How to guarantee the point's position is fixed in the Graphics and Image

Question

SeedRandom[8]
plot = ParametricPlot[
  Evaluate@BezierFunction[temPoint = RandomReal[{-50, 50}, {5, 2}]][
    t], {t, 0, 1}, AxesOrigin -> {0, 0}]

We can hightlight a random point in this graphics.

SeedRandom[3]
Show[plot, 
 Epilog -> {Red, AbsolutePointSize[10], 
   Point[randomPoint = 
     RandomReal /@ RegionBounds@DiscretizeGraphics[plot]]}]

We can change the plot to image.

Image[plot]

Then its coordinate system will be a image coordinate system.To guarantee the randomPoint be the original graphics' position in the new coordinate system,I use RescalingTransform try to implement this target.

HighlightImage[
 Image[plot], {Red, PointSize[.03], 
  Point[RescalingTransform[
     First@Values[AbsoluteOptions[plot, PlotRange]], 
     Transpose[{{0, 0}, ImageDimensions@Image[plot]}]]@randomPoint]}]

But as you see,there is a error in this method obviously.

Answer 1

This isn't elegant, but I think it should work pretty reliably.

Start with the base plot:

plot = ParametricPlot[
  Evaluate@BezierFunction[temPoint = RandomReal[{-50, 50}, {5, 2}]][
    t], {t, 0, 1}, AxesOrigin -> {0, 0}]

Create a version that has the point, but wrap the point in Annotation, with type "Region":

plot2 = Show[plot, 
  Epilog -> {Red, AbsolutePointSize[10], 
    Annotation[
     Point[randomPoint = 
       RandomReal /@ RegionBounds@DiscretizeGraphics[plot]], "Point", 
     "Region"]}]

Rasterize the second one, and extract the regions, converting to the center of the point:

{px, py} = Mean[{"Point", "Region"} /. Rasterize[plot2, "Regions"]];

Convert the main plot to an image:

image = Image[plot];

Get the dimensions of the image:

{ix, iy} = ImageDimensions[image];

Highlight the image, keeping in mind that the region annotations use a coordinate system that is flipped vertically from the usual one used by Image:

HighlightImage[
 Image[plot], {Red, PointSize[.03], Point[{px, iy - py}]}]