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Optimization and search in discrete spacesOwn investigation
Dr. Arno Formella
Departamento de InformáticaUniversidad de Vigo
10/05/07
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Three applications
approximate point set match in 2D and 3D
selection of node B sites in UMTS wireless networks
approximation of point sets with shapes
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First applicationpsm
approximate Point Set Match in 2D and 3D
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¿Dónde está Wally?
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Motivationpsm
searching of patterns(relatively small sets of two– or three–dimensional points),within search spaces(relatively large point sets)
comparing point sets
key wordsgeometric pattern matching, structure comparison, point setmatching, structural alignment, object recognition
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Joint work with
Thorsten Pöschel
some ideas from: Kristian Rother, Stefan Günther
Humboldt Universität—Charité Berlinhttp://www.charite.de/bioinf/people.html
Current project: superimpose (Raphael Bauer)
there will be other algorithms to compare with
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Search of a substructure in a proteinsearch space
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Search of a substructure in a proteinsearch pattern
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¿Dónde está Wally?
Dr. Arno Formella (Universidad de Vigo) OED 10/05/07 9 / 163
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Informal problem description
given a search space and
a search pattern,
find the location within the spacewhich represents best the pattern
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Extensions
find the best part of the patternwhich can be represented within the search space
allow certain types of deformation of the pattern
find similar parts within the same point set
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Formal problem description
search space:S = {s0, s1, . . . , sn−1} ⊂ IRd , |S| = n
search pattern:P = {p0, p1, . . . , pk−1} ⊂ IRd , |P| = k ≤ n
dimension d = 2 or d = 3
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Search and alignment
the aligning process can be separated in two partsfind the matching points in the pattern and the search spacefind the necessary transformation to move the pattern to its location
an approximate alignment must be qualified
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Matching
a matching is a function that assigns to each point of the searchpattern a different point of the search space
µ : P −→ S injective, i.e.,
if pi 6= pj then µ(pi) 6= µ(pj)
let’s write: µ(pi) = s′i and µ(P) = S′
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Transformations
transformations which maintain distances:translation, rotation and reflection
transformations which maintain angles:translation, rotation, reflection and scaling
deforming transformations:shearing, projection, and others (local deformations)
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Congruent and similar transformations
rigid motion transformation(euclidean transformation or congruent transformation)only translation and rotation
similar transformationrigid motion transformation with scaling
we may allow reflections as well (L–matches)
let T be a transformation (normally congruent)
we transform the pattern
let’s write: T (pi) = p′i and T (P) = P ′
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Alignment
a matching µ together with a transformation T is an alignment(µ, T )
rigid motion transformation: congruent alignment
with scaling: similar alignment
with reflection: L–alignment
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One–dimensional example
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Distance of an alignment
let (µ, T ) be an alignment of P in S
we can measure the distances between transformed points of thepattern and their partners in the search space
i.e., the distances
di = d(T (pi), µ(pi)) = d(p′i , s′i )
obviously, if di = 0 for all ithen the alignment is perfect
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Examples of different distances of an alignment
root mean square distance (RMS)
d =
√1n
∑i
(p′i − s′i )2
average distance (AVG)
d =1n
∑i
|p′i − s′i |
maximum distance (MAX)
d = maxi|p′i − s′i |
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Quality of an alignment
there are many interesting distance measures
a distance d has its value in [0,∞[
we use the quality Q of an alignment Q = 1/(1 + d)
(other possibility: Q = exp(−d))
hence: Q = 1 perfect alignment, Q ∈]0, 1]
and: Q < 1 approximate alignment
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Formal problem description
given a search space S, and
given a search pattern P
given a distance measure
find an alignment (µ, T ) of P in Swith minimum distance d (or maximum quality Q)
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Perfect congruent alignments
congruent alignments in IR3 (Boxer 1999):
O(n2.5 4√
log∗ n + kn1.8(log∗ n)O(1)︸ ︷︷ ︸output
log n)
for small k the first term is dominant
log∗ n is smallest l such that
22...2}
l−veces≥ n log∗ n = 5 =⇒ n ≈ 265000
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Perfect similar alignments
similar alignments in IR3 (Boxer 1999):
O(n3 + kn2.2︸ ︷︷ ︸output
log n)
