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Notebook[{
Cell[CellGroupData[{
Cell["Nonseparable 2D Wavelet Filters", "Title"],
Cell["Susumu Sakakibara", "Author"],
Cell["\<\
School of Infromation Environment, Tokyo Denki University, \
Muzaigakuendai 2-1200, Inzai-city, Chiba-ken, Japan 270-1382
susumu@sie.dendai.ac.jp\
\>", "TextAboutAuthor"],
Cell[TextData[{
"In order to improve nonisotropic nature of the conventional tensor DWT on \
images [1], we have constructed new 2D nonseparable biorthogonal wavelets \
[3,4]. Filters are defined on the regular triangular Bravais lattice, which \
are constructed by lifting [2] polyphase components in a symmetrical \
arrangement. A nobel feature of our construction is that there are four \
independent sulattices, which corresponds to the even and odd indexed phases \
in the 1D polyphase representation. Correspondingly, there are one LP filter \
and three HP filters. We show some details of construction of the linear \
prediction triangular filters, and application to a simple image data. ",
Cell[BoxData[
\(TraditionalForm\`L\^1\)]],
"norm is evenly distributed over three detail components, implying that the \
isotropy of images are well respected."
}], "Abstract"],
Cell[BoxData[{
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\(Needs["\"]\)}], "Input",
CellLabel->"In[1]:="],
Cell[CellGroupData[{
Cell["Discrete Wavelet Transform (DWT)", "SectionFirst"],
Cell[TextData[{
"A single decomposition step of DWT of 1D data with size ",
Cell[BoxData[
\(TraditionalForm\`2\^j\)]],
" is realized by the following Mallat matrix:"
}], "Text"],
Cell[BoxData[
\(MallatMatrix[j_, s_] :=
Join[Table[
h[Mod[k - 2 l, 2\^j] - s], {l, 0, 2\^\(j - 1\) - 1}, {k, s,
2\^j - 1 + s}],
Table[g[Mod[k - 2 l, 2\^j]], {l, 0, 2\^\(j - 1\) - 1}, {k, 0,
2\^j - 1}]]; MallatMatrix[j_] := MallatMatrix[j, 0]\)], "Input",
CellLabel->"In[6]:="],
Cell[TextData[{
"which performs a filtering followed by downsampling with LP filter ",
StyleBox["h", "Input"],
" and HP filter ",
StyleBox["g", "Input"],
"."
}], "Text"],
Cell[TextData[{
"For example, consider DWT of the data of size ",
Cell[BoxData[
\(TraditionalForm\`\(\(2\^3\)\(\ \)\)\)]]
}], "Text"],
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CellLabel->"In[7]:="],
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}, Open ]],
Cell["with the Haar filters", "Text"],
Cell[BoxData[
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CellLabel->"In[8]:="],
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CellLabel->"In[9]:="],
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FontSize->9]
}, Open ]],
Cell["\<\
By multiplying this Matrix to the data vector yiels the DWT \
decompotion\
\>", "Text"],
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CellLabel->"In[10]:="],
Cell[BoxData[
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a[1]\/\@2, \(-\(a[2]\/\@2\)\) + a[3]\/\@2, \(-\(a[4]\/\@2\)\) +
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}, Open ]],
Cell[TextData[{
"consisting of coarse and detail components of size ",
Cell[BoxData[
\(TraditionalForm\`2\^2 = \ 4\)]],
", respectively. The inverse transform, reconstruction, is of course \
realized by the inverse matrix"
}], "Text"],
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Cell["For 2D image data", "Text"],
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Cell["the DWT is usually carried out in the tensor product form, ", "Text"],
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}, Closed]]
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Cell[CellGroupData[{
Cell["Lifting scheme", "SectionFirst"],
Cell["\<\
In the lifting scheme, the DWT decomposition is carried out by the \
following procedure. First, the data is decomposed into even and odd indexed \
samples, the step called the Lazy transform.\
\>", "Text"],
Cell[CellGroupData[{
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"The ",
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" component is subtracted by the predictor, which is assumed to be ",
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"."
