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Notebook[{
Cell[CellGroupData[{
Cell["B-Spline Interpolation of Data with Regular Sampling", "Title"],
Cell[TextData[StyleBox["Philippe Th\[EAcute]venaz and Michael Unser\nSwiss \
Federal Institute of Technology Lausanne, Switzerland\n\
http://bigwww.epfl.ch/",
FontSlant->"Italic"]], "Text"],
Cell["\<\
The purpose of the five routines provided here is to illustrate a \
signal-processing approach to the interpolation of regularly-sampled data The \
relevant theory is explained in more details in the following two references:\
\
\>", "Text"],
Cell["\<\
[1]\tM. Unser, A. Aldroubi and M. Eden,
\t\"B-Spline Signal Processing: Part I\[LongDash]Theory,\"
\tIEEE Transactions on Signal Processing, vol. 41, no. 2, pp. 821-833,
\tFebruary 1993.
[2]\tM. Unser, A. Aldroubi and M. Eden,
\t\"B-Spline Signal Processing: Part II\[LongDash]Efficient Design and \
Applications,\"
\tIEEE Transactions on Signal Processing, vol. 41, no. 2, pp. 834-848,
\tFebruary 1993. \
\>", "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Theory", "Section"],
Cell[CellGroupData[{
Cell["Interpolation Coefficients", "Subsection"],
Cell[TextData[{
"Given a set of samples ",
Cell[BoxData[
\(TraditionalForm\`s\)]],
", the fundamental task of interpolation is to compute the following \
equation for any real, possibly non-integer value of ",
Cell[BoxData[
\(TraditionalForm\`x\)]],
":"
}], "Text"],
Cell[BoxData[
\(f[x] = \[Sum]\+\(k = \(-\[Infinity]\)\)\%\[Infinity]\(
c\_\(\(\[LeftDoubleBracket]\)\(k\)\(\[RightDoubleBracket]\)\)\) \
\[CurlyPhi][x - k]\)], "Input"],
Cell[TextData[{
"\nFor a specific integer value ",
Cell[BoxData[
\(TraditionalForm\`j\)]],
", it is also desired that the interpolated function ",
Cell[BoxData[
\(TraditionalForm\`f[j]\)]],
" takes the same value as the sample ",
Cell[BoxData[
\(TraditionalForm\`s\_\(\(\[LeftDoubleBracket]\)\(j\)\(\
\[RightDoubleBracket]\)\)\)]],
" of the set of available samples we want to interpolate"
}], "Text"],
Cell[BoxData[
\(s\_\(\(\[LeftDoubleBracket]\)\(j\)\(\[RightDoubleBracket]\)\) = \(f[
j] = \[Sum]\+\(k = \(-\[Infinity]\)\)\%\[Infinity]\(
c\_\(\(\[LeftDoubleBracket]\)\(k\)\(\[RightDoubleBracket]\)\)\
\) \[CurlyPhi][j - k]\)\)], "Input"],
Cell[TextData[{
"We discourage the traditional approach one usually follows in order to \
satisfy this equation. Traditionally, one makes sure that two conditions are \
met. Firstly, set ",
Cell[BoxData[
\(TraditionalForm\`c\_\(\(\[LeftDoubleBracket]\)\(k\)\(\
\[RightDoubleBracket]\)\) =
s\_\(\(\[LeftDoubleBracket]\)\(k\)\(\[RightDoubleBracket]\)\)\)]],
" for all ",
Cell[BoxData[
\(TraditionalForm\`k\)]],
". Secondly, select a function ",
Cell[BoxData[
\(TraditionalForm\`\[CurlyPhi]\)]],
" that takes value ",
Cell[BoxData[
\(TraditionalForm\`0\)]],
" for all integer arguments, except at the origin where it must equal unity \
(for any non-integer arguments, ",
Cell[BoxData[
\(TraditionalForm\`\[CurlyPhi]\)]],
" is free to take any value it may like, although some choices are better \
than others). This would do the trick, and would be the classical approach to \
interpolation."