searching approximate alignments and/or partial alignmentsis a much more complex problem
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Perfect alignmentsideas
choose one triangle, e.g. (p0, p1, p2), of P
search for all congruent triangles in S(and their corresponding transformations)
verify the rest of the points of P(after having applied the transformation)
the run time is not proportional to n3 (in case of congruence)because we can enumerate the triangles of S in a sophisticatedmanner and there are not as many possibilities
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Approximate alignments
as stated, we work in two stepswe search for adequate matchings µ(according to a certain tolerance)we calculate the optimal transformations T(according to a certain distance measure)
we select the best alignment(s)
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Optimal Transformation
let S′ = µ(P) be a matching
let d be a distance measure
we look for the optimal rigid motion transformation T ,(only translation and rotation), such that
d(T (P), µ(P)) = d(P ′, S′) is minimal
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Root mean square distance
d =
√1n
∑i
d(p′i , s′i )2
=
√1n
∑i
(U · pi + t − s′i )2
U 3x3 rotation matrix, i.e., orthonormal
t translation vector
Objective: find U and t such that d is minimal
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A little bit of math
we observe: t and U are independent
with the partial derivative of d according t
∂d∂t
= 2 ·∑
i
(U · pi + t − s′i ) = 2U∑
i
pi + 2nt − 2∑
i
s′i
we obtain
t = −U1n
∑i
pi +1n
∑i
s′i
= −U · pc + s′c
where pc and s′c are the centroids of both sets
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Optimal Rotation
with the above, d can be written as
d =
√1n
∑i
(U · pi + t − s′i )2
=
√1n
∑i
(U · (pi − pc)− (s′i − s′c))2
where U is a matrix with restrictions (has to be orthonormal)
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Extremal points of functions with restrictions
one converts the problem with restrictions
with the help of LAGRANGE multiplies into
a problem without restrictions
which exhibits the same extremal points
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Let’s skip the details
basically, we calculate first and second derivative according to theentries uij of U
we search for the extremal points
KABSCH algorithm 1976, 1978
open source code at my home page
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KABSCH algorithm
let S be the matrix of rows containing the s′ilet P be the matrix of rows containing the pi
we compute R = S · P>
we set A = [a0 a1 a2] with ak being the eigenvectors of R>R
we compute B = [‖Ra0‖ ‖Ra1‖ ‖Ra2‖]and finally, we get U = B · A>
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Introduction of scaling
let us introduce a scaling value σ ∈ IR
d =
√1n
∑i
(σU · (pi − pc)− (s′i − s′c))2
let p′′i = U · (pi − pc) be the translated and rotated point pi
let s′′i = s′i − s′c be the centralized point s′i
the solution for the optimal σ: σ =
∑i
〈s′′i , p′′i 〉∑i
〈p′′i , p′′i 〉
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Different distance measures—different alignments
AVG
MAX
RMS
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Optimal transformations for non–derivable distancemeasures
if the function for d is not derivable, e.g., the average
we use a gradient free optimization method(only with evaluations of the function)
recently developed iterative method that is guaranteed toconverge towards a local minimum
algorithm of RODRÍGUEZ/GARCÍA–PALOMARES (2002)
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Idea behind RODRIGUEZ/GARCIA–PALOMARES
let f (x) be the function to be minimized
we iterate contracting and expanding adequately parametershk > 0 and τ > 0 such that
f (x i+1) = f (x i ± hkdk ) ≤ f (x i)− τ2
where dk is a direction taken from a finite set of directions(which depends on the point x i )
with τ −→ 0, x i converges to local optimum(while there are no constraints)
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Rotation in quaternion space
a rotation U · p of the point p with the matrix U can be expressedas
q ? p̄ ? q−1 in quaternion space IH(HAMILTON formula, C∼ IR2, IH ∼ IR4)
where p̄ = (0, p) is the canonical quaternion of the point p
and q = (sin(ϕ/2), cos(ϕ/2)u) is the rotation quaternion(with u ∈ IR3 being the axis and ϕ the angle of rotation)
instead of U with 9 constraint variableswe have u and ϕ, i.e., 4 unconstraint variables
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Matching Algorithms
1 maximal clique detection within the graph of compatible distances2 geometric hashing of the pattern3 distance geometry
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Maximal clique detection
we generate a graph G = (V , E) (graph of compatible distances)
vertices vij ∈ V all pairs (pi , sj)
edges e = (vij , vkl) ∈ E , if d(pi , pk ) ≈ d(sj , sl)
search for maximum cliques in G
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Properties of Clique Detection
the problem is NP–complete(however, we search only for cliques of size ≤ k )
fast algorithms need adjacency matrices
if n = |S| = 5000 and k = |P| = 100 we need 30 GByte(counting only one bit per edge)
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Geometric hashing
preprocessing of the search spacelet’s describe the two–dimensional case
we align each pair (si , sj)with si at the origin and sj in direction xwe insert some information for each other point sk ∈ Sin a hashtable defined on a grid over S
