plotting - Fast DensityPlot

Is it possible to increase the performance of the DensityPlot?

For example, let's try to plot the following "flower"

f[x_, y_] := (x^2 + y^2) Exp[-x^2 - y^2] Sin[10 Sqrt[x^2 + y^2] + 10 ArcTan[x, y]]^4;

DensityPlot[f[x, y], {x, -3, 3}, {y, -3, 3}, PlotPoints -> 200, 
 MaxRecursion -> 3, ColorFunction -> Hue, PlotRange -> All, 
 ColorFunctionScaling -> False, ImageSize -> 600]

This toy example takes about 12 seconds on my laptop, eats about 1GB of RAM while plotting, and the 34MB result slows down the notebook.

The questions are:

1. 
   

   How to increase the speed?

   
   


2. 
   

   How to decrease the MaxMemoryUsed?

   
   


3. 
   

   How to decrease the size of the output?

   
   

Answer

Edit: tested with MMA 11.1, option Exclusions -> None added to recover the previous behavior.

I have found that my approach with textures has different applications:

Now I want to use it for the enhancement of the DensityPlot:

Options[fastDensityPlot] = Append[Options[DensityPlot], Subpoints -> 30];
SyntaxInformation[fastDensityPlot] = SyntaxInformation[DensityPlot];

fastDensityPlot[f_, {x_, xmin_, xmax_}, {y_, ymin_, ymax_}, opts : OptionsPattern[]] :=

 DensityPlot[f, {x, xmin, xmax}, {y, ymin, ymax},
       Evaluate@FilterRules[{opts}, Except@Subpoints]] // Normal // toTriangles // 
   texturize[Function[{#1, #2}, #3] & @@ {x, y, f},
        OptionValue[Subpoints], OptionValue[ColorFunction]]

Here Normal converts GraphicsComplex to separate polygons, toTriangles splits polygons to triangles, and texturize puts textures on every triangle (defined below), f is assumed to be Listable.

f[x_, y_] := (x^2 + y^2) Exp[-x^2 - y^2] Sin[10 Sqrt[x^2 + y^2] + 10 ArcTan[x, y]]^4;

fastDensityPlot[f[x, y], {x, -3, 3}, {y, -3, 3}, PlotPoints -> 10, 
 MaxRecursion -> 2, ColorFunction -> Hue, Subpoints -> 20, 

 PlotRange -> All, ImageSize -> 600, Exclusions -> None]

This image looks a bit better. At the same time fastDensityPlot is ~10 times faster then the regular DensityPlot, MaxMemoryUsed is only 64MB and ByteCount is 10MB.

One can see that fastDensityPlot uses the advantage of the non-equidistant mesh:

fastDensityPlot[f[x, y], {x, -3, 3}, {y, -3, 3}, PlotPoints -> 10, 
 MaxRecursion -> 2, ColorFunction -> Hue, Subpoints -> 20, 
 PlotRange -> All, ImageSize -> 600, Mesh -> All, Exclusions -> None]

Definitions of the above functions:

toTriangles = # /. Polygon[v_ /; Length[v] > 3, ___] :> (Polygon@Append[#, Mean[v]] & /@ 
       Partition[v, 2, 1, 1]) &;

texturize[f_, n_, colf_] := # /. Polygon[{v1_, v2_, v3_}, ___] :> {Texture@
       ImageData@Colorize[
         Image@f[v3[[1]] + (v1[[1]] - v3[[1]]) #1 + (v2[[1]] - v3[[1]]) #2, 
                v3[[2]] + (v1[[2]] - v3[[2]]) #1 + (v2[[2]] - v3[[2]]) #2] 
          &[#, Transpose[#]] &@ConstantArray[Range[-1./n, 1 + 1./n, 1./n], n + 3], 
         ColorFunction -> colf, ColorFunctionScaling -> False], 

      Polygon[{v1, v2, v3}, 
       VertexTextureCoordinates -> {{1 - 1.5/(n + 3), 
          1 - 1.5/(n + 3)}, {1.5/(n + 3), 1.5/(n + 3)}, {1.5/(n + 3), 
          1 - 1.5/(n + 3)}}]} &;

As in the linked answer, I add textures to every triangle with an appropriate rectangular grid. This method is fast because it uses packed arrays.