Wolfram mathematica 粗线条轮廓提取

我怎样才能画出粗线的轮廓，比如下面的向量形式？所谓矢量形式，我指的是一些非光栅或图像的图形原语集合

Graphics[{AbsoluteThickness[100], JoinForm["Round"], CapForm["Round"],
   Line[{{0, 0}, {0, 1}, {1, 1}}]}, ImageSize -> 200]

（来源：）

文档中有以下用于提取文本轮廓的示例，但我还没有找到一种方法来修改它以获取

Line

对象的轮廓

ImportString[ExportString[Style["M8", FontFamily -> "Times", FontSize -> 72],"PDF"], "TextMode" -> "Outlines"]

我在line对象上做了

光栅化
，并从alpha通道中减去了一个稍小的版本。这会产生光栅化伪影，对于
ImageSize->500

还询问了mathgroup

更新
我尝试过通过从
形态学周长
获得的点拟合样条曲线
ListCurvePathPlot
理论上是这样做的，但它打破了像素“阶梯”模式。要使楼梯平滑，需要找到曲线周围点的顺序
FindCurvePath
似乎很有希望，但返回了一个断裂曲线列表
FindShortestTour
理论上也可以做到这一点，但在20x20像素图像的轮廓上，它花费了超过一秒钟的时间
ConvexHull
在圆形零件上做得很好，但会切断非凸面零件

我最终得到的解决方案是在周长点上构建最近邻图，并使用version 8函数
FindeureRincycle
查找形状周围像素的顺序，然后使用
MovingAverage
平滑楼梯，然后使用
ListCurvePathPlot
创建样条线对象。这并不完美，因为仍然有“楼梯”模式的残余，而平均值太高会使重要的角落变得平滑。更好的方法可能是将形状分解为多个凸面形状，使用
ConvexHull
，然后重新组合。同时，这里是我正在使用的

getSplineOutline[pp_, smoothLen_: 2, interOrder_: 3] := (
   (* need to negate before finding perimeter to avoid border *)

   perim = MorphologicalPerimeter@ColorNegate@pp;
   points = 
    Cases[ArrayRules@SparseArray@ImageData[perim], 
     HoldPattern[{a_Integer, b_Integer} -> _] :> {a, b}];
   (* raster coordinate system is upside down, flip the points *)

   points = {1, -1} (# - {0, m}) & /@ points;
   (* make nearest neighbor graph *)

   makeEdges[point_] := {Sort[{point, #}]} & /@ 
     Nearest[DeleteCases[points, point], point];
   edges = Union[Flatten[makeEdges /@ points, 2]];
   graph = Graph[UndirectedEdge @@@ edges];
   tour = FindEulerianCycle[graph] // First;
   smoothed = MovingAverage[tour[[All, 1]], smoothLen];
   g = ListCurvePathPlot[smoothed, InterpolationOrder -> interOrder];
   Cases[g, BSplineCurve[___], Infinity] // First
   );

scale = 200;
pp = Graphics[{AbsoluteThickness[scale/2], JoinForm["Round"], 
    CapForm["Round"], Line[{{0, 0}, {0, 1}, {1, 1}}]}, 
   ImageSize -> scale];
Graphics[getSplineOutline[pp, 3, 3]]



（来源：）
不是答案，只是针对你的光栅化评论

我认为这可能更快（在我的机器中，图像大小为500的图像为0.1秒）


顺便说一句，我尝试使用所有矢量图像格式进行“导出”，但令人惊讶的是，除PDF格式外，大多数矢量图像格式中的舍入格式都丢失了，PDF格式是无用的，因为它在导入时恢复了相同的行定义。
很遗憾，
EdgeForm[]
（如文档中所述）不适用于
行
对象。因此，我们最好不要使用
Line[]
，或者使用某种黑客。我能想到的最简单的方法就是

Graphics[{AbsoluteThickness[100], JoinForm["Round"], CapForm["Round"],
   Line[{{0, 0}, {0, 1}, {1, 1}}], AbsoluteThickness[99], White, 
  Line[{{0, 0}, {0, 1}, {1, 1}}]}, ImageSize -> 200]