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Example: geometric hashing
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Searching with geometric hashing
we simulate an insertion of the points of P into the hashtable
but we count only the non–empty entries
many votes reveal candidates for partial alignments
e.g., if we encounter a pair (pi , pj) such thatfor each other point of the pattern there is a non–empty cell in thehashtablewe have found a perfect candidate
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Properties of geometric hashing
grid size must be selected beforehand
preprocessing time O(nd+1)
searching time O(kd+1)
works only for rigid motion transformations
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Distance geometry
we represent both sets S and P as distance graphs
the vertices of the graphs are the points of the sets
the edges of the graphs hold the distances between thecorresponding vertices
e.g., GP = (P, P × P) complete graph
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The ideas behind psm
we define adequate distance graphs GP and GS
we search for subgraphs G′S of GS that are congruent to the graph
GP
(allowing certain tolerances)
we optimally align GP with the subgraphs of G′S
we select the best one among all hits
we extend the search to work with subgraphs of GP as well
we select a best subgraph as final solution
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The four main steps of psm
construction of the graphswith: exploitation of locallity properties
search of subgraphswith: sophisticated backtracking
alignmentwith: minimization of cost functions
search of partial patternswith: reactive tabu search
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Construction of the graphs
let us assume that the pattern P is small
we construct GP as the complete graph
we generate a dictionary D(ordered data structure)that contains all distances (intervals) between points in P
we consider an edge between two vertices in GSif the distance is present in the dictionary D
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Construction of the dictionary
let dij = d(pi , pj) be the distance between two points of P
the dictionary will contain the interval
[ (1− ε) · dij , (1 + ε/(1− ε) · dij ] ∈ D
where 0 ≤ ε < 1 is an appropriate tolerance
the upper limit can be simplified to (1 + ε)(but we loose the symmetry)
we can join intervals in the dictionary D if they intersect
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Construction of the graphs that way
GP GS
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Fast construction of the graphs
we construct GP as a connected (and rigid) graphmantaining only the short edges
we order the points of S previously in a grid of size similar to thelargest of the intervals
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Construction of the graphs that way
GP GS
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Matching
let us assume (at the beginning) that GP is a complete graph
we order the points of GP according to any order e.g.(p0, . . . , pk−1)
we apply a backtracking algorithmthat tries to encounter for earch pi a partner si
following the established ordering
hence:
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Backtracking for the search
let us assume that we already found a subgraph Gs0,...,si
where the graph Gp0,...,pi can be matchedwe look for candidates si+1 for the next point pi+1
that must be neighbors of the point si within GS
that must not be matched already andthat have similar distances to the sj (j ≤ i) as the pi+1 to the pj
(j ≤ i)
while there is a candidate we advance with i
if there are no more candidates for si+1, si cannot be a partner forpi neither (i.e.: backtracking)
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Termination of the algorithm
the algorithm informs each timea candidate for for pk−1 has been foundthe algorithm terminates when
there are no more candidates for p0 orthe first solution has been found
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Optimizations of the basic algorithm
reduction of the edges in GP
implies: reduction of the edges in GS
good ordering of the pi
implies: reduction of the number of candidates
consideration of the type of point (e.g. element type of the atom)implies: reduction of the number of candidates
all heuristics imply:the backtracking advances faster
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Search for partial alignments
find the subset of the points of the pattern that can be matchedbest to some points in the search space
NP–complete
there are |P(P)| = 2k possibilities to choose a subsetwe apply:
genetic algorithmreactive tabu search
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Genetic Algorithm
maintain graph GS as complete graph
genome: sequence of bits indicating si a point belongs to theactual pattern or not
crossover: two point crossover
mutation: flip
selection: roulette wheel
cost function: distance and size of alignment
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Termination of the GA
it is not that easy
once the first solution has been found
once a sufficently good solution has been found
after a certain number of iterations
once diversity of population is too low
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Problems with GA
GS must be a complete graph¿You know a crossover operation for non–complete graphs?
