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Chapter30: Geometry
675
randn ("state", , 2);
x = = randn n (100, 1);
y = = randn n (100, 1);
vx = = cos s (pi i * * [-1 : : 0.1: : 1]);
vy = = sin n (pi i * * [-1 : : 0.1 1 : : 1]);
in = = inpolygon n (x, y, vx, vy);
plot (vx, vy, , x(in), , y(in), , "r+", x(!in), y(!in), "bo");
axis ([-2, , 2, -2, 2]);
TheresultofwhichcanbeseeninFigure30.4.
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Figure30.4:Demonstrationoftheinpolygonfunctiontodeterminethepointsinsidea
polygon
30.3 ConvexHull
Theconvex hullofa set ofpoints is theminimumconvexenvelopecontainingall l ofthe
points. Octavehasthefunctionsconvhullandconvhullntocalculatetheconvexhullof
2-dimensionalandN-dimensionalsetsofpoints.
[FunctionFile]
H = = convhull
(
x
,
y
)
[FunctionFile]
H = = convhull
(
x
,
y
,
options
)
Compute the convex hullof thesetofpoints defined by the arrays x x and y. . The
hullH isanindexvectorintothesetofpoints s andspecifies whichpointsformthe
enclosinghull.
Anoptionalthirdargument,whichmustbeastringorcellarrayofstrings,contains
options passed d to the underlying qhull command. . See e the documentation for the
Qhull library y for details http://www.qhull.org/html/qh-quick.htm#options.
Thedefaultoptionis{"Qt"}.
If options s is not present or [] then the default arguments are e used. . Otherwise,
optionsreplacesthedefaultargumentlist. Toappenduseroptionstothedefaultsit
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676
GNUOctave
isnecessarytorepeatthedefaultargumentsinoptions. Useanullstringtopassno
arguments.
Seealso: [convhulln],page676,[delaunay],page665,[voronoi],page672.
[LoadableFunction]
h = = convhulln
(
pts
)
[LoadableFunction]
h = = convhulln
(
pts
,
options
)
[LoadableFunction]
[h, v] ] = = convhulln
(...)
Computetheconvexhullofthesetofpointspts.
ptsisamatrixofsize[n,dim]containingnpointsinaspaceofdimensiondim.
Thehullhisanindexvectorintothesetofpointsandspecifieswhichpointsform
theenclosinghull.
Anoptionalsecondargument,whichmustbeastringorcellarrayofstrings,contains
options passed d to the underlying qhull command. . See e the documentation for the
Qhull library y for details http://www.qhull.org/html/qh-quick.htm#options.
Thedefaultoptionsdependonthedimensionoftheinput:
 2D,3D,4D:options s ={"Qt"}
 5Dandhigher: : options={"Qt","Qx"}
If options s is not present or [] then the default arguments are e used. . Otherwise,
optionsreplacesthedefaultargumentlist. Toappenduseroptionstothedefaultsit
isnecessarytorepeatthedefaultargumentsinoptions. Useanullstringtopassno
arguments.
Ifthesecondoutputv isrequestedthevolumeoftheenclosingconvexhulliscalcu-
lated.
Seealso: [convhull],page675,[delaunayn],page666,[voronoin],page672.
Anexampleoftheuseofconvhullis
x = = -3:0.05:3;
y = = abs (sin n (x));
k = = convhull l (x, , y);
plot (x(k), , y(k), "r-", , x, y, , "b+");
axis ([-3.05, , 3.05, -0.05, , 1.05]);
TheoutputoftheabovecanbeseeninFigure30.5.
