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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 195594, 4905]*) (*NotebookOutlinePosition[ 219735, 5776]*) (* CellTagsIndexPosition[ 219691, 5772]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Routines", "Subsubtitle"], Cell[CellGroupData[{ Cell["Filter Bank Routines", "Section"], Cell[BoxData[{ \(Off[General::"\"]\), "\n", \(Off[General::"\"]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[{ \(Clear[odd, even, knit]\), "\n", \(odd[y_] := \(Transpose[Partition[y, 2]]\)[\([1]\)]\), "\n", \(even[y_] := \(Transpose[Partition[y, 2]]\)[\([2]\)]\), "\n", \(knit[{o_, e_}] := Flatten[Transpose[{o, e}]]\n\), "\n", \(Clear[dec, rec]\), "\[IndentingNewLine]", \(\(dec[cf_List]\)[data_List] := Module[{k, y, dodd, deven}, deven = even[data]; dodd = odd[data]; y = 0; Do[ y += cf\[LeftDoubleBracket]2\ k + 1\[RightDoubleBracket]\ dodd + cf\[LeftDoubleBracket]2\ k + 2\[RightDoubleBracket]\ deven; deven = RotateLeft[deven]; dodd = RotateLeft[dodd], {k, 0, Length[cf]\/2 - 1}]; \[IndentingNewLine]If[OddQ[Length[cf]], y += Last[cf] dodd]; \ y]\), "\n", \(\(rec[cf_List]\)[data_List] := Module[{k, y, d}, d = data; y = 0; Do[y += {{cf\[LeftDoubleBracket]2\ k + 1\[RightDoubleBracket]}, {cf\[LeftDoubleBracket]2\ k + 2\[RightDoubleBracket]}} . {d}; d = RotateRight[d], {k, 0, Length[cf]\/2 - 1}]; \[IndentingNewLine]If[OddQ[Length[cf]], y += {{Last[cf]}, {0}} . {d}]; \[IndentingNewLine]Flatten[ Transpose[y]]]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Wavelet Routines", "Section"], Cell["1D Single level routines", "Text"], Cell[BoxData[{ \(\(Wave[{cf1_, cf2_}]\)[data_] := Join[\(dec[cf1]\)[data], \(dec[cf2]\)[data]]\), "\n", \(\(invWave[{cf1_, cf2_}]\)[cd_] := Block[{c, d}, \[IndentingNewLine]{c, d} = Partition[cd, Length[cd]/2]; \[IndentingNewLine]\(rec[cf1]\)[ c] + \(rec[cf2]\)[d]]\)}], "Input"], Cell["2D Single level routines", "Text"], Cell[BoxData[{ \(getC[cd_, level_] := Take[cd, Length[cd]/2^level]\), "\[IndentingNewLine]", \(getCD[cd_, level_] := getC[cd, level - 1]\), "\[IndentingNewLine]", \(dropC[cd_, level_] := Drop[cd, Length[cd]/2^level]\), "\[IndentingNewLine]", \(dropCD[cd_, level_] := dropC[cd, level - 1]\), "\n", \(\(Wave[cf_]\)[cd_, level_] := Join[\(Wave[cf]\)[getC[cd, level]], dropC[cd, level]]\), "\[IndentingNewLine]", \(\(invWave[cf_]\)[cd_, level_] := Join[\(invWave[cf]\)[getCD[cd, level]], dropCD[cd, level]]\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(Wave2D[cf_]\)[cd_, level_] := Transpose[ Map[\(Wave[cf]\)[#, level] &, Transpose[ Map[\(Wave[cf]\)[#, level] &, cd]]]]\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(invWave2D[cf_]\)[cd_, level_] := Transpose[ Map[\(invWave[cf]\)[#, level] &, Transpose[Map[\(invWave[cf]\)[#, level] &, cd]]]]\)}], "Input"], Cell["Multilevel Routines", "Text"], Cell[BoxData[{ \(\(WT[cf_]\)[cd_, oldlevel_, newlevel_] := Module[{n, newcd}, \n\t\tnewcd = cd; \n\t\tWhich[oldlevel < newlevel, \ Do[\ newcd = \(Wave[cf]\)[newcd, n], \ {n, oldlevel, newlevel - 1}], \n\t\t\tnewlevel < oldlevel, \ Do[\ newcd = \(invWave[cf]\)[newcd, n], \ {n, oldlevel, newlevel + 1, \(-1\)}]]; \n\t\tnewcd]\), "\[IndentingNewLine]", \(\(WT[cf_, cft_]\)[cd_, oldlevel_, newlevel_] := Which[oldlevel < newlevel, \[IndentingNewLine]\(WT[cf]\)[cd, oldlevel, newlevel], newlevel < oldlevel, \(WT[cft]\)[cd, oldlevel, newlevel], \[IndentingNewLine]oldlevel == newlevel, cd]\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(WT2D[cf_]\)[cd_, oldlevel_, newlevel_] := Module[{n, newcd}, \n\t\tnewcd = cd; \n\t\tWhich[oldlevel < newlevel, \ Do[\ newcd = \(Wave2D[cf]\)[newcd, n], \ {n, oldlevel, newlevel - 1}], \n\t\t\tnewlevel < oldlevel, \ Do[\ newcd = \(invWave2D[cf]\)[newcd, n], \ {n, oldlevel, newlevel + 1, \(-1\)}]]; \n\t\tnewcd]\), "\[IndentingNewLine]", \(\(WT2D[cf_, cft_]\)[cd_, oldlevel_, newlevel_] := Which[oldlevel < newlevel, \[IndentingNewLine]\(WT2D[cf]\)[cd, oldlevel, newlevel], newlevel < oldlevel, \(WT2D[cft]\)[cd, oldlevel, newlevel], \[IndentingNewLine]oldlevel == newlevel, cd]\[IndentingNewLine]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Auxiliary Routines", "Section"], Cell["Partition into different levels 1D", "Text"], Cell[BoxData[ \(separate[cf_, n_] := \[IndentingNewLine]Block[{m, cd}, \ \[IndentingNewLine]\t If[IntegerQ[ Length[cf]/ 2^n], \[IndentingNewLine]\t\tcd = {cf}; \ \[IndentingNewLine]\t\ \tm = Length[cf]/2; \[IndentingNewLine]\t\tDo[ cd = Join[Partition[cd[\([1]\)], m], Drop[cd, 1]]; m = m/2, {n}]; \[IndentingNewLine]\t\tcd, \[IndentingNewLine]\t Print["\" n]]]\)], "Input"], Cell["2D partition", "Text"], Cell[BoxData[{ \(block[a_, {m1_, m2_}, {n1_, n2_}] := Transpose[ Take[Transpose[Take[a, {m1, m2}]], {n1, n2}]]\[IndentingNewLine]\), "\n", \(blockpartition[a_] := Block[{mrow, mcol}, \[IndentingNewLine]{mrow, mcol} = Dimensions[a]/2; \[IndentingNewLine]{block[ a, {1, mrow}, {1, mcol}], {block[ a, {1, mrow}, {mcol + 1, 2 mcol}], \[IndentingNewLine]block[ a, {mrow + 1, 2 mrow}, {1, mcol}], block[a, {mrow + 1, 2 mrow}, {mcol + 1, 2 mcol}]}}]\[IndentingNewLine]\), "\n", \(separate2D[cf_, n_] := \[IndentingNewLine]Block[{cd}, \ \[IndentingNewLine]\t\tcd = \ {cf}; \ \[IndentingNewLine]\t\t{mrow, mcol} = Dimensions[cf]/ 2; \[IndentingNewLine]\t\tDo[\(cd = Join[blockpartition[cd[\([1]\)]], Drop[cd, 1]];\), {n}]; \[IndentingNewLine]\t\tcd]\t\)}], "Input"], Cell[BoxData[{ \(normSq[cd_] := Flatten[cd] . Flatten[cd]\), "\n", \(norm[cd_] := Sqrt[\ normSq[cd]]\), "\n", \(ignoreFirstMap[f_, cd_] := Join[{cd[\([1]\)]}, Map[f, Drop[cd, 1], {2}]]\), "\[IndentingNewLine]", \(dMap[f_, cd_, n_] := Flatten[ignoreFirstMap[f, separate[cd, n]]]\), "\n", \(onezero[a_] := If[a == 0, 0, 1]\), "\n", \(SetAttributes[onezero, Listable]\), "\n", \(CountNonZero[a_List] := Module[{b}, b = onezero[Flatten[Chop[a]]]; \ b . b]\)}], "Input"], Cell[BoxData[ \(ShowImage[data_] := Show[Graphics[Raster[data/Max[data]]], AspectRatio \[Rule] Automatic]\)], "Input"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Filter Coefficients", "Subsubtitle"], Cell[CellGroupData[{ Cell["Orthogonal Filters", "Section"], Cell[BoxData[{ \(H2 = {{1\/\@2, 1\/\@2}, {\(-\(1\/\@2\)\), 1\/\@2}} // N; h = {1 + \@3, 3 + \@3, 3 - \@3, 1 - \@3}\/\(4\ \@2\) // N;\), "\n", \(\(g = Table[\((\(-1\))\)\^k\ h\[LeftDoubleBracket]4 - k\[RightDoubleBracket], {k, 0, 3}];\)\), "\n", \(\(D4 = {h, g};\)\n\)}], "Input"], Cell[BoxData[ \(Dau[ p_] := \(\(Dau[p]\)\(=\)\(Module[{Q, \ asol, \ R, h, L}, P[y_] = Sum[Binomial[p - 1 + k, k] y^k, {k, 0, p - 1}]; \n fac[aa_] = \((1 - aa*z)\)/\((1 - aa)\); \n Q[z_] = P[\((2 - z - 1/z)\)/4] // Expand; \n asol = NSolve[Q[z] == 0]; \na[k_] := z /. asol[\([k]\)]; \n R[z_] = \(\(Product[ If[Abs[a[k]] < 1, fac[a[k]], 1], {k, 1, Length[asol]}] // N\) // Expand\) // Chop; \n h = CoefficientList[ N[Sqrt[2]] \((\((z + 1)\)/2)\)^p\ R[z] // Expand, z]; \[IndentingNewLine]L = Length[h]; \[IndentingNewLine]{h, Table[\((\(-1\))\)\^k\ h\[LeftDoubleBracket] L - k\[RightDoubleBracket], {k, 0, L - 1}]}]\)\(\[IndentingNewLine]\)\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Biorthogonal Filters", "Section"], Cell[CellGroupData[{ Cell["FBI 9-7", "Subsection"], Cell[BoxData[{ \(\(FBIh7 = {0, \(-0.064538882628938438636933954159912`19.7033\), \ \(-0.040689417609558436723762363091178`18.5453\), 0.418092273222212200837356135209162`19.0788, 0.78848561640566439784836908831229`19.2056, 0.418092273222212200837356135209162`19.0788, \ \(-0.040689417609558436723762363091178`18.5453\), \ \(-0.064538882628938438636933954159912`19.7033\)};\)\), \ "\[IndentingNewLine]", \(\(FBIh9 = {0.037828455506995461393080045762034`18.939, \ \(-0.023849465019380001913171591052783`17.6371\), \ \(-0.1106244044184234088485556911`17.6003\), 0.3774028556126537641135926`17.7409, 0.8526986790094034193117981757`17.9686, 0.3774028556126537641135926`17.7409, \ \(-0.1106244044184234088485556911`17.6003\), \ \(-0.023849465019380001913171591052783`17.6371\), 0.037828455506995461393080045762034`18.939};\)\), "\ \[IndentingNewLine]", \(\(FBIg7 = {0, 0.037828455506995461393080045762034`18.939, 0.023849465019380001913171591052783`17.6371, \ \(-0.1106244044184234088485556911`17.6003\), \ \(-0.3774028556126537641135926`17.7409\), 0.8526986790094034193117981757`17.9686, \ \(-0.3774028556126537641135926`17.7409\), \ \(-0.1106244044184234088485556911`17.6003\), 0.023849465019380001913171591052783`17.6371, 0.037828455506995461393080045762034`18.939};\)\), "\ \[IndentingNewLine]", \(\(FBIg9 = {0, 0, 0.064538882628938438636933954159912`19.7033, \ \(-0.040689417609558436723762363091178`18.5453\), \ \(-0.418092273222212200837356135209162`19.0788\), 0.78848561640566439784836908831229`19.2056, \ \(-0.418092273222212200837356135209162`19.0788\), \ \(-0.040689417609558436723762363091178`18.5453\), 0.064538882628938438636933954159912`19.7033};\)\), "\ \[IndentingNewLine]", \(\(FBI7 = {FBIh7, FBIg7};\)\), "\[IndentingNewLine]", \(\(FBI9 = {FBIh9, FBIg9};\)\)}], "Input"] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Examples", "Subsubtitle"], Cell[CellGroupData[{ Cell["Use of Wavelet Routines", "Section"], Cell[CellGroupData[{ Cell["Single level", "Subsection"], Cell[CellGroupData[{ Cell["-Orthogonal filters", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[{ \(wt0 = Range[32]\), "\[IndentingNewLine]", \(wt1 = \(Wave[D4]\)[wt0]\), "\[IndentingNewLine]", 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.41661 Mdot .14472 .41819 Mdot .14658 .41976 Mdot .14844 .42133 Mdot .1503 .4229 Mdot .15216 .42446 Mdot .15402 .42601 Mdot .15588 .42757 Mdot .15774 .42912 Mdot .1596 .43066 Mdot .16146 .4322 Mdot .16332 .43373 Mdot .16518 .43527 Mdot .16704 .43679 Mdot .1689 .43831 Mdot .17076 .43983 Mdot .17262 .44134 Mdot .17448 .44285 Mdot .17634 .44435 Mdot .1782 .44585 Mdot .18006 .44734 Mdot .18192 .44882 Mdot .18378 .45031 Mdot .18564 .45178 Mdot .1875 .45325 Mdot .18936 .45472 Mdot .19122 .45618 Mdot .19308 .45763 Mdot .19494 .45908 Mdot .1968 .46053 Mdot .19866 .46196 Mdot .20052 .4634 Mdot .20238 .46482 Mdot .20424 .46625 Mdot .2061 .46766 Mdot .20796 .46907 Mdot .20982 .47047 Mdot .21168 .47187 Mdot .21354 .47326 Mdot .2154 .47465 Mdot .21726 .47603 Mdot .21912 .4774 Mdot .22098 .47877 Mdot .22284 .48013 Mdot .2247 .48148 Mdot .22656 .48283 Mdot .22842 .48417 Mdot .23028 .48551 Mdot .23214 .48683 Mdot .234 .48816 Mdot .23586 .48947 Mdot .23772 .49078 Mdot .23958 .49208 Mdot .24144 .49338 Mdot .2433 .49466 Mdot .24516 .49595 Mdot .24702 .49722 Mdot .24888 .49849 Mdot .25074 .49975 Mdot .2526 .501 Mdot .25446 .50225 Mdot .25632 .50349 Mdot .25818 .50472 Mdot .26004 .50594 Mdot .2619 .50716 Mdot .26376 .50837 Mdot .26563 .50957 Mdot .26749 .51077 Mdot .26935 .51195 Mdot .27121 .51313 Mdot .27307 .51431 Mdot .27493 .51547 Mdot .27679 .51663 Mdot .27865 .51778 Mdot .28051 .51892 Mdot .28237 .52005 Mdot .28423 .52118 Mdot .28609 .5223 Mdot .28795 .52341 Mdot .28981 .52451 Mdot .29167 .5256 Mdot .29353 .52669 Mdot .29539 .31181 Mdot .29725 .09478 Mdot .29911 .09372 Mdot .30097 .09267 Mdot .30283 .09162 Mdot .30469 .09059 Mdot .30655 .08956 Mdot .30841 .08854 Mdot .31027 .08753 Mdot .31213 .08652 Mdot .31399 .08553 Mdot .31585 .08454 Mdot .31771 .08356 Mdot .31957 .08259 Mdot .32143 .08163 Mdot .32329 .08068 Mdot .32515 .07974 Mdot .32701 .0788 Mdot .32887 .07788 Mdot .33073 .07696 Mdot .33259 .07605 Mdot .33445 .07515 Mdot .33631 .07426 Mdot .33817 .07338 Mdot .34003 .07251 Mdot .34189 .07165 Mdot .34375 .07079 Mdot .34561 .06995 Mdot .34747 .06911 Mdot .34933 .06828 Mdot .35119 .06747 Mdot .35305 .06666 Mdot .35491 .06586 Mdot .35677 .06507 Mdot .35863 .06429 Mdot .36049 .06352 Mdot .36235 .06275 Mdot .36421 .062 Mdot .36607 .06126 Mdot .36793 .06052 Mdot .36979 .0598 Mdot .37165 .05908 Mdot .37351 .05838 Mdot .37537 .05768 Mdot .37723 .057 Mdot .37909 .05632 Mdot .38095 .05565 Mdot .38281 .05499 Mdot .38467 .05435 Mdot .38653 .05371 Mdot .38839 .05308 Mdot .39025 .05246 Mdot .39211 .05185 Mdot .39397 .05125 Mdot .39583 .05066 Mdot .39769 .05008 Mdot .39955 .04951 Mdot .40141 .04895 Mdot .40327 .0484 Mdot .40513 .04786 Mdot .40699 .04733 Mdot .40885 .04681 Mdot .41071 .0463 Mdot .41257 .0458 Mdot .41443 .04531 Mdot .41629 .04483 Mdot .41815 .04436 Mdot .42001 .0439 Mdot .42188 .04345 Mdot .42374 .04301 Mdot .4256 .04258 Mdot .42746 .31181 Mdot .42932 .58186 Mdot .43118 .58226 Mdot .43304 .58265 Mdot .4349 .58303 Mdot .43676 .5834 Mdot .43862 .58376 Mdot .44048 .58411 Mdot .44234 .58444 Mdot .4442 .58477 Mdot .44606 .58509 Mdot .44792 .5854 Mdot .44978 .58569 Mdot .45164 .58598 Mdot .4535 .58626 Mdot .45536 .58652 Mdot .45722 .58678 Mdot .45908 .58702 Mdot .46094 .58726 Mdot .4628 .58748 Mdot .46466 .5877 Mdot .46652 .5879 Mdot .46838 .58809 Mdot .47024 .58828 Mdot .4721 .58845 Mdot .47396 .58861 Mdot .47582 .58876 Mdot .47768 .5889 Mdot .47954 .58903 Mdot .4814 .58915 Mdot .48326 .58926 Mdot .48512 .58936 Mdot .48698 .58945 Mdot .48884 .58953 Mdot .4907 .5896 Mdot .49256 .58966 Mdot .49442 .5897 Mdot .49628 .58974 Mdot .49814 .58977 Mdot .5 .58978 Mdot .50186 .58979 Mdot .50372 .58978 Mdot .50558 .58977 Mdot .50744 .58974 Mdot .5093 .5897 Mdot .51116 .58966 Mdot .51302 .5896 Mdot .51488 .58953 Mdot .51674 .58945 Mdot .5186 .58936 Mdot .52046 .58926 Mdot .52232 .58915 Mdot .52418 .31181 Mdot .52604 .03472 Mdot .5279 .03486 Mdot .52976 .03501 Mdot .53162 .03517 Mdot .53348 .03535 Mdot .53534 .03553 Mdot .5372 .03572 Mdot .53906 .03592 Mdot .54092 .03614 Mdot .54278 .03636 Mdot .54464 .0366 Mdot .5465 .03684 Mdot .54836 .0371 Mdot .55022 .03736 Mdot .55208 .03764 Mdot .55394 .03793 Mdot .5558 .03822 Mdot .55766 .03853 Mdot .55952 .03885 Mdot .56138 .03918 Mdot .56324 .03951 Mdot .5651 .03986 Mdot .56696 .04022 Mdot .56882 .04059 Mdot .57068 .04097 Mdot .57254 .04136 Mdot .5744 .04176 Mdot .57626 .04216 Mdot .57813 .04258 Mdot .57999 .04301 Mdot .58185 .04345 Mdot .58371 .0439 Mdot .58557 .04436 Mdot .58743 .04483 Mdot .58929 .04531 Mdot .59115 .0458 Mdot .59301 .0463 Mdot .59487 .04681 Mdot .59673 .04733 Mdot .59859 .04786 Mdot .60045 .0484 Mdot .60231 .04895 Mdot .60417 .04951 Mdot .60603 .05008 Mdot .60789 .05066 Mdot .60975 .05125 Mdot .61161 .05185 Mdot .61347 .05246 Mdot .61533 .05308 Mdot .61719 .05371 Mdot .61905 .05435 Mdot .62091 .05499 Mdot .62277 .05565 Mdot .62463 .05632 Mdot .62649 .057 Mdot .62835 .05768 Mdot .63021 .05838 Mdot .63207 .05908 Mdot .63393 .0598 Mdot .63579 .06052 Mdot .63765 .06126 Mdot .63951 .062 Mdot .64137 .06275 Mdot .64323 .06352 Mdot .64509 .06429 Mdot .64695 .06507 Mdot .64881 .06586 Mdot .65067 .06666 Mdot .65253 .06747 Mdot .65439 .06828 Mdot .65625 .06911 Mdot .65811 .06995 Mdot .65997 .07079 Mdot .66183 .07165 Mdot .66369 .07251 Mdot .66555 .07338 Mdot .66741 .07426 Mdot .66927 .07515 Mdot .67113 .07605 Mdot .67299 .07696 Mdot .67485 .07788 Mdot .67671 .0788 Mdot .67857 .07974 Mdot .68043 .08068 Mdot .68229 .08163 Mdot .68415 .08259 Mdot .68601 .08356 Mdot .68787 .08454 Mdot .68973 .08553 Mdot .69159 .08652 Mdot .69345 .08753 Mdot .69531 .08854 Mdot .69717 .08956 Mdot .69903 .09059 Mdot .70089 .09162 Mdot .70275 .09267 Mdot .70461 .09372 Mdot .70647 .09478 Mdot .70833 .31181 Mdot .71019 .52669 Mdot .71205 .5256 Mdot .71391 .52451 Mdot .71577 .52341 Mdot .71763 .5223 Mdot .71949 .52118 Mdot .72135 .52005 Mdot .72321 .51892 Mdot .72507 .51778 Mdot .72693 .51663 Mdot .72879 .51547 Mdot .73065 .51431 Mdot .73251 .51313 Mdot .73438 .51195 Mdot .73624 .51077 Mdot .7381 .50957 Mdot .73996 .50837 Mdot .74182 .50716 Mdot .74368 .50594 Mdot .74554 .50472 Mdot .7474 .50349 Mdot .74926 .50225 Mdot .75112 .501 Mdot .75298 .49975 Mdot .75484 .49849 Mdot .7567 .49722 Mdot .75856 .49595 Mdot .76042 .49466 Mdot .76228 .49338 Mdot .76414 .49208 Mdot .766 .49078 Mdot .76786 .48947 Mdot .76972 .48816 Mdot .77158 .48683 Mdot .77344 .48551 Mdot .7753 .48417 Mdot .77716 .48283 Mdot .77902 .48148 Mdot .78088 .48013 Mdot .78274 .47877 Mdot .7846 .4774 Mdot .78646 .47603 Mdot .78832 .47465 Mdot .79018 .47326 Mdot .79204 .47187 Mdot .7939 .47047 Mdot .79576 .46907 Mdot .79762 .46766 Mdot .79948 .46625 Mdot .80134 .46482 Mdot .8032 .4634 Mdot .80506 .46196 Mdot .80692 .46053 Mdot .80878 .45908 Mdot .81064 .45763 Mdot .8125 .45618 Mdot .81436 .45472 Mdot .81622 .45325 Mdot .81808 .45178 Mdot .81994 .45031 Mdot .8218 .44882 Mdot .82366 .44734 Mdot .82552 .44585 Mdot .82738 .44435 Mdot .82924 .44285 Mdot .8311 .44134 Mdot .83296 .43983 Mdot .83482 .43831 Mdot .83668 .43679 Mdot .83854 .43527 Mdot .8404 .43373 Mdot .84226 .4322 Mdot .84412 .43066 Mdot .84598 .42912 Mdot .84784 .42757 Mdot .8497 .42601 Mdot .85156 .42446 Mdot .85342 .4229 Mdot .85528 .42133 Mdot .85714 .41976 Mdot .859 .41819 Mdot .86086 .41661 Mdot .86272 .41503 Mdot .86458 .41344 Mdot .86644 .41185 Mdot .8683 .41026 Mdot .87016 .40866 Mdot .87202 .40706 Mdot .87388 .40546 Mdot .87574 .40385 Mdot .8776 .40224 Mdot .87946 .40062 Mdot .88132 .39901 Mdot .88318 .39738 Mdot .88504 .39576 Mdot .8869 .39413 Mdot .88876 .3925 Mdot .89063 .39087 Mdot .89249 .38923 Mdot .89435 .38759 Mdot .89621 .38595 Mdot .89807 .3843 Mdot .89993 .38266 Mdot .90179 .38101 Mdot .90365 .37935 Mdot .90551 .3777 Mdot .90737 .37604 Mdot .90923 .37438 Mdot .91109 .37272 Mdot .91295 .37105 Mdot .91481 .36938 Mdot .91667 .36771 Mdot .91853 .36604 Mdot .92039 .36437 Mdot .92225 .36269 Mdot .92411 .36101 Mdot .92597 .35933 Mdot .92783 .35765 Mdot .92969 .35597 Mdot .93155 .35428 Mdot .93341 .3526 Mdot .93527 .35091 Mdot .93713 .34922 Mdot .93899 .34753 Mdot .94085 .34584 Mdot .94271 .34414 Mdot .94457 .34245 Mdot .94643 .34075 Mdot .94829 .33906 Mdot .95015 .33736 Mdot .95201 .33566 Mdot .95387 .33396 Mdot .95573 .33226 Mdot .95759 .33056 Mdot .95945 .32886 Mdot .96131 .32715 Mdot .96317 .32545 Mdot .96503 .32375 Mdot .96689 .32204 Mdot .96875 .32034 Mdot .97061 .31863 Mdot .97247 .31693 Mdot .97433 .31522 Mdot .97619 .31352 Mdot .02567 .32021 Mdot .02753 .32566 Mdot .02939 .32851 Mdot .03125 .33205 Mdot .03311 .33301 Mdot .03497 .33466 Mdot .03683 .337 Mdot .03869 .33916 Mdot .04055 .33872 Mdot .04241 .33898 Mdot .04427 .33993 Mdot .04613 .3407 Mdot .04799 .34216 Mdot .04985 .34344 Mdot .05171 .34453 Mdot .05357 .34567 Mdot .05543 .34422 Mdot .05729 .34346 Mdot .05915 .3434 Mdot .06101 .34315 Mdot .06287 .34359 Mdot .06473 .34385 Mdot .06659 .34392 Mdot .06845 .34405 Mdot .07031 .34486 Mdot .07217 .34549 Mdot .07403 .34594 Mdot .07589 .34643 Mdot .07775 .34674 Mdot .07961 .3471 Mdot .08147 .34751 Mdot .08333 .34791 Mdot .08519 .35412 Mdot .08705 .35878 Mdot .08891 .36188 Mdot .09077 .3654 Mdot .09263 .36735 Mdot .09449 .36972 Mdot .09635 .37252 Mdot .09821 .3752 Mdot .10007 .37631 Mdot .10193 .37785 Mdot .10379 .37981 Mdot .10565 .38165 Mdot .10751 .38391 Mdot .10937 .38606 Mdot .11124 .3881 Mdot .1131 .39017 Mdot .11496 .39067 Mdot .11682 .3916 Mdot .11868 .39294 Mdot .12054 .39417 Mdot .1224 .39582 Mdot .12426 .39736 Mdot .12612 .39879 Mdot .12798 .40024 Mdot .12984 .40212 Mdot .1317 .40388 Mdot .13356 .40553 Mdot .13542 .40721 Mdot .13728 .40877 Mdot .13914 .41037 Mdot .141 .412 Mdot .14286 .41362 Mdot .14472 .41738 Mdot .14658 .42056 Mdot .14844 .42317 Mdot .1503 .42594 Mdot .15216 .42813 Mdot .15402 .43047 Mdot .15588 .43297 Mdot .15774 .43543 Mdot .1596 .43732 Mdot .16146 .43936 Mdot .16332 .44155 Mdot .16518 .4437 Mdot .16704 .446 Mdot .1689 .44827 Mdot .17076 .45049 Mdot .17262 .45272 Mdot .17448 .45438 Mdot .17634 .4562 Mdot .1782 .45817 Mdot .18006 .46009 Mdot .18192 .46217 Mdot .18378 .46421 Mdot .18564 .46621 Mdot .1875 .46822 Mdot .18936 .47038 Mdot .19122 .4725 Mdot .19308 .47458 Mdot .19494 .47667 Mdot .1968 .47872 Mdot .19866 .48078 Mdot .20052 .48285 Mdot .20238 .48492 Mdot .20424 .48095 Mdot .2061 .4786 Mdot .20796 .47786 Mdot .20982 .4767 Mdot .21168 .47715 Mdot .21354 .47716 Mdot .2154 .47674 Mdot .21726 .47644 Mdot .21912 .47776 Mdot .22098 .47865 Mdot .22284 .4791 Mdot .2247 .47966 Mdot .22656 .4798 Mdot .22842 .48004 Mdot .23028 .48041 Mdot .23214 .48074 Mdot .234 .4827 Mdot .23586 .48422 Mdot .23772 .4853 Mdot .23958 .4865 Mdot .24144 .48727 Mdot .2433 .48815 Mdot .24516 .48915 Mdot .24702 .49012 Mdot .24888 .49066 Mdot .25074 .49131 Mdot .2526 .49208 Mdot .25446 .49281 Mdot .25632 .49367 Mdot .25818 .49449 Mdot .26004 .49528 Mdot .2619 .49608 Mdot .26376 .49888 Mdot .26563 .50114 Mdot .26749 .50287 Mdot .26935 .50474 Mdot .27121 .50607 Mdot .27307 .50755 Mdot .27493 .50917 Mdot .27679 .51076 Mdot .27865 .51643 Mdot .28051 .521 Mdot .28237 .52448 Mdot .28423 .52826 Mdot .28609 .52432 Mdot .28795 .52245 Mdot .28981 .50915 Mdot .29167 .49891 Mdot .29353 .50942 Mdot .29539 .29841 Mdot .29725 .07968 Mdot .29911 .07841 Mdot .30097 .10193 Mdot .30283 .11882 Mdot .30469 .07881 Mdot .30655 .05405 Mdot .30841 .07943 Mdot .31027 .09137 Mdot .31213 .08988 Mdot .31399 .09199 Mdot .31585 .08066 Mdot .31771 .07293 Mdot .31957 .0688 Mdot .32143 .06371 Mdot .32329 .07219 Mdot .32515 .07703 Mdot .32701 .07823 Mdot .32887 .08041 Mdot .33073 .07895 Mdot .33259 .07847 Mdot .33445 .07896 Mdot .33631 .07919 Mdot .33817 .07579 Mdot .34003 .07336 Mdot .34189 .0719 Mdot .34375 .07019 Mdot .34561 .06944 Mdot .34747 .06844 Mdot .34933 .06717 Mdot .35119 .06598 Mdot .35305 .06728 Mdot .35491 .06791 Mdot .35677 .06787 Mdot .35863 .06801 Mdot .36049 .06748 Mdot .36235 .06714 Mdot .36421 .06697 Mdot .36607 .06675 Mdot .36793 .06587 Mdot .36979 .06516 Mdot .37165 .06463 Mdot .37351 .06406 Mdot .37537 .06366 Mdot .37723 .06322 Mdot .37909 .06273 Mdot .38095 .06225 Mdot .38281 .06008 Mdot .38467 .05837 Mdot .38653 .05711 Mdot .38839 .05573 Mdot .39025 .0548 Mdot .39211 .05375 Mdot .39397 .05257 Mdot .39583 .05143 Mdot .39769 .05074 Mdot .39955 .04994 Mdot .40141 .049 Mdot .40327 .04811 Mdot .40513 .04709 Mdot .40699 .0461 Mdot .40885 .04515 Mdot .41071 .04418 Mdot .41257 .04831 Mdot .41443 .05108 Mdot .41629 .05248 Mdot .41815 .05425 Mdot .42001 .04453 Mdot .42188 .03788 Mdot .42374 .03432 Mdot .4256 .02993 Mdot .42746 .34452 Mdot .42932 .57365 Mdot .43118 .55533 Mdot .43304 .60332 Mdot .4349 .57711 Mdot .43676 .57078 Mdot .43862 .58433 Mdot .44048 .59256 Mdot .44234 .58433 Mdot .4442 .58052 Mdot .44606 .58111 Mdot .44792 .58052 Mdot .44978 .58434 Mdot .45164 .58698 Mdot .4535 .58843 Mdot .45536 .59021 Mdot .45722 .5882 Mdot .45908 .58721 Mdot .46094 .58723 Mdot .4628 .58698 Mdot .46466 .58774 Mdot .46652 .58823 Mdot .46838 .58844 Mdot .47024 .58873 Mdot .4721 .58853 Mdot .47396 .58846 Mdot .47582 .58852 Mdot .47768 .58855 Mdot .47954 .58871 Mdot .4814 .58883 Mdot .48326 .58892 Mdot .48512 .58902 Mdot .48698 .58925 Mdot .48884 .58944 Mdot .4907 .5896 Mdot .49256 .58977 Mdot .49442 .5899 Mdot .49628 .59004 Mdot .49814 .59019 Mdot .5 .59034 Mdot .50186 .58989 Mdot .50372 .5896 Mdot .50558 .58947 Mdot .50744 .5893 Mdot .5093 .58929 Mdot .51116 .58923 Mdot .51302 .58914 Mdot .51488 .58905 Mdot .51674 .59119 Mdot .5186 .59273 Mdot .52046 .59367 Mdot .52232 .59478 Mdot .52418 .27567 Mdot .52604 .04236 Mdot .5279 .06249 Mdot .52976 .01472 Mdot .53162 .04129 Mdot .53348 .04795 Mdot .53534 .03468 Mdot .5372 .02675 Mdot .53906 .03399 Mdot .54092 .03717 Mdot .54278 .03628 Mdot .54464 .03647 Mdot .5465 .0368 Mdot .54836 .0371 Mdot .55022 .03736 Mdot .55208 .03763 Mdot .55394 .03786 Mdot .5558 .0381 Mdot .55766 .03836 Mdot .55952 .03861 Mdot .56138 .0393 Mdot .56324 .03987 Mdot .5651 .04033 Mdot .56696 .04081 Mdot .56882 .04118 Mdot .57068 .04158 Mdot .57254 .04201 Mdot .5744 .04244 Mdot .57626 .04274 Mdot .57813 .04308 Mdot .57999 .04345 Mdot .58185 .04381 Mdot .58371 .0442 Mdot .58557 .04458 Mdot .58743 .04496 Mdot .58929 .04534 Mdot .59115 .04625 Mdot .59301 .04702 Mdot .59487 .04765 Mdot .59673 .04831 Mdot .59859 .04883 Mdot .60045 .04939 Mdot .60231 .04999 Mdot .60417 .05058 Mdot .60603 .05102 Mdot .60789 .0515 Mdot .60975 .05202 Mdot .61161 .05253 Mdot .61347 .05308 Mdot .61533 .05362 Mdot .61719 .05415 Mdot .61905 .05468 Mdot .62091 .05486 Mdot .62277 .05514 Mdot .62463 .0555 Mdot .62649 .05585 Mdot .62835 .05628 Mdot .63021 .0567 Mdot .63207 .05708 Mdot .63393 .05748 Mdot .63579 .05796 Mdot .63765 .05843 Mdot .63951 .05886 Mdot .64137 .05931 Mdot .64323 .05973 Mdot .64509 .06015 Mdot .64695 .06058 Mdot .64881 .06101 Mdot .65067 .06383 Mdot .65253 .06601 Mdot .65439 .06755 Mdot .65625 .06926 Mdot .65811 .07033 Mdot .65997 .07157 Mdot .66183 .07298 Mdot .66369 .07435 Mdot .66555 .07508 Mdot .66741 .07597 Mdot .66927 .07704 Mdot .67113 .07807 Mdot .67299 .07926 Mdot .67485 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dWoo0P00;goo000JOol00`00Oomoo`3@Ool2000aOol001Yoo`03001oogoo0Goo000JOol00`00Oomoo`36Ool2000kOol001Yoo`03001o ogoo0Ool00`00Oomoo`2bOol0 00Moo`04001oogoo0008Ool01000Oomoo`000goo00<007ooOol0CWoo00<007ooOol0/Woo0007Ool0 1000Oomoo`002Goo00<007ooOol00goo0P00CWoo00<007ooOol0/goo0007Ool01000Oomoo`002Woo 00<007ooOol00Woo00<007ooOol0CGoo00<007ooOol0/goo0007Ool01000Oomoo`001goo00D007oo Oomoo`0000=oo`03001oogoo04eoo`03001oogoo0;=oo`001goo00@007ooOol000Moo`05001oogoo Ool00003Ool00`00Oomoo`1"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.0105157, -0.0178569, \ 0.00039754, 0.00145395}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]], Cell["\<\ 1. At what threshold is the compression ratio equal to .1? What is the RMS error?