.
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Model with absorption two horizontal
projections
The worst case analysis
Theorem For each n, n ? 3 i) the worst
performance of the reconstruction algorithm is on
the word 10 n-1 , when the single switch of
maximal length is required ii) the number of
checks for the suitable switch positions of 10
n-1 is 3\2 n k , with k constant.
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Model with absorption two horizontal
projections
Example Let us consider the two words w1 1
(0 0)6 and w2 1 (0 0)3 0 1 (0 0)1 0 0 having
the same length and compute the number of check
operations when performing all the possible
switches inside them.
w1 1 0 0 0 0 0 0 0 0 0 0 0 0 the elements of
the word w1 are checked one time, till
position 13 w1 0 1 0 1 0 1 0 1 0 1 0 1 1 after
each switch, a second check and a second
switch is performed.
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Model with absorption two horizontal
projections
w2 1 0 0 0 0 0 0 0 1 0 0 0 0 the elements of
the word w2 are checked one time, till
position 5 w2 1 0 0 0 0 0 0 0 0 1 0 1 1 a
second check occurs in position 3 w2 1 0 0 0 0
0 0 0 0 1 0 1 1 the elements of w2 are
finished to be checked from position 7 w2 0
1 0 1 0 1 0 1 1 1 0 1 1 three further checks and
switches occurs.
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Matrices under the Microscope
M. Nivat, Sous-ensembles homogènes de Z2 et
pavages du plan, C. R. Acad. Sci. Paris, Ser. I
335 (2002), 83-86. A.Frosini, M. Nivat, On a
tomographic equivalence between (0,1) matrices,
Pure Mathematics and Applications,16 (2005),
251-265. A. Frosini, M. Nivat, Binary matrices
under the microscope A tomographical problem,
Theoret. Comput. Sci., 370 (2007), 201-217
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Matrices under the Microscope
A different concept of projection given a
binary matrix A and a rectangular window of
dimension p x q
If A has dimension m x n, then Rp,q(A) has
dimension (m p 1) x (n q 1)
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Matrices under the Microscope
Example p 2, q 3
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Matrices under the Microscope
Example p 2, q 3
R2,3(A)
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Matrices under the Microscope
Example p 2, q 3
A
R2,3(A)
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Matrices under the Microscope
The matrix ?p,q(A) of dimension (m p) x (n
q) is defined as
?p,q(A) i , j A i , j A i p, j q
- A i p, j - A i , j q
63
Matrices under the Microscope
The matrix ?p,q(A) of dimension (m p) x (n
q) is defined as
?p,q(A) i , j A i , j A i p, j q
- A i p, j - A i , j q
?2,3(A)
If any element of ?p,q(A) is zero, then A is said
to be smooth
?1,1(Rp,q(A)) ?p,q(A)
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The complete reconstruction algorithm
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Dagstuhl 97