more precisely:we need a crossover (and mutation) operation that maintains aspecific property of the graphs (e.g., connectivity, rigidness)
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Reactive Tabu Search
we start with an admissible solution
we search for possibilities to improve the current solution
if we can: we choose one randomlyif we cannot:
we search for possibilities to reduce the current solutionif we can: we again try improvementsif we cannot: we jump to another admissible solution
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The Tabu criterion
we avoid repetitive movements taking advantage of a memory thatstores intermediate solutions
i.e.: we mark certain movements as tabu for a certain number ofiterations
reactive means: we adapt the tabu period dynamically
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Quality of a partial alignment
evaluation of the cost of a solution:number of aligned points plus quality of the alignment
remember: quality Q ∈]0, 1], but we will use Q ≥ threshold
hence, maximal quality: |P|+ 1
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Reactive Tabu Search for psm
representation of the problem:sets of indices of the matched points
search for candidates to improve (add):(rigidly) connected neighbors within graph GS
search for candidates to reduce (drop):any point of the current solution that mantains the graph GSconnected (and rigid)
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¿When do we terminate?
not that simple
once we found a sufficiently good solution
once we have run a certain number of iterations
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Search for the largest common pattern
P S
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Search for the largest common pattern
p/P p/S
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Search for the largest common pattern
s/P s/S
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Search for pattern with deformation
instead of the complete graph use a connected sparse graph
parts of the graph could be rigid
the graph may specify hinges or torsion axis
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psm
command line tool with configuration file
GUI
web–site to perform searches
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Softwareold version
almost 13.000 specific lines of C++uses the libraries:
Leda (library of efficient data structures and algorithms)gsl (Gnu scientific library)GAlib (genetic algorithms, MIT)Gtkmm (graphical user interface)own libraries
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Softwarenewer version
almost 11.000 specific lines of C++uses the libraries:
mtl (matrix template library)Gtkmm (graphical user interface)own libraries
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psm
psm
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Possible extensions
enumerate more rigorously all locations(up to now we have concentrated on the best solution)
extend the properties of the graphs defining deformations of thepattern (e.g. torsion of parts, restriction of angles)
allow local tolerances (e.g. per edge)
improve the user interface
more applications
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optimUMTS
optimUMTS
Optimization of wireless UMTS networks
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Joint work with
Fernando AguadoDepartamento de Teoría de la SeñalUniversidad de Vigo
Luis MendoUniversidad Politécnica de Madrid
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Objectives
given a set of possible nodes B (base stations)
find optimal subset
to guarantee certain services (bandwidth)
to an estimated user distribution
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Principal algorithm
LOAD
GENERATE
COMPUTEASSIGNMENT
DownlinkUplink
static user distributionpath lossUMTS settings
terrain datanode B locations
traffic description
best set B−nodes best set B−nodesassignment
GA HBB
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Cartography of Madrid
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Calculation of attenuation matrix α
simple logarithmic decay
L(m, k) = 100.1·(32.2+35.1·log(d(m,k)))
α(m, k) =Geff(m) ·Geff(k)
L(m, k)
simplied Xia model
mixed model: close—Xia, far—simple
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Coverage around a node B
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Traffic distribution
static distribution
grid of estimated user with activation percentage
polygons with Poisson process per service
uniform distibution everywhere or only on streets
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Calculation of SIR
uplink SIR γUL:
γUL(m, k) = (Eb/N0)UL · b0(k)/B0
downlink SIR γDL:
γDL(m, k) = (Eb/N0)DL · b0(k)/B0
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Calculation of noise
uplink noise NT (m) at transmitter m:
F (m) = 100.1·NF (m)
N(m) = kB · Tamb · B0 · F (m)
downlink noise NR(k) at receiver k :
F (k) = 100.1·NF (k)
N(k) = kB · Tamb · B0 · F (k)