\ \>", "Subsection"], Cell["2. Redo with H2=Dau[1], Dau[5], Dau[10] and compare.", "Subsection"], Cell["\<\ 3. Change the number of jumps to 20 and compare the results for D4=Dau[2] and for Dau[10]. \ \>", "Subsection"], Cell["\<\ 4. If time allows investigate how the optimal filter depends on the length of the smooth sections (i.e. number of jumps). \ \>", "Subsection"] }, Closed]], Cell[CellGroupData[{ Cell["Problem 1: Denoising", "Section"], Cell[BoxData[ \(<< Statistics`ContinuousDistributions`\)], "Input"], Cell[CellGroupData[{ Cell["\<\ D4: Estimate best threshold. Try with different levels of \ noise.\ \>", "Subsubsection"], Cell[BoxData[{ \(\(n = 256;\)\), "\n", \(\(dst = NormalDistribution[0, .2];\)\), "\n", \(\(c0 = \t{N[ Table[Sign[j - n/2] Sin[Pi*j/\((n - 1)\)], {j, 0, n - 1}]]};\)\), "\n", \(\(nc0 = c0 + {RandomArray[dst, n]};\)\)}], "Input"], Cell[BoxData[ \(ListPlot[c0[\([1]\)], \ PlotJoined -> True]\)], "Input"], Cell[BoxData[ \(ListPlot[nc0[\([1]\)], \ PlotJoined -> True]\)], "Input"], Cell[BoxData[{ \(\(c6 = \(WT[D4]\)[c0, 6];\)\), "\n", \(\(nc6 = \(WT[D4]\)[nc0, 6];\)\), "\n", \(\(dnc6 = dMap[Chop[#, .7] &, nc6];\)\), "\n", \(ListPlot[Flatten[Drop[c6, 1]], \ PlotRange -> All]\), "\n", \(ListPlot[Flatten[Drop[nc6, 1]], \ PlotRange -> All]\), "\n", \(ListPlot[Flatten[Drop[dnc6, 1]], \ PlotRange -> All]\)}], "Input"], Cell[BoxData[{ \(\(dnc0 = \(WT[D4]\)[dnc6, 0];\)\), "\n", \(ListPlot[dnc0[\([1]\)], PlotJoined -> True]\), "\n", \(norm[dnc0 - c0]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[" Fourier transform.", "Subsubsection"], Cell[BoxData[{ \(\(dnft = Chop[Fourier[nc0[\([1]\)]], .5];\)\), "\n", \(ListPlot[dnc0 = Chop[InverseFourier[dnft]]]\), "\n", \(norm[{dnc0} - c0]\)}], "Input"], Cell["\<\ After the following table is generated, close the group of graphics \ cells and select Animate Graphics from the Cell Menu. \ \>", "Text"], Cell[BoxData[ \(\(\(\t\)\(tab1 = Table[\n\t\tdnc6 = dMap[Chop[#, t] &, nc6]; \n\t\tdnc0 = \(WT[D4]\)[dnc6, 0]; \n\t\tListPlot[dnc0[\([1]\)], \ PlotRange -> All, PlotJoined -> True]; \n\t{t, norm[dnc0 - c0]}\t, \n{t, 0, 1.5, .05}];\)\(\n\)\)\)], "Input"], Cell["\<\ Here is a plot of the error (square root of sum of squares) between \ the denoised signal and the original signal as a function of the threshold \ for the Chop function. (The plot will be slightly different each time, \ but a threshold of .6 to .9 seems to be optimal almost surely.)\ \>", "Text"], Cell[BoxData[ \(ListPlot[tab1, PlotJoined -> True]\)], "Input"], Cell["\<\ Repeat with the Fourier denoising technique. After the following \ table is generated, close the group of graphics cells and select Animate \ Graphics from the Cell Menu. Note that in this case there is significant smoothing of the edge (in contrast to \ the wavelet case). \ \>", "Text"], Cell[BoxData[ \(\(tab2 = Table[dnft = Chop[Fourier[nc0[\([1]\)]], t]; \n ListPlot[dnc0 = Chop[InverseFourier[dnft]], \ PlotJoined -> True]; \n{t, norm[{dnc0} - c0]}\t, {t, 0, 1.5, .05}];\)\)], "Input"], Cell[BoxData[ \(ListPlot[tab2, PlotJoined -> True]\)], "Input"] }, Closed]], Cell["Redo with H2 and D6 and compare.", "Subsubsection"] }, Closed]], Cell[CellGroupData[{ Cell["Problem: Image Compression", "Section"], Cell[CellGroupData[{ Cell["Image ", "Subsection"], Cell[BoxData[ \(\(img = {{116, 70, 35, 63, 60, 15, 8, 14, 16, 30, 18, 14, 32, 43, 37, 14, 23, 39, 48, 45, 47, 52, 51, 46, 67, 69, 79, 74, 51, 59, 62, 71, 72, 58, 32, 23, 55, 97, 90, 53, 24, 27, 32, 56, 119, 168, 176, 183, 176, 167, 159, 145, 124, 108, 97, 84, 92, 101, 109, 111, 116, 120, 109, 90, 86, 78, 70, 62, 54, 50, 57, 69, 77, 99, 129, 158, 181, 198, 213, 223, 211, 199, 199, 213, 224, 224, 226, 232, 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