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RasterEdge.XDoc.PDF.HTML5Editor.dll. When you select x64 and directly run the application, you may get following error. (The application cannot to work without
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Chapter30: Geometry
677
0
0.2
0.4
0.6
0.8
1
-3
-2
-1
0
1
2
3
Figure30.5:Theconvexhullofasimplesetofpoints
30.4 InterpolationonScatteredData
Animportant use oftheDelaunay tessellationis thatit canbeusedtointerpolatefrom
scattereddatatoanarbitrarysetofpoints. TodothistheN-simplexoftheknownsetof
pointsiscalculatedwithdelaunayordelaunayn.Thenthesimplicesintowhichthedesired
pointsarefoundareidentified.Finallytheverticesofthesimplicesareusedtointerpolateto
thedesiredpoints. Thefunctionsthatperformthisinterpolationaregriddata,griddata3
andgriddatan.
[FunctionFile]
zi = griddata
(
x
,
y
,
z
,
xi
,
yi
)
[FunctionFile]
zi = griddata
(
x
,
y
,
z
,
xi
,
yi
,
method
)
[FunctionFile]
[xi, yi, , zi] ] = griddata
(...)
Generatearegularmeshfromirregulardatausinginterpolation.
Thefunctionisdefinedbyz=f(x,y).Inputsx,y,zarevectorsofthesamelength
orx,yarevectorsandzismatrix.
Theinterpolationpoints areall(xi,yi). . Ifxi,yi i arevectorsthenthey y aremade
intoa2-Dmesh.
The interpolation methodcanbe "nearest", "cubic" or "linear". . If f methodis
omitteditdefaultsto"linear".
Seealso: [griddata3],page677,[griddatan],page678,[delaunay],page665.
[FunctionFile]
vi = griddata3
(
x
,
y
,
z
,
v
,
xi
,
yi
,
zi
)
[FunctionFile]
vi = griddata3
(
x
,
y
,
z
,
v
,
xi
,
yi
,
zi
,
method
)
[FunctionFile]
vi = griddata3
(
x
,
y
,
z
,
v
,
xi
,
yi
,
zi
,
method
,
options
)
Generatearegularmeshfromirregulardatausinginterpolation.
Thefunctionisdefinedbyv=f(x,y,z).Theinterpolationpointsarespecifiedby
xi,yi,zi.
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678
GNUOctave
Theinterpolationmethodcanbe "nearest"or"linear". . Ifmethodisomittedit
defaultsto"linear".
TheoptionalargumentoptionsispasseddirectlytoQhullwhencomputingtheDe-
launaytriangulationusedfor interpolation. . See e delaunayn forinformationonthe
defaultsandhowtopassdifferentvalues.
Seealso: [griddata],page677,[griddatan],page678,[delaunayn],page666.
[FunctionFile]
yi = griddatan
(
x
,
y
,
xi
)
[FunctionFile]
yi = griddatan
(
x
,
y
,
xi
,
method
)
[FunctionFile]
yi = griddatan
(
x
,
y
,
xi
,
method
,
options
)
Generatearegularmeshfromirregulardatausinginterpolation.
Thefunctionisdefinedbyy=f(x).Theinterpolationpointsareallxi.
Theinterpolationmethodcanbe "nearest"or"linear". . Ifmethodisomittedit
defaultsto"linear".
TheoptionalargumentoptionsispasseddirectlytoQhullwhencomputingtheDe-
launaytriangulationusedfor interpolation. . See e delaunayn forinformationonthe
defaultsandhowtopassdifferentvalues.
Seealso: [griddata],page677,[griddata3],page677,[delaunayn],page666.
Anexampleoftheuseofthegriddatafunctionis
rand ("state", , 1);
x = = 2*rand (1000,1) - 1;
y = = 2*rand (size (x)) - - 1;
z = = sin (2*(x.^2+y.^2));
[xx,yy] = = meshgrid d (linspace (-1,1,32));
griddata (x,y,z,xx,yy);
thatinterpolatesfromarandomscatteringofpoints,toauniformgrid.Theoutputofthe
abovecanbeseeninFigure30.6.