好的，我不确定这是否值得，但我们来看看：一种使用图像变换、最小二乘法和数据聚类的方法

Clear["Global`*"];
(*Functions for Least Square Circle \
from  http://www.dtcenter.org/met/users/docs/write_ups/circle_fit.pdf*)


t[x_] := Plus[#, -Mean[x]] & /@ x;
Suu[x_] := Sum[i[[1]]^2, {i, t[x]}];
Svv[x_] := Sum[i[[2]]^2, {i, t[x]}];
Suv[x_] := Sum[i[[1]] i[[2]], {i, t[x]}];
Suvv[x_] := Sum[i[[1]] i[[2]]^2, {i, t[x]}];
Svuu[x_] := Sum[i[[2]] i[[1]]^2, {i, t[x]}];
Suuu[x_] := Sum[i[[1]]^3, {i, t[x]}];
Svvv[x_] := Sum[i[[2]]^3, {i, t[x]}];
s[x_] := Solve[{uc Suu[x] + vc Suv[x] == 1/2 (Suuu[x] + Suvv[x]), 
    uc Suv[x] + vc Svv[x] == 1/2 (Svvv[x] + Svuu[x])}, {uc, vc}];
(*Utility fun*)
ppfilterCoords[x_, k_] := Module[{ppflat},
   ppflat = 
    Flatten[Table[{i, j, ImageData[x][[i, j]]}, {i, k[[1]]}, {j, 
       k[[2]]}], 1];
   Take[#, 2] & /@ Select[ppflat, #[[3]] == 0 &]
   ];
(*Start*)
thk = 100;
pp = Graphics[{AbsoluteThickness[100], JoinForm["Round"], 
   CapForm["Round"], Line[{{0, 0}, {0, 1}, {2, 1}, {2, 2}}]}, 
  ImageSize -> 300]
(*
pp=Graphics[{AbsoluteThickness[thk],JoinForm["Round"],CapForm["Round"]\
,Line[{{0,0},{0,3},{1,3},{1,0}}]},ImageSize->300];
*)
pp1 = ColorNegate@MorphologicalPerimeter@pp;
(* Get vertex in pp3*)
pp3 = Binarize[ColorNegate@HitMissTransform[pp1,
     { {{1, -1}, {-1, -1}}, {{-1, 1}, {-1, -1}},
      {{-1, -1}, {1, -1}}, {{-1, -1}, {-1, 1}}}], 0];
k = Dimensions@ImageData@pp3;

clus = FindClusters[ppfilterCoords[pp3, k],(*get circles appart*)
   Method -> {"Agglomerate", "Linkage" -> "Complete"}, 
   DistanceFunction -> (If [EuclideanDistance[#1, #2] <= thk/2, 0, 
       EuclideanDistance[#1, #2]] &)]; 
(*Drop Spurious clusters*)
clus = Select[clus, Dimensions[#][[1]] > 10 &];
(*Calculate centers*)
centerOffset = Flatten[{uc, vc} /. s[#] & /@ clus, 1];
(*coordinates correction*)
center = {-1, 1} Plus[#, {0, k[[2]]}] & /@ -N[
     centerOffset + Mean /@ clus, 2];
Print["Circles Centers ", center];
(*get radius from coordinates. All radius are equal*)
radius = Max[Table[
     {Max[First /@ clus[[i]]] - Min[First /@ clus[[i]]],
      Max[Last /@ clus[[i]] - Min[Last /@ clus[[i]]]]}
     , {i, Length[clus]}]]/2;
Print["Circles Radius ", radius];

(*Now get the straight lines*)
(*horizontal lines*)
const = 30;(*a number of aligned pixels for line detection*)
ph = ColorNegate@
  HitMissTransform[ColorNegate@pp1, {Table[1, {const}]}];
(*vertical lines *)
pv = ColorNegate@
   HitMissTransform[ColorNegate@pp1, {Table[{1}, {const}]}];
(*if there are diagonal lines add patterns accordingy*)
(*coordinates correction function*)
corr[x_, k_] := {-1, 1} Plus[-x, {0, k[[2]]}];
dfunH[x_, y_] := Abs[x[[1]] - y[[1]]];
dfunV[x_, y_] := Abs[x[[2]] - y[[2]]];
(*Get clusters for horiz*)
clusH = FindClusters[ppfilterCoords[ph, k],(*get lines appart*)
   Method -> {"Agglomerate", "Linkage" -> "Complete"}, 
   DistanceFunction -> dfunH];
hlines = Table[{Line[{corr[First[i], k] + {1, const/2 - 1}, 
      corr[Last[i], k] + {1, -const/2 - 1}}]}, {i, clusH}];

clusV = FindClusters[ppfilterCoords[pv, k],(*get lines appart*)
   Method -> {"Agglomerate", "Linkage" -> "Complete"}, 
   DistanceFunction -> dfunV];
vlines = Table[{Line[{corr[First[i], k] - {const/2 - 1, 1}, 
      corr[Last[i], k] + {const/2 - 1, -1}}]}, {i, clusV}];
Graphics[{vlines, hlines, 
  Table[Circle[center[[i]], radius], {i, Length@clus}]}]