}], "Text"],
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Cell[TextData[{
"so that the total energy is the same as the original signal ",
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"."
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Cell["\<\
In the lifting scheme, the linear prediction filters are obtained \
by modifying the predicter and updator as \
\>", "Text"],
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\(\({c, d} = Transpose[Partition[simpleData1D, 2]];\)\)], "Input",
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\(c = c + \((d + RotateRight[d])\)/4\)], "Input",
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Cell[BoxData[
\({a[0] +
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Cell[BoxData[
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"This is the decomposed coarse ",
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" component, by the linear predition filter, usually called CDF(2,2).",
"\n",
"By replacing ",
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Cell[BoxData[
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Cell[BoxData[
\(TraditionalForm\`\(g\&~\)\)]],
"."
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Cell[CellGroupData[{
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\(h\&^[x_] =
c\[LeftDoubleBracket]2\[RightDoubleBracket] /. {a[n_] \[Rule]
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Cell["\<\
In order to construct our filters on the regular triangular \
lattice, let us define the primitive translation vectors\
\>", "Text"],
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CellLabel->"In[28]:="],
Cell[TextData[{
"which generate the trianglar lattice called the Bravais lattice in two \
dimension. The 2 dimensional signal ",
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\>"],
ImageRangeCache->{{{185, 402.562}, {700.375, 471.125}} -> {-8.39287, \
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}, Open ]],
Cell["\<\
It is crucial to observe that the lattice sites may be devided into \
four independent sublattices, each of which is obtained by shifting zero or \
one unit of primitive translation vectors followed by scaling the original \
lattice by factor of two. To show this, we define the function that splits \
the lattice sites into the four sublattices.\
\>", "Text"],
Cell[BoxData[
\(Lazy[mat_] :=
Module[{size},
size = Length[mat]; {Take[mat, {1, size, 2}, {1, size, 2}],
Take[mat, {1, size, 2}, {2, size, 2}],
Take[RotateRight[mat], {1, size, 2}, {1, size, 2}],
Take[RotateLeft[Map[RotateRight, mat]], {1, size, 2}, {1, size,
2}]}]\)], "Input",
CellLabel->"In[33]:="],
Cell["\<\
It is easier to understand the four sublattices by plotting them in \
different colors.\
\>", "Text"],
Cell[BoxData[
\(\(Module[{lazyLattice}, lazyLattice = Lazy[plattice];
allSites = Graphics[{GrayLevel[ .3], PointSize[0.035], plattice}];
sites0 = Graphics[{PointSize[0.02], lazyLattice[\([1]\)]}];
sites1 =
Graphics[{PointSize[0.02], RGBColor[1, 0, 0],
lazyLattice[\([2]\)]}];
sites2 =
Graphics[{PointSize[0.02], RGBColor[0, 1, 0],
lazyLattice[\([3]\)]}];
sites3 =
Graphics[{PointSize[0.02], RGBColor[0, 0, 1],
lazyLattice[\([4]\)]}];];\)\)], "Input",
CellLabel->"In[34]:="],
Cell[CellGroupData[{
Cell[BoxData[
\(\(Show[{allSites, sites0\ (*\ Gray\ *) , sites1\ (*\ Red\ *) ,
sites2 (*\ Green\ *) , sites3\ (*\ Blue\ *) },
AspectRatio \[Rule] Automatic, Axes \[Rule] Automatic];\)\)], "Input",\
CellLabel->"In[35]:="],
Cell[GraphicsData["PostScript", "\<\
%!