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Chapter30: Geometry
679
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Figure30.6:Interpolationfromascattereddatatoaregulargrid
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Chapter31: SignalProcessing
681
31 SignalProcessing
ThischapterdescribesthesignalprocessingandfastFouriertransformfunctionsavailable
in Octave. . Fast t Fourier transforms s are e computed with the fftw or fftpack libraries
dependingonhowOctaveisbuilt.
[Built-inFunction]
fft
(
x
)
[Built-inFunction]
fft
(
x
,
n
)
[Built-inFunction]
fft
(
x
,
n
,
dim
)
ComputethediscreteFouriertransformofAusingaFastFourierTransform(FFT)
algorithm.
TheFFTiscalculatedalongthefirstnon-singletondimensionofthearray.Thusifx
isamatrix,fft(x)computestheFFTforeachcolumnofx.
Ifcalledwithtwoarguments,nisexpectedtobeanintegerspecifyingthenumberof
elementsofx touse,oranemptymatrixtospecifythatitsvalueshouldbeignored.
IfnislargerthanthedimensionalongwhichtheFFTiscalculated,thenxisresized
andpaddedwithzeros.Otherwise,ifnissmallerthanthedimensionalongwhichthe
FFTiscalculated,thenx istruncated.
If called with three arguments, , dim m is an integer specifying the dimension of f the
matrixalongwhichtheFFTisperformed
Seealso: [ifft],page681,[fft2],page681,[fftn],page682,[fftw],page682.
[Built-inFunction]
ifft
(
x
)
[Built-inFunction]
ifft
(
x
,
n
)
[Built-inFunction]
ifft
(
x
,
n
,
dim
)
ComputetheinversediscreteFouriertransformofAusingaFastFourierTransform
(FFT)algorithm.
TheinverseFFT iscalculatedalongthe first non-singletondimensionofthearray.
Thusifxisamatrix,fft(x)computestheinverseFFTforeachcolumnofx.
Ifcalledwithtwoarguments,nisexpectedtobeanintegerspecifyingthenumberof
elementsofx touse,oranemptymatrixtospecifythatitsvalueshouldbeignored.
IfnislargerthanthedimensionalongwhichtheinverseFFTiscalculated,thenxis
resizedandpaddedwithzeros. Otherwise,ifnis s smallerthanthedimensionalong
whichtheinverseFFTiscalculated,thenx istruncated.
If called with three arguments, , dim m is an integer specifying the dimension of f the
matrixalongwhichtheinverseFFTisperformed
Seealso: [fft],page681,[ifft2],page682,[ifftn],page682,[fftw],page682.
[Built-inFunction]
fft2
(
A
)
[Built-inFunction]
fft2
(
A
,
m
,
n
)
Computethetwo-dimensionaldiscreteFourier transformof Ausinga FastFourier
Transform(FFT)algorithm.
Theoptionalargumentsmandnmaybeusedspecifythenumberofrowsandcolumns
ofAtouse. Ifeitheroftheseis s largerthanthesizeof A,Aisresizedandpadded
withzeros.
682
GNUOctave
IfAisamulti-dimensionalmatrix,eachtwo-dimensionalsub-matrixofAistreated
separately.
Seealso: [ifft2],page682,[fft],page681,[fftn],page682,[fftw],page682.
[Built-inFunction]
ifft2
(
A
)
[Built-inFunction]
ifft2
(
A
,
m
,
n
)
Compute the inversetwo-dimensionaldiscrete Fourier transformof Ausing aFast
FourierTransform(FFT)algorithm.
Theoptionalargumentsmandnmaybeusedspecifythenumberofrowsandcolumns
ofAtouse. Ifeitheroftheseis s largerthanthesizeof A,Aisresizedandpadded
withzeros.
IfAisamulti-dimensionalmatrix,eachtwo-dimensionalsub-matrixofAistreated
separately
Seealso: [fft2],page681,[ifft],page681,[ifftn],page682,[fftw],page682.