清除[“全局”*”]；
（*最小二乘圆函数）\
从…起http://www.dtcenter.org/met/users/docs/write_ups/circle_fit.pdf*)
t[x]：=加上[#，-平均值[x]]&/@x；
Suu[x]：=Sum[i[[1]]^2，{i，t[x]}；
Svv[x_]：=Sum[i[[2]]^2，{i，t[x]}]；
Suv[x_]：=Sum[i[[1]]i[[2]]，{i，t[x]}；
Suvv[x_]：=Sum[i[[1]]i[[2]]^2，{i，t[x]}；
Svuu[x_]：=Sum[i[[2]]i[[1]]^2，{i，t[x]}；
Suuu[x]：=Sum[i[[1]]^3，{i，t[x]}；
Svvv[x_]：=Sum[i[[2]]^3，{i，t[x]}；
s[x_]：=Solve[{uc Suu[x]+vc Suv[x]==1/2（Suuu[x]+Suvv[x]），
uc-Suv[x]+vc-Svv[x]==1/2（Svvv[x]+Svuu[x]），{uc，vc}；
（*实用乐趣*）
ppfilterCoords[x_，k_]：=模块[{ppflat}，
ppflat=
展平[表[{i，j，ImageData[x][[i，j]]}，{i，k[[1]]}，{j，
k[[2]}]，1]；
取[#，2]&/@Select[ppflat，[[3]]==0&]
];
（*开始*）
厚度=100；
pp=图形[{绝对厚度[100]，接缝形式[“圆形”]，
CapForm[“Round”]，第[{0，0}，{0，1}，{2，1}，{2，2}]}行，
图像大小->300]
(*
pp=图形[{绝对厚度[thk]，接缝形式[“圆形”]，帽形[“圆形”]\
，行[{0,0}，{0,3}，{1,3}，{1,0}]}，ImageSize->300]；
*)
pp1=ColorNegate@MorphologicalPerimeter@聚丙烯；
（*获取pp3*中的顶点）
pp3=二值化[ColorNegate@HitMissTransform[pp1，
{ {{1, -1}, {-1, -1}}, {{-1, 1}, {-1, -1}},
{{-1, -1}, {1, -1}}, {{-1, -1}, {-1, 1}}}], 0];
k=Dimensions@ImageData@pp3；
clus=FindClusters[ppfilterCoords[pp3，k]，（*get circles appart*）
方法->{“凝聚”，“连接”->“完成”}，
距离函数->（如果[欧几里德距离[#1，#2]10&]；
（*计算中心*）
centerOffset=Flatten[{uc，vc}/.s[#]&/@clus，1]；
（*坐标校正*）
中心={-1,1}加上[#，{0,k[[2]}]&/@-N[
中心偏移量+平均值/@clus，2]；
打印[“圆-中心”，中心]；
（*从坐标中获取半径。所有半径都相等*）
半径=最大值[表[
{Max[First/@clus[[i]]]-Min[First/@clus[[i]]]，
Max[Last/@clus[[i]]]-Min[Last/@clus[[i]]]]]
，{i，长度[clus]}]]/2；
打印[“圆半径”，半径]；
（*现在获取直线*）
（*水平线*）
const=30；（*用于行检测的对齐像素数*）
ph=颜色否定@
希特勒变换[ColorNegate@pp1，{表[1，{const}]}]；
（*垂直线*）
pv=颜色否定@
希特勒变换[ColorNegate@pp1，{Table[{1}，{const}]}]；
（*如果有对角线，则添加相应的图案*）
（*坐标校正函数*）
corr[x_，k_]：={-1，1}加[-x，{0，k[[2]}]；
dfunH[x_u，y_u]：=Abs[x[[1]]-y[[1]]；
dfunV[x_u，y_u]：=Abs[x[[2]]-y[[2]]；
（*为horiz*获取群集）
clusH=FindClusters[ppfilterCoords[ph，k]，（*获取行程序*）
方法->{“凝聚”，“连接”->“完成”}，
距离功能->dfunH]；
hlines=Table[{Line[{corr[First[i]，k]+{1，const/2-1}，
corr[Last[i]，k]+{1，-const/2-1}}}，{i，clusH}]；
clusV=FindClusters[ppfilterCoords[pv，k]，（*获取行程序*）
方法->{“凝聚”，“连接”->“完成”}，
距离功能->dfunV]；
vlines=表[{Line[{corr[First[i]，k]{const/2-1，1}，
corr[Last[i]，k]+{const/2-1，-1}}}，{i，clusV}]；
图形，
表[圆[中心[[i]]，半径]，{i，Length@clus}]}]



