TiKZ: How to apply a pgftransformnonlinear to imported image using includegraphics

Question

I've written a non linear transform and am able to transform figures drawn in TiKZ correctly (transformed the rectangle on the left to the arc on the right) but doing the same on \includegraphics doesn't work. How can I apply the same transformation on it ? Here's the code I have for transformation

%!tikz editor 1.0
\documentclass[tikz]{standalone}
\usepackage{tikz}
\usepackage[graphics, active, tightpage]{preview}
\PreviewEnvironment{tikzpicture}

%!tikz preamble begin
\usepgfmodule{nonlineartransformations}
\usepgfmodule[nonlineartransformations]
\usepackage{mathtools}

\makeatletter
\newcommand*{\length}{50}%
\newcommand*{\width}{20}%
\newcommand*{\PI}{3.14}%
\newcommand*{\Rad}{30}%
\newcommand*{\rad}{27.5}%
\newcommand*{\slant}{20}%

\pgfmathsetmacro{\alpha}{(2*\PI*(\Rad-\rad))/\slant}%
\pgfmathsetmacro{\tconst}{(\rad*\slant)/(\Rad-\rad)}%

\tikzset{declare function={zeta(\x,\y) = \tconst + ((\x*\slant)/\length);}}
\tikzset{declare function={theta(\x,\y)=(\alpha*\y)/\width;}}
\tikzset{declare function={shi(\x)=\x+60;}}

\def\polartransformation
{   
    \pgfmathsetmacro{\costheta}{cos(deg(theta(\pgf@x,\pgf@y)))}

    \pgfmathsetmacro{\sintheta}{sin(deg(theta(\pgf@x,\pgf@y)))}

    \pgfmathsetmacro{\resz}{zeta(\pgf@x,\pgf@y)}


    \pgfmathsetmacro{\xcoord}{\resz*\costheta}

    \pgfmathsetmacro{\ycoord}{\resz*\sintheta}

    \setlength{\pgf@x}{\xcoord pt}
    \setlength{\pgf@y}{\ycoord pt}
}
\makeatother
%!tikz preamble end


\begin{document}

%!tikz source begin
\begin{tikzpicture}
\draw [black, left color=white, right color=gray ] (100pt,20pt) rectangle (150pt,0pt);
{
        \pgftransformnonlinear{\polartransformation}
        \pgfsettransformnonlinearflatness{0.2pt}
        \draw [black, left color=white, right color=gray ] (0pt,20pt) rectangle (50pt,0pt);
}
\end{tikzpicture}
%!tikz source end
\end{document}

Answer 1

This is almost completely copied from this very nice answer except that I changed the map to a polar transformation.

In order to explain the changes, recall first that fx and fy are the images of the transformation, i.e.

(x,y) \mapsto (fx(x,y),fy(x,y)) .

The other functions fxx, fxy, fyx and fyy are derived from these functions (and are just the derivatives of the function in the lattice sense). In order to obtain a polar transformation, one may interpret x as the angle and y as the radius. Then a possible choice for a polar transformation is

fx(x,y) = -(y+10)*cos(x*5) ,
fy(x,y) = (y+10)*sin(x+y) ,

where the numerical constants 10 and 5 are just chosen by hand to get a reasonable output. If you shift the argument of the trigonometric function you will rotate the image, and so on and so forth.

\documentclass[border=9,tikz]{standalone}
\begin{document}

\pgfmathdeclarefunction{fx}{2}{\pgfmathparse{-(#2+10)*sin(#1*5)}}
\pgfmathdeclarefunction{fy}{2}{\pgfmathparse{(#2+10)*cos(#1*5)}}

\pgfmathdeclarefunction{fxx}{2}{\pgfmathparse{fx(#1+1,#2)-fx(#1,#2)}}
\pgfmathdeclarefunction{fxy}{2}{\pgfmathparse{fy(#1+1,#2)-fy(#1,#2)}}
\pgfmathdeclarefunction{fyx}{2}{\pgfmathparse{fx(#1,#2+1)-fx(#1,#2)}}
\pgfmathdeclarefunction{fyy}{2}{\pgfmathparse{fy(#1,#2+1)-fy(#1,#2)}}


\begin{tikzpicture}

    \path(-15,-5)(15,18);
    \foreach\i in{-10,...,9}{
        \foreach\j in{-5,...,4}{
            \pgfmathsetmacro\aa{fxx(\i,\j)}
            \pgfmathsetmacro\ab{fxy(\i,\j)}
            \pgfmathsetmacro\ba{fyx(\i,\j)}
            \pgfmathsetmacro\bb{fyy(\i,\j)}
            \pgfmathsetmacro\xx{fx (\i,\j)}
            \pgfmathsetmacro\yy{fy (\i,\j)}
            \pgflowlevelobj{
                \pgfsettransformentries{\aa}{\ab}{\ba}{\bb}{\xx cm}{\yy cm}
            }{
                \fill[black!10](1,0)--(0,0)--(0,1);
                \clip(1,0)--(0,0)--(0,1)--cycle;
                \tikzset{shift={(-\i,-\j)}}
                \path(0,0)node{\includegraphics[width=20cm,height=10cm]{example-image-duck}};
            }
            \pgfmathsetmacro\aa{fxx(\i  ,\j+1)}
            \pgfmathsetmacro\ab{fxy(\i  ,\j+1)}
            \pgfmathsetmacro\ba{fyx(\i+1,\j  )}
            \pgfmathsetmacro\bb{fyy(\i+1,\j  )}
            \pgfmathsetmacro\xx{fx (\i+1,\j+1)}
            \pgfmathsetmacro\yy{fy (\i+1,\j+1)}
            \pgflowlevelobj{
                \pgfsettransformentries{\aa}{\ab}{\ba}{\bb}{\xx cm}{\yy cm}
            }{
                \clip(0,0)--(-1,0)--(0,-1)--cycle;
                \tikzset{shift={(-\i-1,-\j-1)}}
                \path(0,0)node{\includegraphics[width=20cm,height=10cm]{example-image-duck}};
            }
        }
    }
\end{tikzpicture}

\end{document}