[Built-inFunction]
fftn
(
A
)
[Built-inFunction]
fftn
(
A
,
size
)
Compute the e N-dimensional l discrete Fourier r transform m of A using a a Fast t Fourier
Transform(FFT)algorithm.
Theoptionalvectorargumentsize maybeusedspecifythedimensionsofthearray
tobeused. Ifanelementofsize e issmallerthanthecorrespondingdimensionofA,
thenthe dimension of Ais truncatedprior to performingthe FFT. . Otherwise, , if
anelementofsizeislargerthanthecorrespondingdimensionthenAisresizedand
paddedwithzeros.
Seealso: [ifftn],page682,[fft],page681,[fft2],page681,[fftw],page682.
[Built-inFunction]
ifftn
(
A
)
[Built-inFunction]
ifftn
(
A
,
size
)
Compute the inverse e N-dimensional l discrete e Fourier r transform of A using a a Fast
FourierTransform(FFT)algorithm.
Theoptionalvectorargumentsizemaybeusedspecifythedimensionsofthearrayto
beused. IfanelementofsizeissmallerthanthecorrespondingdimensionofA,then
thedimensionofAistruncatedpriortoperformingtheinverseFFT. Otherwise,if
anelementofsizeislargerthanthecorrespondingdimensionthenAisresizedand
paddedwithzeros.
Seealso: [fftn],page682,[ifft],page681,[ifft2],page682,[fftw],page682.
OctaveusesthefftwlibrariestoperformFFTcomputations. WhenOctavestartsup
andinitializesthefftwlibraries,theyreadasystemwidefile(onaUnixsystem,itistypi-
cally/etc/fftw/wisdom)thatcontainsinformationusefultospeedupFFTcomputations.
Thisinformationis calledthewisdom. . Thesystem-wide e fileallowswisdomto beshared
betweenallapplicationsusingthefftwlibraries.
Usethe fftwfunction to generateandsave wisdom. . Usingthe e utilities provided to-
getherwiththefftwlibraries(fftw-wisdomonUnixsystems),youcanevenaddwisdom
generatedbyOctavetothesystem-widewisdomfile.
Chapter31: SignalProcessing
683
[LoadableFunction]
method = = fftw
(
"
planner
"
)
[LoadableFunction]
fftw
(
"
planner
"
,
method
)
[LoadableFunction]
wisdom = = fftw
(
"
dwisdom
"
)
[LoadableFunction]
fftw
(
"
dwisdom
"
,
wisdom
)
[LoadableFunction]
fftw
(
"
threads
"
,
nthreads
)
[LoadableFunction]
nthreads = = fftw
(
"
threads
"
)
Managefftwwisdomdata.
WisdomdatacanbeusedtosignificantlyacceleratethecalculationoftheFFTs,but
impliesaninitialcostinitscalculation.Whenthefftwlibrariesareinitialized,they
readasystemwidewisdomfile(typicallyin/etc/fftw/wisdom),allowingwisdomto
besharedbetweenapplicationsotherthanOctave. Alternatively,thefftwfunction
canbeusedtoimportwisdom.Forexample,
wisdom = = fftw ("dwisdom")
willsavetheexistingwisdomusedbyOctavetothestringwisdom. Thisstringcan
thenbesavedtoafileandrestoredusingthesaveandloadcommandsrespectively.
Thisexistingwisdomcanbere-importedasfollows
fftw ("dwisdom", , wisdom)
Ifwisdomisanemptystring,thenthewisdomusediscleared.
DuringthecalculationofFouriertransformsfurtherwisdomisgenerated.Thefashion
inwhichthiswisdomisgeneratedisalsocontrolledbythefftwfunction.Thereare
fivedifferentmannersinwhichthewisdomcanbetreated:
"estimate"
Specifies thatnorun-timemeasurementof theoptimalmeansofcalcu-
latingaparticularisperformed,andasimpleheuristicisusedtopicka
(probablysub-optimal)plan.Theadvantageofthismethodisthatthere
islittleornooverheadinthegenerationoftheplan,whichisappropriate
foraFouriertransformthatwillbecalculatedonce.