编辑

更新：

仅使用几何图形

当然，这一个应该能够击败使用欧尔'笛卡尔几何
Clear["Global`*"];
(*Functions for Least Square Circle \
from  http://www.dtcenter.org/met/users/docs/write_ups/circle_fit.pdf*)


t[x_] := Plus[#, -Mean[x]] & /@ x;
Suu[x_] := Sum[i[[1]]^2, {i, t[x]}];
Svv[x_] := Sum[i[[2]]^2, {i, t[x]}];
Suv[x_] := Sum[i[[1]] i[[2]], {i, t[x]}];
Suvv[x_] := Sum[i[[1]] i[[2]]^2, {i, t[x]}];
Svuu[x_] := Sum[i[[2]] i[[1]]^2, {i, t[x]}];
Suuu[x_] := Sum[i[[1]]^3, {i, t[x]}];
Svvv[x_] := Sum[i[[2]]^3, {i, t[x]}];
s[x_] := Solve[{uc Suu[x] + vc Suv[x] == 1/2 (Suuu[x] + Suvv[x]), 
    uc Suv[x] + vc Svv[x] == 1/2 (Svvv[x] + Svuu[x])}, {uc, vc}];
(*Utility fun*)
ppfilterCoords[x_, k_] := Module[{ppflat},
   ppflat = 
    Flatten[Table[{i, j, ImageData[x][[i, j]]}, {i, k[[1]]}, {j, 
       k[[2]]}], 1];
   Take[#, 2] & /@ Select[ppflat, #[[3]] == 0 &]
   ];
(*Start*)
thk = 100;
pp = Graphics[{AbsoluteThickness[100], JoinForm["Round"], 
   CapForm["Round"], Line[{{0, 0}, {0, 1}, {2, 1}, {2, 2}}]}, 
  ImageSize -> 300]
(*
pp=Graphics[{AbsoluteThickness[thk],JoinForm["Round"],CapForm["Round"]\
,Line[{{0,0},{0,3},{1,3},{1,0}}]},ImageSize->300];
*)
pp1 = ColorNegate@MorphologicalPerimeter@pp;
(* Get vertex in pp3*)
pp3 = Binarize[ColorNegate@HitMissTransform[pp1,
     { {{1, -1}, {-1, -1}}, {{-1, 1}, {-1, -1}},
      {{-1, -1}, {1, -1}}, {{-1, -1}, {-1, 1}}}], 0];
k = Dimensions@ImageData@pp3;

clus = FindClusters[ppfilterCoords[pp3, k],(*get circles appart*)
   Method -> {"Agglomerate", "Linkage" -> "Complete"}, 
   DistanceFunction -> (If [EuclideanDistance[#1, #2] <= thk/2, 0, 
       EuclideanDistance[#1, #2]] &)]; 
(*Drop Spurious clusters*)
clus = Select[clus, Dimensions[#][[1]] > 10 &];
(*Calculate centers*)
centerOffset = Flatten[{uc, vc} /. s[#] & /@ clus, 1];
(*coordinates correction*)
center = {-1, 1} Plus[#, {0, k[[2]]}] & /@ -N[
     centerOffset + Mean /@ clus, 2];
Print["Circles Centers ", center];
(*get radius from coordinates. All radius are equal*)
radius = Max[Table[
     {Max[First /@ clus[[i]]] - Min[First /@ clus[[i]]],
      Max[Last /@ clus[[i]] - Min[Last /@ clus[[i]]]]}
     , {i, Length[clus]}]]/2;
Print["Circles Radius ", radius];