}], "Text"],
Cell[TextData[{
"Now, B-splines have very attractive properties, which makes them very good \
candidates in the role of the function ",
Cell[BoxData[
\(TraditionalForm\`\[CurlyPhi]\)]],
". Unfortunately, they don't have all the qualities in the world; in \
particular, they may take a non-zero value for integer arguments, and they \
may differ from unity at the origin. Thus, one must depart from traditional \
interpolation so as to use B-splines for interpolating data. Instead, one \
must determine the suitable set of coefficients ",
Cell[BoxData[
\(TraditionalForm\`c\_\(\(\[LeftDoubleBracket]\)\(k\)\(\
\[RightDoubleBracket]\)\) \[NotEqual]
s\_\(\(\[LeftDoubleBracket]\)\(k\)\(\[RightDoubleBracket]\)\) =
f[k]\)]],
" such that ",
Cell[BoxData[
\(TraditionalForm\`f[
j] = \[Sum]\(c\_\(\(\[LeftDoubleBracket]\)\(k\)\(\
\[RightDoubleBracket]\)\)\) \[CurlyPhi][j - k]\)]],
" still holds true for any integer ",
Cell[BoxData[
\(TraditionalForm\`j\)]],
". This time, setting ",
Cell[BoxData[
\(TraditionalForm\`c\_\(\(\[LeftDoubleBracket]\)\(k\)\(\
\[RightDoubleBracket]\)\) =
s\_\(\(\[LeftDoubleBracket]\)\(k\)\(\[RightDoubleBracket]\)\)\)]],
" ",
StyleBox["doesn't",
FontSlant->"Italic"],
" do the trick. We need instead some routine that produces a set of \
coefficients ",
Cell[BoxData[
\(TraditionalForm\`c\)]],
" out of a set of samples ",
Cell[BoxData[
\(TraditionalForm\`s\)]],
". This is the role of the routine ",
StyleBox["BSplineCoefficients[]", "Input",
FontWeight->"Bold"],
". The core of this routine consists in a forward-backward recursive filter \
(for more details, see the references [1, 2])."
}], "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell["Border Conditions", "Subsection"],
Cell[TextData[{
"So far, we have taken for granted that the number of samples ",
Cell[BoxData[
\(TraditionalForm\`s\)]],
" was infinite, same with the number of coefficients ",
Cell[BoxData[
\(TraditionalForm\`c\)]],
". Obviously, the amount of RAM in your computer is only so much, which \
asks for a finite number of data. Reconciliation of these contradictory \
requirements is the object of the present section."
}], "Text"],
Cell[TextData[{
"Let us suppose you have ",
Cell[BoxData[
\(TraditionalForm\`m\)]],
" samples at your disposal. Given only those samples you know of, if we \
find a way to predict (extrapolate) the value of any coefficient outside the \
range of your known data, then we are back to the first (infinite) \
formulation, without its associated cost. This is known as an implicit \
scheme. Extrapolating the samples ",
Cell[BoxData[
\(TraditionalForm\`s\)]],
" is less relevant than extrapolating the coefficients ",
Cell[BoxData[
\(TraditionalForm\`c\)]],
", for the former do not explicitly appear in what we compute in practice, \
that is, ",
Cell[BoxData[
\(TraditionalForm\`\[Sum]\(c\_\(\(\[LeftDoubleBracket]\)\(k\)\(\
\[RightDoubleBracket]\)\)\) \[CurlyPhi][x - k]\)]],
"."
}], "Text"],
Cell[TextData[{
"There is no limit to the number of ways one may select to perform \
extrapolation. After all, any single one is but a wild guess, even though \
some are more informed than others. We advocate here an extrapolation scheme \
where coefficients are mirrored around their extremities, namely, around ",
Cell[BoxData[
\(TraditionalForm\`c\_\(\(\[LeftDoubleBracket]\)\(1\)\(\
\[RightDoubleBracket]\)\)\)]],
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Cell[BoxData[
\(TraditionalForm\`c\_\(\(\[LeftDoubleBracket]\)\(m\)\(\
\[RightDoubleBracket]\)\)\)]],
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Cell[BoxData[
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Cell[BoxData[
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" is from the interval ",
Cell[BoxData[
\(TraditionalForm\`\([1, m]\)\)]],
". Moreover, it imposes the same doubly-mirrored structure on the samples \
",
Cell[BoxData[
\(TraditionalForm\`s\_\(\(\[LeftDoubleBracket]\)\(k\)\(\
\[RightDoubleBracket]\)\)\)]],
", and also on the whole curve ",
Cell[BoxData[
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" which now satisfies ",
Cell[BoxData[
\(TraditionalForm\`f[1 - x] = f[1 + x]\)]],
" and ",
Cell[BoxData[
\(TraditionalForm\`f[m - x] = f[m + x]\)]],
", ",
Cell[BoxData[
\(TraditionalForm\`\[ForAll] x \[Element] \[DoubleStruckCapitalR]\)]],
". An important benefit of this scheme is that there is no need to \"invent\
\" special values (e.g., zero); all extrapolated values are replicated from \
those of the known interval. Moreover, this scheme ensures that no \
discontinuities (or strong changes of value) of ",
Cell[BoxData[
\(TraditionalForm\`f[x]\)]],
" arise at the extremities of the known interval."