%%Creator: Mathematica
%%AspectRatio: .57735
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
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%%IncludeFont: Courier
/Courier findfont 10 scalefont setfont
% Scaling calculations
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[.74943 .2369 3 0 ]
[.93084 .2369 -3 -9 ]
[.93084 .2369 3 0 ]
[.55553 .06799 -12 -4.5 ]
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[.55553 .1587 -12 -4.5 ]
[.55553 .1587 0 4.5 ]
[.55553 .3401 -6 -4.5 ]
[.55553 .3401 0 4.5 ]
[.55553 .43081 -6 -4.5 ]
[.55553 .43081 0 4.5 ]
[.55553 .52151 -6 -4.5 ]
[.55553 .52151 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .57735 0 0 ]
] MathScale
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StyleBox["t[1]", "Input"],
" shifted sublattice (red dots), the ",
StyleBox["t[2]", "Input"],
" shifted sublattice (green dots), the ",
StyleBox["t[3]", "Input"],
" shifted sublattice (blue dots) add up to form the original lattice (gray \
circles). These sublattices correspond to the even and odd phases of 1D data \
vectors."
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Cell["Triangular Filter", "SectionFirst"],
Cell["\<\
In order to extend the same steps as in the 1D linear prediction \
filters, let us define the functions that shift data in three \
directions.\
\>", "Text"],
Cell[BoxData[{
\(\(\(Shift[0]\)[m_] := m;\)\), "\[IndentingNewLine]",
\(\(\(Shift[1]\)[m_] := Map[RotateRight, m];\)\), "\[IndentingNewLine]",
\(\(\(Shift[\(-1\)]\)[m_] :=
Map[RotateLeft, m];\)\), "\[IndentingNewLine]",
\(\(\(Shift[2]\)[m_] := RotateLeft[m];\)\), "\[IndentingNewLine]",
\(\(\(Shift[\(-2\)]\)[m_] := RotateRight[m];\)\), "\[IndentingNewLine]",
\(\(\(Shift[3]\)[m_] :=
RotateRight[Map[RotateLeft, m]];\)\), "\[IndentingNewLine]",
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RotateLeft[Map[RotateRight, m]];\)\)}], "Input",
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Cell["\<\
The triangular decomposition of the 2D data is carried out by the \
function\
\>", "Text"],
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\(TriTwistDWT[data2D_] :=
Module[{c0, d1, d2, d3, td1, td2, td3}, {c0, td1, td2, td3} =
Lazy[data2D]; \[IndentingNewLine]d1 = \(-td1\) + td2 + td3;
d2 = td1 - td2 + td3; d3 = td1 + td2 - td3;
d1 = d1 - \((c0 + \(Shift[\(-1\)]\)[c0])\)/2;
d2 = d2 - \((c0 + \(Shift[\(-2\)]\)[c0])\)/2;
d3 = d3 - \((c0 + \(Shift[\(-3\)]\)[c0])\)/2;
c0 = c0 + \((d1 + \(Shift[1]\)[d1] + d2 + \(Shift[2]\)[d2] +
d3 + \(Shift[3]\)[d3])\)/8; {2 c0, d1/2, d2/2, d3/2} //
Expand]\)], "Input",
CellLabel->"In[43]:="],
Cell[TextData[{
"which is essentially a straightforward generalization of the 1D DWT, \
except for the twist of three detail components in the third line of the \
above definition. Without the twist, we would get ",
StyleBox["TriStraightDWT", "Input"],
", but in this paper I would like to confine myself to the twisted version. \
As always, the inverse transform is given by tracing the same operations \
backward."
}], "Text"],
Cell[TextData[{
"The data on the lattice is decomposed by the function ",
StyleBox["TriTwistDWT", "Input"],
"."
}], "Text"],
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CellLabel->"In[44]:="],
Cell[TextData[{
"which consists of four components, LP and three HP components. When the \
data ",
StyleBox["c[n, m]", "Input"],
" are replaced by exponetials, the four components yields the transfer \
functions of LP and three HP filters. There are four LP transfer functions in \
the first element, which are all equivalent due to periodicity. By choosing \
an appropriate one, which turns out to be the (3,3) element here, they are"
}], "Text"],
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}, Open ]],
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The array of filter coefficients on the lattice can by visulized as\
\
\>", "Text"],
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coef = Transpose[{coef,
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expl /. {x \[Rule] 0, y \[Rule] 1}}]}];
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