(*Now get the straight lines*)
(*horizontal lines*)
const = 30;(*a number of aligned pixels for line detection*)
ph = ColorNegate@
  HitMissTransform[ColorNegate@pp1, {Table[1, {const}]}];
(*vertical lines *)
pv = ColorNegate@
   HitMissTransform[ColorNegate@pp1, {Table[{1}, {const}]}];
(*if there are diagonal lines add patterns accordingy*)
(*coordinates correction function*)
corr[x_, k_] := {-1, 1} Plus[-x, {0, k[[2]]}];
dfunH[x_, y_] := Abs[x[[1]] - y[[1]]];
dfunV[x_, y_] := Abs[x[[2]] - y[[2]]];
(*Get clusters for horiz*)
clusH = FindClusters[ppfilterCoords[ph, k],(*get lines appart*)
   Method -> {"Agglomerate", "Linkage" -> "Complete"}, 
   DistanceFunction -> dfunH];
hlines = Table[{Line[{corr[First[i], k] + {1, const/2 - 1}, 
      corr[Last[i], k] + {1, -const/2 - 1}}]}, {i, clusH}];

clusV = FindClusters[ppfilterCoords[pv, k],(*get lines appart*)
   Method -> {"Agglomerate", "Linkage" -> "Complete"}, 
   DistanceFunction -> dfunV];
vlines = Table[{Line[{corr[First[i], k] - {const/2 - 1, 1}, 
      corr[Last[i], k] + {const/2 - 1, -1}}]}, {i, clusV}];
Graphics[{vlines, hlines, 
  Table[Circle[center[[i]], radius], {i, Length@clus}]}]

k[pp_] := Module[{ED(*TODO: make all symbols local*)}, (
    (*follows some analytic geometry *)
    (*Functions to calcu|late borderlines*)
    linesIncrUpDown[{x0_, y0_}, {x1_, y1_}] := 
     thk/2 {-(y1 - y0), (x1 - x0)}/ED[{x0, y0}, {x1, y1}];
    lineUp[{{x0_, y0_}, {x1_, y1_}}] := 
     Plus[linesIncrUpDown[{x0, y0}, {x1, y1}], #] & /@ {{x0, y0}, {x1,y1}};
    lineDown[{{x0_, y0_}, {x1_, y1_}}] := 
     Plus[-linesIncrUpDown[{x0, y0}, {x1, y1}], #] & /@ {{x0,y0}, {x1, y1}};
    (*Distance from line to point*)
    distanceLinePt[{{x1_, y1_}, {x2_, y2_}}, {x0_, y0_}] := 
     Abs[(x2 - x1) (y1 - y0) - (x1 - x0) (y2 - y1)]/ED[{x1, y1}, {x2, y2}];
    (*intersect between two lines without overflows for verticals*)
    intersect[{{{x1_, y1_}, {x2_, y2_}}, {{x3_, y3_}, {x4_, 
         y4_}}}] := {((x3 - x4) (-x2 y1 + x1 y2) + (x1 - x2) (x4 y3 - 
          x3 y4))/(-(x3 - x4) (y1 - y2) + (x1 - x2) (y3 - 
          y4)), (-(x2 y1 - x1 y2) (y3 - y4) + (y1 - y2) (x4 y3 - 
          x3 y4))/(-(x3 - x4) (y1 - y2) + (x1 - x2) (y3 - y4))};
    l2C := #[[1]] + I #[[2]] & ; (*list to complex for using Arg[]*);
    ED = EuclideanDistance; (*shorthand*)


    thk = Cases[pp, AbsoluteThickness[x_] -> x, Infinity][[1]];
    lines = Cases[pp, Line[x_] -> x, Infinity][[1]];
    isz = Cases[pp, Rule[ImageSize, x_] -> x, Infinity][[1]];
    (*now get the scale *)
    {minX, maxX} = {Min[#], Max[#]} &@Transpose[lines][[1]];
    (*scale graphDiam +thk= isz *)
    scale = (isz - thk)/(maxX - minX);
    (*calculate absolute positions for lines*)
    absL = (lines) scale + thk/2;
    (*now we already got the centers for the circles*)
    (*Calculate both lines Top Down*)
    luT = Table[Line[lineUp[absL[[i ;; i + 1]]]], {i, Length[absL] - 1}];
    luD = Table[Line[lineDown[absL[[i ;; i + 1]]]], {i, Length[absL] - 1}];
    (*Calculate intersection points for Top and Down lines*)
    iPuT =Table[intersect[{luT[[i, 1]], luT[[i + 1, 1]]}], {i,Length@luT - 1}];
    iPuD =Table[intersect[{luD[[i, 1]], luD[[i + 1, 1]]}], {i,Length@luD - 1}];

