Made in response to a recent question posted at the  Mathematica Stack Exchange.

Mathematica code:
img = 
 ImageCrop@DensityPlot[
    Sin[2 x - 20 Log[2 (Sin[y]^2 + 1), 2]],
 {x, 0, 16 Pi}, {y, 0, 32 Pi},
 PlotPoints -> 250, ColorFunction -> "SunsetColors", 
 Frame -> False, ImageSize -> 600]LogPolar[x_, y_] := {Log[Sqrt[x^2 + y^2]], ArcTan[x, y]}d = ImageDimensions[img][[1]]
Manipulate[
  ImageResize[
   ImageTransformation[
    ImageTake[
     img, 
    {1, 14*d/16}, {1 + (2 - 2 t)*d/32, (32 - 2 t)*d/32}], 
    LogPolar[#[[1]], #[[2]]] &, DataRange -> {{-Pi, Pi}, {-Pi, Pi}}],
   500],
  {t, 0, 6/7, 1/7}]

Made in response to a recent question posted at the Mathematica Stack Exchange.

Mathematica code:

img = 
 ImageCrop@DensityPlot[
    Sin[2 x - 20 Log[2 (Sin[y]^2 + 1), 2]],
 {x, 0, 16 Pi}, {y, 0, 32 Pi},
 PlotPoints -> 250, ColorFunction -> "SunsetColors", 
 Frame -> False, ImageSize -> 600]

LogPolar
[x_, y_] := {Log[Sqrt[x^2 + y^2]], ArcTan[x, y]}

d = ImageDimensions[img][[1]]
Manipulate
[ ImageResize[ ImageTransformation[ ImageTake[ img, {1, 14*d/16}, {1 + (2 - 2 t)*d/32, (32 - 2 t)*d/32}], LogPolar[#[[1]], #[[2]]] &, DataRange -> {{-Pi, Pi}, {-Pi, Pi}}], 500], {t, 0, 6/7, 1/7}]
 
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