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mkRect[size_]:=Rectangle[{25/2-size,25/2-size},{25/2+size,25/2+size}]​​squares=Graphics[{Opacity[.5],Orange,mkRect[3.75],Blue,mkRect[7.5]},PlotRange{{0,25},{0,25}}]​​grid=Show[squares,GridLines{Range[0,25],Range[0,25]}]
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tf=ImageTransformation[squares,(rd=#-0.5;0.5+rd(Norm[rd]+5Norm[rd]^3-3Norm[rd]^5))&,1000,Resampling{"Lanczos",0.5}]​​ImageTransformation[grid,(rd=#-0.5;0.5+rd(Norm[rd]+5Norm[rd]^3-3Norm[rd]^5))&,1000,Resampling{"Lanczos",0.5}]
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out=Rasterize[Show[tf,GridLines{Range[0,#,#/25],Range[0,#2,#2/25]}&@@ImageDimensions[tf],GridLinesStyleDirective[Black],Method{"GridLinesInFront"True}],RasterSize2000,ImageSize540]
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