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i = Import@"https://i.sstatic.net/8I3B1.jpg"; f[{{tmin_, tmax_}, {rmin_, rmax_}}, ___] := Module[{l = Join[{{0, 0}}, Table[{Cos@t, Sin@t}, {t, tmin, tmax, (tmax-tmin)/100}]]}, {Texture[i], EdgeForm[], Polygon[l, VertexTextureCoordinates -> 1/2 Transpose[Transpose[l] + {1, 1}]]}] Framed@PieChart[{1, 2, 3, 4, 5, 6}, ChartElementFunction -> f] 

i = Import@"https://i.sstatic.net/8I3B1.jpg"; f[{{tmin_, tmax_}, {rmin_, rmax_}}, ___] := Module[{l = Join[{{0, 0}}, Table[{Cos@t, Sin@t}, {t, tmin, tmax, (tmax-tmin)/100}]]}, {Texture[i], EdgeForm[], Polygon[l, VertexTextureCoordinates -> 1/2 Transpose[Transpose[l] + {1, 1}]]}] Framed@PieChart[{1, 2, 3, 4, 5, 6}, ChartElementFunction -> f] 

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You may also want to get a better visual feedback:

Module[{cd = ColorData[3, "ColorList"]}, f[{{tmin_, tmax_}, {rmin_, rmax_}}, ___] := Module[{l = Join[{{0, 0}}, Table[{Cos@t, Sin@t}, {t, tmin, tmax, (tmax - tmin)/100}]]}, cd = RotateLeft@cd; {Texture[ImageCompose[i, {Graphics[{cd[[1]], Disk[]}], 0.5}]], EdgeForm[], Polygon[l, VertexTextureCoordinates -> 1/2 Transpose[Transpose[l] + {1, 1}]]}] ] Framed@PieChart[{1, 2, 3, 4, 5, 6}, ChartElementFunction -> f] 

i = Import@"https://i.sstatic.net/8I3B1.jpg" f[{{tmin_, tmax_}, {rmin_, rmax_}}, ___] :=  Module[{l = Join[{{0, 0}, {Cos@tmin, Sin@tmin}},  Table[{Cos[t], Sin[t]}, {t, tmin, tmax, (tmax - tmin)/30}], {{Cos@tmax, Sin@tmax}}]},    {Texture[i], EdgeForm[], Polygon[l, VertexTextureCoordinates ->  1/2 Transpose[Transpose[l] + {1, 1}]]}] Framed@PieChart[{1, 2, 3, 4, 5, 6}, ChartElementFunction -> f] 

i = Import@"https://i.sstatic.net/8I3B1.jpg"; f[{{tmin_, tmax_}, {rmin_, rmax_}}, ___] := Module[{l = Join[{{0, 0}}, Table[{Cos@t, Sin@t}, {t, tmin, tmax, (tmax-tmin)/100}]]}, {Texture[i], EdgeForm[],  Polygon[l, VertexTextureCoordinates -> 1/2 Transpose[Transpose[l] + {1, 1}]]}]   Framed@PieChart[{1, 2, 3, 4, 5, 6}, ChartElementFunction -> f] 

i = Import@"https://i.sstatic.net/8I3B1.jpg" f[{{tmin_, tmax_}, {rmin_, rmax_}}, ___] := Module[{l = Join[{{0, 0}, {Cos@tmin, Sin@tmin}}, Table[{Cos[t], Sin[t]}, {t, tmin, tmax, (tmax - tmin)/30}], {{Cos@tmax, Sin@tmax}}]}, {Texture[i], Polygon[l, VertexTextureCoordinates -> 1/2 Transpose[Transpose[l] + {1, 1}]]}] Framed@PieChart[{1, 2, 3, 4, 5, 6}, ChartElementFunction -> f] 

i = Import@"https://i.sstatic.net/8I3B1.jpg" f[{{tmin_, tmax_}, {rmin_, rmax_}}, ___] := Module[{l = Join[{{0, 0}, {Cos@tmin, Sin@tmin}}, Table[{Cos[t], Sin[t]}, {t, tmin, tmax, (tmax - tmin)/30}], {{Cos@tmax, Sin@tmax}}]}, {Texture[i], EdgeForm[], Polygon[l, VertexTextureCoordinates -> 1/2 Transpose[Transpose[l] + {1, 1}]]}] Framed@PieChart[{1, 2, 3, 4, 5, 6}, ChartElementFunction -> f]