image processing - How do I transform a rasterized graphic's coordinates back to its original ListPlot data coordinates?

From this question I successfully made an elliptical fit for my data. However, when I try and collect the datapoints within the ellipse via this question the coordinates of my ellipse correspond to an ellipse of a rasterized image of my listplot data and not the original. How can I transform or scale the ellipse that fits the rasterized image to the original listplot.

Ok, with all that said heres an example.

data = RandomReal[NormalDistribution[], {100000, 2}]
p = ListPlot[data, ImageSize -> 4000];

f = FillingTransform@ColorNegate@Binarize@p // DeleteSmallComponents
{c, s, t} = 1 /. ComponentMeasurements[f, {"Centroid", "SemiAxes", "Orientation"}]
Show[Rasterize[p], Graphics[{Red, Rotate[Circle[c, s],t]}]]

where I get a nice image:

however, the coordinates c, s, t of the ellipse are pixle coordiantes corresponding to the rasterized image rather than data coordinates.

So when I need the ellipses parameter to do any calculations I get bunk results.

The image processing approach would be the best as I am filtering out the most dense cluster of data.

Thanks so much.

Answer

(*Generate Data and fit*)
data1 = RandomReal[NormalDistribution[10, 1], {10^4}];(*test data*)
data2 = RandomReal[NormalDistribution[20, 5], {10^4}];(*test data*)
data  = Transpose@{data1, data2};
r     = RotationTransform[Pi/8];
data3 = r /@ data;

(*we need to specify PlotRange due to a kown bug in AbsoluteOptions[] *)
prange = {Min@#, Max@#} & /@ {First@#, Last@#} &@Transpose@data3;


p = ListPlot[data3, Axes -> None, PlotRange -> prange];
f = FillingTransform@ColorNegate@Binarize@p // DeleteSmallComponents;
{co, so, to} = 1 /. ComponentMeasurements[f, {"Centroid", "SemiAxes", "Orientation"}];

(*Transform Image to Graphic coordinates*)
c = Rescale[co[[#]], {1, ImageDimensions[f][[#]]}, prange[[#]]] & /@ {1, 2};
s = Rescale[so[[#]], {0, Norm@ImageDimensions@f}, {0, Norm@(Differences/@ prange)}] & /@ {1, 2};
t = -ArcTan@Rescale[Tan@to, {0, 1/Divide @@ ImageDimensions[f]}, 
                            {0, 1/First@(Divide @@ Differences /@ prange)}];

(* Replot graphic*)

{s1, s2} = s;
{cx, cy} = c;
f0 = Sqrt[s1 s1 - s2 s2];
f1 = {cx + f0 Cos[t], cy - f0 Sin[t]};
f2 = {cx - f0 Cos[t], cy + f0 Sin[t]};
r = 2 Sqrt[f0 f0 + s2 s2];

sd = Select[data3, EuclideanDistance[#, f1] + EuclideanDistance[#, f2] < r &];

Show[p, Graphics[{Red, PointSize[Large], Point@sd}], Axes -> True]