We define a restricted domain as the discrete set of points representing any convex, four-connected, filled polygon whose (i) vertices lie on the lattice points, (ii) interior angles are multiples of 45°, and (iii) number of sides are at most eight. We describe the boundary code and discrete half-plane representation and use them for representing restricted domains. Morphological operations of dilation and n-fold dilation on the restricted domains with structuring elements that are also restricted domains are expressed in terms of the above representations. We give algorithms for these operations and prove that they are of O(1) complexity and hence are independent of the size of the objects.