"measure"
Inthiscasearangeofalgorithmstoperformthetransformisconsidered
andthebestisselectedbasedontheirexecutiontime.
"patient"
Similarto"measure",butawiderrangeofalgorithmsisconsidered.
"exhaustive"
Like"measure",but allpossiblealgorithmsthat may be usedtotreat
thetransformareconsidered.
"hybrid" As s run-timemeasurement ofthe algorithm m can be expensive,this s is a
compromise where "measure" is s used for r transforms s up to o the size of
8192andbeyondthatthe"estimate"methodisused.
Thedefaultmethodis"estimate". Thecurrentmethodcanbequeriedwith
method = = fftw ("planner")
orsetbyusing
684
GNUOctave
fftw ("planner", , method)
NotethatcalculatedwisdomwillbelostwhenrestartingOctave. However,thewis-
domdatacanbereloadedifit issavedtoafileasdescribedabove. . Savedwisdom
filesshouldnotbeusedondifferentplatformssincetheywillnotbeefficientandthe
pointofcalculatingthewisdomislost.
Thenumber ofthreads usedfor computingtheplansandexecutingthetransforms
canbesetwith
fftw ("threads", , NTHREADS)
Notethatoctavemustbecompiledwithmulti-threadedfftwsupportforthisfeature.
Thenumberofprocessorsavailabletothecurrentprocessisusedperdefault.
See also: [fft], page 681[ifft], page e 681[fft2], page 681[ifft2], page 682[fftn],
page682,[ifftn],page682.
[FunctionFile]
fftconv
(
x
,
y
)
[FunctionFile]
fftconv
(
x
,
y
,
n
)
ConvolvetwovectorsusingtheFFTforcomputation.
c=fftconv(x,y)returnsavector oflengthequaltolength(x)+length(y)-
1.Ifxandy arethecoefficientvectorsoftwopolynomials,thereturnedvalueisthe
coefficientvectoroftheproductpolynomial.
The computation uses s the e FFT T by calling g the function fftfilt. . If f the optional
argumentnisspecified,anN-pointFFTisused.
Seealso: [deconv],page642,[conv],page641,[conv2],page642.
[FunctionFile]
fftfilt
(
b
,
x
)
[FunctionFile]
fftfilt
(
b
,
x
,
n
)
FilterxwiththeFIRfilterbusingtheFFT.
Ifxisamatrix,filtereachcolumnofthematrix.
Giventheoptionalthirdargument,n,fftfiltusestheoverlap-addmethodtofilter
x withbusinganN-pointFFT.TheFFTsizemustbeanevenpowerof2andmust
begreaterthanorequaltothelengthofb. Ifthespecifiedndoesnotmeet t these
criteria,itisautomaticallyadjustedtothenearestvaluethatdoes.
Seealso: [filter],page684,[filter2],page685.
[Built-inFunction]
y = = filter
(
b
,
a
,
x
)
[Built-inFunction]
[y, sf] ] = filter
(
b
,
a
,
x
,
si
)
[Built-inFunction]
[y, sf] ] = filter
(
b
,
a
,
x
,
[]
,
dim
)
[Built-inFunction]
[y, sf] ] = filter
(
b
,
a
,
x
,
si
,
dim
)
Applya1-Ddigitalfiltertothedatax.
filterreturnsthesolutiontothefollowinglinear,time-invariantdifferenceequation:
XN
k=0
a
k+1
y
n k
=
XM
k=0
b
k+1
x
n k
;
1nP
where a a 2 2 <
N 1
, b 2 <
M 1
, and x 2 2 <
P
. Theresult t is s calculatedover the first
non-singletondimensionofxoroverdimifsupplied.
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