How to imitate the opacity plot in EditColorFunction

Question

I wish to imitate the opacity plot as shown below:

Edit2: (ref: https://www.wolfram.com/mathematica/new-in-9/3d-volumetric-image-processing/edit-color-function-palette-for-image3d.html)

Here is what I am up to now:

makeColorFn[colors_, opacities_, vars_] := 
  Transpose[{vars, MapThread[Append, {colors, opacities}]}] /. 
    body_ :> (Blend[body, #] &)

colors = 
  {RGBColor[0.761959, 0.470832, 0.940597], RGBColor[0.927848, 0.742785, 0.615138], 
   RGBColor[0.929162, 0.95034, 664815], RGBColor[0.431296, 0.709773, 0.927077]};

opacities = {0.632, 0.938, 0.906, 0.864};

keys = {0, 1/3, 2/3, 1};

mesh = 
  ParametricPlot[{v, u}, {v, 0, 1}, {u, 0, 1}, 
    MeshFunctions -> {#2 &, #1 &}, 
    MeshShading -> {{Gray, White}, {White, Gray}}, Mesh -> {20, 20}, 
    FrameLabel -> {"Intensity", Opacity}, 
    ColorFunction -> "", 
    PlotRange -> {0, 1}]

The above code was taken from here and gives me the following plot:

If I now plot the color function on top of this background,

ListLinePlot[Transpose[{keys, opacities}], 
   Prolog -> mesh[[1]], 
   Mesh -> All, MeshStyle -> Directive[PointSize[Large], colors], 
   PlotRange -> Full, Filling -> Axis, 
   ColorFunction -> ColorTransferFunction[colors, opacities, keys], 
   Frame -> True, FrameLabel -> {"Intensity", "Opacity"}]

And the corresponding result is:

Firstly, I need the plot to be scaled accordingly as in fig 2.

Secondly, I want the control points to be colored according to the colors and opacities at those points and not a single color (blue) as I have got.

[Optional: How can I remove the checkered background above the control points?]

Update

I have updated the code for plotting as follows:

DynamicModule[{points = Transpose[{keys, opacities}]},
  ListLinePlot[points, 
    Epilog -> {PointSize[0.03], Point[points, VertexColors -> colors]},
    Prolog -> mesh[[1]],
    PlotRange -> {{0., 1.}, {0., 1.}},
    Frame -> True,
    FrameLabel -> {"Intensity", "Opacity"},
    Filling -> Axis,
    ColorFunction -> ColorTransferFunction[colors, opacities, keys]]]

and got the result:

Answer 1

Using Texture just like Kuba this is the lazy-person way to solve your issue. I just used your points because I didn't want to be clever or anything. I'm using Kuba's bg and his correction for the standard AspectRatio because it was too much work to adapt mesh. If you want different aspect ratios you'll need to correct the VertexTextureCoordinates appropriately.

DynamicModule[{points = Transpose[{keys, opacities}]},
 ListLinePlot[
  points,
  Epilog -> {
    PointSize[0.03],
    Point[points,
     VertexColors -> colors
     ]
    },
  Prolog -> {
    Texture@bg,
    With[{l =
       Join[
        {
         {0, 0}
         },
        points,
        {
         {1, 0},
         {0, 0}
         }]},
     Polygon[l,
      VertexTextureCoordinates ->
       Map[{GoldenRatio, 1}*# &, l]
      ]
     ]
    },
  PlotRange -> {{0., 1.}, {0., 1.}},
  Frame -> True,
  FrameLabel -> {"Intensity", "Opacity"},
  Filling -> Axis,
  ColorFunction -> (Append[RandomColor[], RandomReal[]] &)]
 ]

Looks like this (note that I used RandomColor because I didn't know your ColorTransferFunction:

Answer 2

bg = ImageResize[Image @ ArrayPad[{{1, .8}, {.8, 1}}, 5, "Periodic"], 400]

Firstly, I need the plot to be scaled accordingly as in fig 2.

Most of *Plot* functions have 1/GoldenRatio as an AspectRatio, it affects the texture too, we need to compensate for that by adjusting VertexTextureCoordinates:

Graphics[Disk[{0, 0}, .5]
  , AspectRatio -> 1/GoldenRatio
  , PlotRange -> 1
  , Frame -> True
  , Prolog -> {
        Texture[bg], Polygon[
            Scaled /@ {{0, 0}, {1, 0}, {1, 1}, {0, 1}}
          , VertexTextureCoordinates -> {
                {0, 0}, {GoldenRatio, 0}, {GoldenRatio, 1}, {0, 1}
            }
        ]
    }
]

Secondly, I want the control points to be colored according to the colors and opacities at those points and not a single color (blue) as I have got.

Solved by OP.

[Optional: How can I remove the checkered background above the control points?]

Striaghtorward way is to apply Texture only to a polygon which consists of your points and bottom corners of the plot. But it is easier to control VertexTextureCoordinates for fixed rectangular polygon so the other way is to draw a white one over it:

... Prolog -> {
        ... 
      , White, Polygon[{{-1, .5}, {.5, 0}, {1, 1}, {-1, 1}}]
    }
...