    (*beware drawArc has side effects as modifies luT and luD*)
    drawArc[i_] := Module[{s},
      Circle[absL[[i]], thk/2,
       Switch[i,

        1 , (*first point*)
        If[ ED[absL[[i + 1]],absL[[i]] + {Cos[s = ((#[[2]] + #[[1]])/2)], Sin[s]}] <
            ED[absL[[i + 1]],absL[[i]] + {Cos[s + Pi], Sin[s + Pi]}], # + Pi, #]
            &@{Min@#, Max@#} &@
         Mod[ {Arg[l2C @((luD[[i]])[[1, 1]] - absL[[i]])],
               Arg[l2C @((luT[[i]])[[1, 1]] - absL[[i]])]}, 2 Pi],

        Length@absL,(*last point*)
        If[ED[absL[[i - 1]], absL[[i]] + {Cos[s = ((#[[2]] + #[[1]])/2)], Sin[s]}] <
           ED[absL[[i - 1]], absL[[i]] + {Cos[s + Pi], Sin[s + Pi]}], # + Pi, #] 
           &@{Min@#, Max@#} &@
         Mod[{Arg[l2C @((luD[[i - 1]])[[1, 2]] - absL[[i]])], 
              Arg[l2C@((luT[[i - 1]])[[1, 2]] - absL[[i]])]}, 2 Pi],

        _,(*all middle points*)
        (* here I must chose which lines to intersect luD or luT.
        the correct answer is the line farthest to the previous point*)


        If[
         distanceLinePt[luD[[i, 1]], absL[[i - 1]]] > 
         distanceLinePt[luT[[i, 1]], absL[[i - 1]]],
         (*shorten the other lines*)
         luT[[i - 1, 1, 2]] = luT[[i, 1, 1]] = iPuT[[i - 1]]; lu = luD;
         ,
         (*shorten the other lines*)
         luD[[i - 1, 1, 2]] = luD[[i, 1, 1]] = iPuD[[i - 1]]; 
         lu = luT;];
        (If[ED[absL[[i - 1]], absL[[i]] + {Cos[s = ((#[[2]] + #[[1]])/2)], Sin[s]}] <
            ED[absL[[i - 1]], absL[[i]] + {Cos[s + Pi], Sin[s + Pi]}], {#[[2]]-2 Pi, #[[1]]}, #]) 
          &@{Min@#, Max@#} &@
         {Arg[l2C @((lu[[i - 1]])[[1, 2]] - absL[[i]])], 
          Arg[l2C@((lu[[i]])[[1, 1]] - absL[[i]])]}
        ] ] ];
    );
   Graphics[{Black, Table[drawArc[i], {i, Length@absL}], Red, luT, Blue, luD},
     ImageSize -> isz] ];

isz = 250;
pp[1] = Graphics[{AbsoluteThickness[50], JoinForm["Round"], 
    CapForm["Round"], Line[{{0, 0}, {1, 0}, {0, 1}, {1, 1}}]}, 
   ImageSize -> isz];
pp[2] = Graphics[{AbsoluteThickness[50], JoinForm["Round"], 
    CapForm["Round"], 
    Line[{{0, 0}, {1, 0}, {0, -1}, {0.7, -1}, {0, -4}, {2, -3}}]}, 
   ImageSize -> isz];
pp[3] = Graphics[{AbsoluteThickness[50], JoinForm["Round"], 
    CapForm["Round"], 
    Line[{{0, 0}, {0, 1}, {1, 1}, {2, 0}, {2, 3}, {5, 5}, {5, 1}, {4, 
       1}}]}, ImageSize -> isz];
pp[4] = Graphics[{AbsoluteThickness[50], JoinForm["Round"], 
    CapForm["Round"], 
    Line[{{0, 0}, {0, 1}, {1, 1}, {1, 0}, {1/2, 0}}]}, 
   ImageSize -> isz];
GraphicsGrid[Table[{pp[i], k@pp[i]}, {i, 4}]]