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Calculation of power (Hanly and Mendo)
initial power P0(k) for receiver k :
t0(m, k) =γUL(m, k)
α(m, k)· N(m) P0(k) = min
m{t(m, k)}
power Pi(k) for receiver k in iteration i :
si = Pi−1 · α(m)
ti(m, k) =γUL(m, k)
α(m, k)· ((si − Pi−1(k) · α(m, k)) · a(k) + N(m))
Pi(k) = minm{ti(m, k)}
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Termination criteria for iteration
∃k with P(k) > Pmax(k) =⇒ no assignment
i > Imax =⇒ no assignment
maxk
{Pi(k)
Pi−1(k),Pi−1(k)
Pi(k)
}≤ ∆ =⇒ assignment possible
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Calculation of assignment
A(k) = m with t(m, k) = minm{t(m, k)}
P(k) = minm{t(m, k)}
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Validation of assignment
uplink:
Pmin(k) ≤ P(k) ≤ Pmax(k)
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preliminaries
downlink:
β(n, k , m) =
{ρ(m, k) if m = n1 otherwise
calculation of SSIR γ̃:
γ̃(m, k) =γDL(m, k)
1 + ρ(m, k) · γDL(m, k)
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System to solve for transmitter power calculation
H(m, n) = δm,n −∑
k∈A−1(m)
α(n, k) · β(n, k , m) · γ̃(m, k)
α(m, k)
ν(m) = Pplt(m) +∑
k∈A−1(m)
γ̃(m, k) · N(k)
α(m, k)
H · T = ν
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Validation of power restrictions
validation of maximum power of transmitter m:
0 < T (m) ≤ Tmax(m)
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Validation of power restrictions
calculation of downlink power of receiver k :
P(k) =γ̃(m, k)
α(A(k), k)
(M∑
m=1
α(m, k) · β(m, k , A(k)) · T (m) + N(k)
)
validation of transmitter maximum channel power:
P(k) ≤ Pchn
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Optimization with heuristic backtracking
Observation:
if there is no assignment with certain m nodes B
then there is no assignment with less nodes B
hence
start with all nodes B
eliminate nodes B til optimum found with branch&bound
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Termination with heuristic backtracking
stop if stagnation occurs(if within a subtree all minimum solutions are at the same depth)
finds optimum solution
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Heuristics
ordering of nodes B plays an important role in finding fast goodsolutions
reorder nodes B for backtracking according to heuristics
for instance: eccentricity, random, number of initial connections,etc.
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Optimization with genetic algorithm (GA)
steady-state incremental evolution(in each iteration two new decendents are generated)
selection: roulette wheel
mutation: flip
crossover: two–point–cyclic
quality: number of nodes B plus power as tiebreak
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Genom generation I
matrix representation of nodes B
exploiting locality properties
using allele: usable, unusable, used, fixed
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Genom generation II
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Genom generation III
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Crossover I
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Crossover II
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Evolution
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Assignment and heuristic branch result
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GA results with and without downlink
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Software
almost 22.000 specific lines of C++uses the libreries:
Leda (library of efficient data structures and algorithms)GAlib (genetic algorithms, MIT)own libraries
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Further research and implementations
use of MonteCarlo method
to find best subset of nodes B
for several user distributions
i.e., find best subset to satisfy different scenarios
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shaprox
Approximation of point sets with shapes
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Informal problem description
given a set of points in the plane
construct a geometric figure interpolating the sample points
that reasonably capture the shape of the point set
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Applications
pattern recognition
object definition in geographic information systems
CAD/CAM services
vectorization tasks
curve reconstruction in image analysis
single–computation pose estimation
geometric indexing into pictorial databases
shape tracking etc.
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Examplepoint set
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Exampleinitial shapes
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Exampleadapted shapes
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Approximation task
Three steps:
clustering of the points to identify the individual parts of a set ofshapes,
generating of an initial guess of the individual shapes,
adapting the individual shapes to the underlying point setaccording to some distance metric.
