morph surface

i need to create an animation that will deform a flat wall into something similar to the next image >


i need to control the animation with something similar to “lerp”
from flat (lerp 0.0) to the shape of the image (lerp 1.0)

Can you suggest how i could make this effect ?

previously i was trying to use tesellation on the material but that wont allow to have a convex shape ,
will this has to be done with morph target or does it exist a way to morph a plane into a complex shape ?