}], "Text"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell["Implementation", "Section"],
Cell[TextData[{
"We provide five routines. Since this count is rather low, we proceed by \
the way of function definitions rather than by the way of a full-blown ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" package. The drawback is that you have to pay attention to dependencies, \
and you need to ensure that the definitions are made in the same order as \
they are described here."
}], "Text"],
Cell[CellGroupData[{
Cell["1. Border Conditions", "Subsubsection"],
Cell[TextData[{
"Given an integer index value ",
StyleBox["i", "Input"],
", this routine returns an index lying in the range ",
StyleBox["Interval[{k1,k2}]", "Input"],
". The mapping satisfies mirror boundary conditions."
}], "Text"],
Cell[BoxData[
\(MirroredIndex[i_, k1_, k2_] :=
Module[{m = k2 - k1 + 1, j}, \[IndentingNewLine]If[m \[Equal] 1, k1,
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k1 + If[j < m, j, 2 m - 2 - j]]]\)], "Input",
InitializationCell->True]
}, Closed]],
Cell[CellGroupData[{
Cell["2. Auxilliary Function", "Subsubsection"],
Cell[TextData[{
"Given a strictly positive real argument ",
StyleBox["x", "Input"],
" and a real exponent ",
StyleBox["a", "Input"],
", this routine returns ",
StyleBox["x^a", "Input"],
". If the argument is strictly negative, the value ",
StyleBox["0", "Input"],
" is returned. When the argument is null, special rules apply."
}], "Text"],
Cell[BoxData[
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}, Closed]],
Cell[CellGroupData[{
Cell["3. B-Spline", "Subsubsection"],
Cell[TextData[{
"Given a real argument ",
StyleBox["x", "Input"],
" and a positive or null integer parameter ",
StyleBox["n", "Input"],
" called the B-spline degree, this routine evaluates the value of the \
B-spline."
}], "Text"],
Cell[BoxData[
\(BSpline[x_, n_] :=
Module[{k}, \[Sum]\+\(k = 0\)\%\(n + 1\)\(\((\(-1\))\)\^k\/\(n!\)\)
Binomial[n + 1, k]
OneSidedPower[x - k + \(n + 1\)\/2, n]]\)], "Input",
InitializationCell->True]
}, Closed]],
Cell[CellGroupData[{
Cell["4. Interpolation Coefficients", "Subsubsection"],
Cell[TextData[{
"Given a non-empty list ",
StyleBox["s", "Input"],
" of real-valued data samples, and given a positive or null integer \
B-spline degree ",
StyleBox["n", "Input"],
", this routine returns a list of interpolation coefficients. The returned \
list has the same length than ",
StyleBox["s", "Input"],
" and assumes double mirroring of the data."