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Brain storming
distances, metrics, optimization, local and globalminima, discrete–continuous, partially plain functions,multi-objetive optimization, local decisions,multi–scale, simplification, VORONOI–diagram,DELAUNAY–diagram, graph analysis, similaritydetection, clasification (with and without supervision),filtering
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Polygonal approximationtype of clustering
Given a set of points of a plane curve,
construct a polygonal structure
interpolating the sample points
that reasonably captures the shape of the point set.
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Three approaches for polygonal approximation
α–shapes
crust
curve approximation
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Alpha–shapesapplication
The algorithm is able to detect
the outer boundary of a set of points
which covers more or less evenly distributed the interior of ashape.
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Alpha–shapesalgorithm
Compute the DELAUNAY triangulation of the point set.
Eliminate all triangles of the resulting graph which have a radiuslarger than α times the minimum radius.
The final shape is given by the outer edges of the remaininggraph.
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VORONOI–diagram
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VORONOI–diagram with more points
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DELAUNAY–diagram
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Alpha–shapesExample
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Alpha–shapesshort comings
Need of suitable alpha.
Alpha is constant over the entire point set.
Interior points needed.
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CRUSTapplication
The algorithm is able to reconstruct
a curve
that is sampled sufficiently dense
especially smooth curves, i.e., possibly many component curveswithout branches, endpoints, or self–intersections.
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CRUSTalgorithm
The crust
is the set of edges
selected from the Delaunay triangulation of the initial point set
extended by its Voronoi points
where both endpoints of the edges belong to the initial set.
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CRUSTExample
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CRUSTExample
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Curve reconstructionalgorithm
Computes the GABRIEL graph as a subgraph of the Delaunaygraph
(an edge between two input points belongs to the Gabriel graph ifa disk with this edge as diameter does not contain any other inputpoint).
Eliminate the edges which do not fulfill the local granularityproperty,
i.e., at each point only the two shortest edges are maintained.
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Curve reconstructionapplication
The method works well if there exists a regular interpolant,
that is, a polygonal closed curve such that the local granularity,
defined as minimum distance to an input point,
at each point of the curve is strictly smaller
than the local thickness at that point,
defined as the distance to the medial axis of the shape.
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Skeleton
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Skeleton
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Simplification problemobjective
Given polygonal chain (or polygon) P (with n vertices),
approximate P by another one Q whose vertices are a subset of kvertices of P.
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Simplification problemtwo variants
min–# problem: minimize the number of vertices of anapproximating polygonal chain (or closed polygon) with the errorwithin a given bound;
min–ε problem: minimize the error of an approximating polygonalchain (or closed polygon) consisting of a given number of vertices.
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Simplification problemsolutions
Both problems can be solved in optimal time O(n2).
There exist near–optimal algorithms for solving the min–#problem for the Euclidean distance which for practical problemsoutperform the optimal algorithms.
There exist a genetic algorithm to cope with the min–# problemwhich found near optimal solutions in the presented experiments.
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shaproxshape description
A shape S is defined by a number of points and certain parameters:
line: a point L ∈ IR2 and an angle ϕ ∈ IR;
circumference: a center point C ∈ IR2 and a radius % ∈ IR;
set of circumferences: set of pairs of center points andcorresponding radii, i.e., (C,R) ⊂ IR2 × IR;
polyline or polygon: an ordered set of corner pointsQ = {Q1, Q2, . . . , Qk}, Qj ∈ IR2, j = 1, . . . , k , where the onlydifference between the two shapes is that for a polygon the lastand first corner are connected;
rounded box: a line segment defined by two points Q1, Q2 ∈ IR2,an aspect ratio α ∈ IR, and a corner radius % ∈ IR.
