I am interested in photographing buildings such that the vertical lines are preserved. As shown in the image on the right.

I understand that this can be achieved with a tilt-shift lens. However, my Fujifilm X100 has a fixed lens, so I wont be able to do this with my current camera.

So I am interested in achieving a similar result using software. Note that I am not after that "minature" look with shallow depth of field at a long distance. This is what most software that is billed as "fake tilt-shift" seems to reproduce.

I simply want my vertical lines to be preserved.

Is this as simple as projecting a rectangular image onto a trapezoid? Just squashing the bottom edge of the rectangle? Or is it more involved than this? Is the projected shape more complex that a trapezoid perhaps?

I don't have access to Photoshop, but do use Gimp and Inkscape. And if needed, can write my own software. To clarify, I am after the actual geometric transformation.