}], "Text"],
Cell[BoxData[
\(BSplineCoefficients[s_, n_] :=
If[\((n \[Equal] 0)\) || \((n \[Equal] 1)\),
Return[s], \[IndentingNewLine]Module[{k, i, m = Length[s], x, p,
q = Quotient[n, 2], c, z}, \[IndentingNewLine]p =
N[Roots[0 == \[Sum]\+\(k = \(-q\)\)\%q\( x\^k\) BSpline[k, n],
x]]; \[IndentingNewLine]p =
Sort[Table[p[\([k, 2]\)], {k, 2 q}]]; \[IndentingNewLine]p =
Take[p, \(-q\)]; \[IndentingNewLine]c =
s \(\[Product]\+\(k = 1\)\%\(Quotient[n, 2]\)\((1 - p\_\(\(\
\[LeftDoubleBracket]\)\(k\)\(\[RightDoubleBracket]\)\))\)\^2\/\(-p\_\(\(\
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\[IndentingNewLine]For[k = 1,
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\[IndentingNewLine]c\_\(\(\[LeftDoubleBracket]\)\(1\)\(\[RightDoubleBracket]\)\
\) = \(\(c\_\(\(\[LeftDoubleBracket]\)\(1\)\(\[RightDoubleBracket]\)\)\) z + \
\(c\_\(\(\[LeftDoubleBracket]\)\(m\)\(\[RightDoubleBracket]\)\)\) z\^m + \
\[Sum]\+\(i = 2\)\%\(m - 1\)\(c\_\(\(\[LeftDoubleBracket]\)\(i\)\(\
\[RightDoubleBracket]\)\)\) \((z\^i + z\^\(2 m - i\))\)\)\/\(z - z\^\(2 m - \
1\)\); \[IndentingNewLine]For[i = 2, i \[LessEqual] m, \(i++\),
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\(c\_\(\(\[LeftDoubleBracket]\)\(i - 1\)\(\[RightDoubleBracket]\)\)\)
z]; \[IndentingNewLine]c\_\(\(\[LeftDoubleBracket]\)\(m\)\(\
\[RightDoubleBracket]\)\) = \(\(-z\) \((c\_\(\(\[LeftDoubleBracket]\)\(m\)\(\
\[RightDoubleBracket]\)\) + \(c\_\(\(\[LeftDoubleBracket]\)\(m - 1\)\(\
\[RightDoubleBracket]\)\)\) z)\)\)\/\(1 - z\^2\); \[IndentingNewLine]For[
i = m - 1, 1 \[LessEqual] i, \(i--\),
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z \((c\_\(\(\[LeftDoubleBracket]\)\(i + 1\)\(\
\[RightDoubleBracket]\)\) -
c\_\(\(\[LeftDoubleBracket]\)\(i\)\(\
\[RightDoubleBracket]\)\))\)]]; \[IndentingNewLine]Return[c]]]\)], "Input",
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}, Closed]],
Cell[CellGroupData[{
Cell["5. Interpolation Evaluation", "Subsubsection"],
Cell[TextData[{
"Given a real argument ",
StyleBox["x", "Input"],
", a non-empty list ",
StyleBox["c", "Input"],
" of real-valued interpolation coefficients, and given its corresponding \
positive or null integer B-spline degree ",
StyleBox["n", "Input"],
", this routine evaluates the interpolated function ",
StyleBox["s", "Input"],
" (out of which ",
StyleBox["c", "Input"],
" has been computed) for the argument ",
StyleBox["x", "Input"],
". The interpolation scheme assumes that ",
StyleBox["c", "Input"],
" can be extended by double-mirroring."
}], "Text"],
Cell[BoxData[
\(BSplineInterpolate[x_, c_,
n_] := \[IndentingNewLine]Module[{k, i, m = Length[c],
s = 0}, \[IndentingNewLine]i = \[LeftFloor]x +
Mod[n + 1, 2]\/2\[RightFloor] -
Quotient[n,
2]; \[IndentingNewLine]\[Sum]\+\(k = 0\)\%n\(
c\_\(\(\[LeftDoubleBracket]\)\(MirroredIndex[i + k, 1,
m]\)\(\[RightDoubleBracket]\)\)\)
BSpline[x - i - k, n]]\)], "Input",
InitializationCell->True]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell["Examples", "Section"],
Cell[CellGroupData[{
Cell["Choose an arbitrary signal", "Text"],
Cell[BoxData[
\(s = {1, 3, \(-1\), 5, 7, 2, 4, 0, 6, 8}\)], "Input"],
Cell["\<\
Plot a graphical representation of the data we want to \
interpolate\
\>", "Text"],
Cell[BoxData[{
\(g =
ListPlot[s,
DisplayFunction \[Rule] Identity]\), "\[IndentingNewLine]",