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shaproxdistance functions: point—shape I
lineδL2(Pi ,S) =
∣∣∣Det(Pi − L, (cos ϕ, sin ϕ)T )∣∣∣
circumferenceδL2(Pi ,S) = |‖Pi − C‖ − %|
set of circumferences
δL2(Pi ,S) = min(Cj ,%j )∈(C,R)
∣∣‖Pi − Cj‖ − %j∣∣
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shaproxdistance functions: point—shape II
polyline or polygon, first we need the distance for a segmentQQj = Qj+1 −Qj :
δL2(Pi , QQj) =‖Pi −Qj‖ if (Pi −Qj)
T QQj < 0‖Pi −Qj+1‖ if (Pi −Qj+1)
T QQj > 0∣∣∣∣Det(QQj ,Pi−Qj )
‖QQj‖
∣∣∣∣ otherwise
and obtain for a polyline or polygon
δL2(Pi ,S) = minQj∈Q
δL2(Pi , QQj)
where for a polyline the index j runs from 1, . . . , k − 1 and for apolygon from 1, . . . , k with Qk+1 = Q1.
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shaproxdistance functions: point—shape III
rounded box
δL2(Pi ,S) =
∣∣∣∣ minj=1,...,4
δL2(Pi , QQj)± %
∣∣∣∣where Q3 = Q2 + α Q>
12 and Q4 = Q1 + α Q>12 being Q>
12 the leftturned perpendicular vector to Q2 −Q1 of same length. +% istaken when the point Pi lies inside the rectangular box throughQ1, . . . , Q4, and −% when Pi lies outside.
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shaproxdistance functions: non–euclidean
vertical distance to a line:
δV (Pi ,S) =∣∣∣P2
i − L2 − (P1i − L1) · sin ϕ/ cos ϕ
∣∣∣length of the segments could have an influence as a weight
δwL2(Pi , QQj) =∣∣∣Det(QQj , Pi −Qj)
∣∣∣
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shaproxdistance functions: point set—shape
RMS
δRMS,L2(P,S) =
√√√√1l
∑Pi∈P
δL2(Pi ,S)2
AVG
δAVG,L2(P,S) =1l
∑Pi∈P
δL2(Pi ,S)
MAXδMAX ,L2(P,S) = max
Pi∈PδL2(Pi ,S)
or δRMS,V (P,S) or δAVG,L1(P,S), etc
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shaproxalgorithm for local search
Algorithm of RODRÍGUEZ/GARCÍA–PALOMARES
derivative–free minimization method
proved convergence for either locally strictly differentiable ornon–smooth locally convex functions.
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shaproxSubsidiary objective
Til now concentrated on the minimization of a single function.
However, in our optimization problem, it can happen that there isno change in the value of the distance function although the shapeis modified.
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shaproxExample: Subsidiary objective
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shaproxSubsidiary objective: idea
Minimize perimeter as well.
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shaproxSubsidiary objective: possible solution
Formulate a convex combination of all objectives,
i.e., optimize the single–objective function
f (x) = β1f1(x) + β2f2(x) + · · ·+ βl fl(x)
where the βj > 0, for j = 1, . . . , l , are strictly positive weights,
fj(·) : IRn −→ IR are the individual objective functions.
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shaproxSubsidiary objective: short coming of possible solution
Explores the Pareto front defined by the weights βj ,
it might be difficult to find weights such that the encounteredminimum is sufficiently close to the minimum considering only theprincipal objective f1(·).
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shaproxExample: convex combination as objective function
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shaproxSubsidiary objective: better solution
modify the comparison f (z) ≤ f (x)− τ2 (in the local searchalgorithm)
with the following iteratively defined comparison function for lobjectives:
[f1(z), . . . , fl(z)] ≤τ2 [f1(x), . . . , fl(x)] ⇐⇒∀j = 1, . . . , l : fj(z) ≤ fj(x)− τ2 or
(j < l and fj(z) ≤ fj(x) and fj+1(z) ≤ fj+1(x)− τ2)
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shaproxExample
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NP–Completeness and non–convexity of the objectivefunction
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shaproxExamples: wedge
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shaproxExamples: simple polygon
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shaproxExamples: convex hull
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shaproxExamples: circumferences
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shaproxExamples: influence of metrics
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shaproxExamples: completing shapes
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shaproxExamples: rounded rectangle
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shaproxExamples: complex polygon
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Software
more than 7.000 specific lines of C++uses the libraries:
ImageMagickGtkmm (graphical user interface)own libraries
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shaprox
shaprox
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¿Questions?
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¡Many thanks!
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