How to texturize a Disk/Circle/Rectangle?

I noticed an example in the document of Texture which used the alpha channel. So I think a disk-shape primitive may be simulated to a limited degree by mapping the image img, which has been set to 100% transparent outside of the circle, onto a rectangle-shape Polygon.

My code:

img = Rasterize[
                DensityPlot[Sin[x] Sin[y],
                            {x, -4, 4}, {y, -3, 3},
                            ColorFunction -> "BlueGreenYellow",
                            Frame -> None, ImageSize -> 100, PlotRangePadding -> 0
              ]];

imgdim = ImageDimensions[img]

alphamask = Array[
                  If[
                     Norm[{#1, #2} - imgdim/2] < imgdim[[1]]/2,
                     1,0]&,
                  imgdim];

alphaimg = MapThread[Append, {img // ImageData, alphamask}, 2];

Graphics[{
          Polygon[{{0, 0}, {1, 0}, {1, 1}, {0, 1}} + .3],
          Texture[alphaimg],
          Polygon[{{0, 0}, {1, 0}, {1, 1}, {0, 1}}, 
                  VertexTextureCoordinates -> {{0, 0}, {1, 0}, {1, 1}, {0, 1}}
                 ],
          Gray, Disk[{0, 0}, .5]
         }]

which gives result like this:

Like RM, I've not been able to texture a Disk primitive. We can create a textured disk using ParametricPlot, however.

ParametricPlot[{r*Cos[t], r*Sin[t]}, {r, 0, 1}, {t, 0, 2 Pi},
  Mesh -> False, BoundaryStyle -> None, Axes -> False,
  PlotStyle -> {Opacity[1], 
   Texture[ExampleData[{"ColorTexture", "LightCherry"}]]}]

My fallback method for the moment is the following: approximate a circle with a polygon, fill the latter with the texture and finally conceal the angular edge with an overlaid Circle. If the whole image is small, the number of nodes of the polygon can be further reduced. One annoying sideeffect is though that the Circle is not antialiased...

img = Rasterize@
   DensityPlot[Sin@x Sin@y, {x, -4, 4}, {y, -3, 3}, 
    ColorFunction -> "BlueGreenYellow", Frame -> None, 
    ImageSize -> 200, PlotRangePadding -> 0];
coord = Block[{n = 100}, 
   Table[{Cos[2 \[Pi] k/n], Sin[2 \[Pi] k/n]}, {k, 0, n - 1}]];
Graphics[{Texture@img, EdgeForm@None, 
  Polygon[coord, VertexTextureCoordinates -> (coord/2 + .5)], Black, 
  Thick, Circle[]}, ImageSize -> 200, Background -> [email protected]]