\(Show[g, Prolog \[Rule] AbsolutePointSize[5],
DisplayFunction \[Rule] $DisplayFunction,
PlotRange \[Rule] {{0, 11}, {\(-2\), 10}}]\)}], "Input"]
}, Closed]],
Cell[CellGroupData[{
Cell[TextData[{
"Compute the B-spline interpolation coefficients for several B-spline \
degrees. Observe that ",
StyleBox["s=c0=c1", "Input"],
"; also, observe that ",
StyleBox["s\[NotEqual]c", "Input"],
Cell[BoxData[
\(TraditionalForm\`n\)]],
" for ",
Cell[BoxData[
\(TraditionalForm\`n > 1\)]]
}], "Text"],
Cell[BoxData[{
\(c0 = BSplineCoefficients[s, 0]\), "\[IndentingNewLine]",
\(c1 = BSplineCoefficients[s, 1]\), "\[IndentingNewLine]",
\(c2 = BSplineCoefficients[s, 2]\), "\[IndentingNewLine]",
\(c3 = BSplineCoefficients[s, 3]\), "\[IndentingNewLine]",
\(c4 = BSplineCoefficients[s, 4]\), "\[IndentingNewLine]",
\(c5 = BSplineCoefficients[s, 5]\)}], "Input"]
}, Closed]],
Cell[CellGroupData[{
Cell["\<\
Check that the interpolated functions reproduce the samples \
exactly, up to numerical inaccuracies\
\>", "Text"],
Cell[BoxData[{
\(\(s0 =
Table[BSplineInterpolate[x, c0, 0], {x, 1,
Length[c0]}];\)\), "\[IndentingNewLine]",
\(\(s1 =
Table[BSplineInterpolate[x, c1, 1], {x, 1,
Length[c1]}];\)\), "\[IndentingNewLine]",
\(\(s2 =
Table[BSplineInterpolate[x, c2, 2], {x, 1,
Length[c2]}];\)\), "\[IndentingNewLine]",
\(\(s3 =
Table[BSplineInterpolate[x, c3, 3], {x, 1,
Length[c3]}];\)\), "\[IndentingNewLine]",
\(\(s4 =
Table[BSplineInterpolate[x, c4, 4], {x, 1,
Length[c4]}];\)\), "\[IndentingNewLine]",
\(\(s5 =
Table[BSplineInterpolate[x, c5, 5], {x, 1,
Length[c5]}];\)\), "\[IndentingNewLine]",
\(Sqrt[\((s - s0)\) . \((s - s0)\)/\((s .
s)\)]\), "\[IndentingNewLine]",
\(Sqrt[\((s - s1)\) . \((s - s1)\)/\((s .
s)\)]\), "\[IndentingNewLine]",
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s)\)]\), "\[IndentingNewLine]",
\(Sqrt[\((s - s3)\) . \((s - s3)\)/\((s .
s)\)]\), "\[IndentingNewLine]",
\(Sqrt[\((s - s4)\) . \((s - s4)\)/\((s .
s)\)]\), "\[IndentingNewLine]",
\(Sqrt[\((s - s5)\) . \((s - s5)\)/\((s . s)\)]\)}], "Input"]
}, Closed]],
Cell[CellGroupData[{
Cell[TextData[{
"Plot the interpolated functions of varying degrees, put them all on the \
same graph. Observe the symmetry around ",
Cell[BoxData[
\(TraditionalForm\`x = 1\)]],
" and around ",
Cell[BoxData[
\(TraditionalForm\`x = m\)]]
}], "Text"],
Cell[BoxData[{
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Table[c =
BSplineCoefficients[s, n]; {Plot[
BSplineInterpolate[x, c, n], {x, 0, Length[c] + 1},
DisplayFunction \[Rule] Identity],
ListPlot[s, DisplayFunction \[Rule] Identity]}, {n, 0,
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PlotRange \[Rule] {{0, 11}, {\(-2\), 10}}, Frame \[Rule] True,
FrameLabel \[Rule] {"\<\>", "\<\>", "\"
PaddedForm[5, 2], "\<\>"}, Axes \[Rule] None,
DisplayFunction \[Rule] $DisplayFunction]\)}], "Input"]
}, Closed]],
Cell[CellGroupData[{
Cell["\<\
Plot the interpolated functions of varying degrees, put them on \
different graphs (you can animate them, if you want)\
\>", "Text"],
Cell[BoxData[
\(Do[Show[h[\([k]\)], Prolog \[Rule] AbsolutePointSize[5],
PlotRange \[Rule] {{0, 11}, {\(-2\), 10}}, Frame \[Rule] True,
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}, Closed]],
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]
*)
(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)