Optimizing recursive RDom

[itaydaon]

I am quite new to halide and used to schedule my algorithm manually.
I managed to reach similar level of scheduling in halide with many examples but not with this one.
I am trying to implement the following integral+transpose function:

  Var x("x"), y("y");
  Func integral("integral");
  integral(x,y) = cast<uint32_t>(input(x, y));
  RDom r2y(1, input.height()-1,"r2y");
  integral(x,r2y) += integral(x, r2y-1);
  output(x, y) = integral(x, y);

The code genration I would like to acheive is somthing like this:

for (x, 0, width) {
	t1 = input(x,0)
	output(0,y) = t1
	for (y ,1, height-1) {
		t1 = input(x,y) + t1
		output(y,x) = t1
	}
}

Most likely I would like to add some vectorization on the x level but the most important goal is to reduce memory accesses and fuse the reduction stage with the in/out copies.
In all my trials, I ended up with full copy from input to integral buffer + read&write from integral buffer per each index + full copy to output

Any ideas on how to write the desired schedule?

Thanks for any suggestions.