The LOT does not decay smoothly to zero at the boundaries, as shown in the figures of the previous section. Thus, discontinuities still occur in images reconstructed using the LOT, although they should be more bland than the discontinuities evident by using unlapped transforms. Smoothing out the decay, should result in less blocking effects, since there should be less sharp discontinuities due to quantization errors.
A lapped transform should still allow for perfect reconstruction of the
image with no quantization. There should be a separate forward and
backward transform, that are not related simply by conjugate transposition as in
the orthogonal transform. A biorthogonal transform can be represented in
matrix form, with a forward analyses matrix 
and a backward synthesis
matrix 
. Orthogonality must still hold in that
The asymmetry of the biorthogonal transform allows for more flexibility in
the shape of the basis functions. A careful modification of the original
LOT 
matrix allows us to reshape the poorly behaved LOT basis
functions. These new synthesis functions decay closer to zero in a
smoother fashion than the LOT basis functions do.
The analyses (synthesis) transformation is changed by multiplying (dividing) the each first
odd DCT coefficient by 
. This leads to an asymmetric transform,
which is equivalent to pre-multiplying all 
terms in the original
LOT matrix by a scaling matrix
and the backwards scaling matrix
This leads to the following synthesis matrix:
and analysis matrix
Now,
where c is the contribution from adjacent blocks. Notice that the overlap condition is valid, in that
where W is the shift matrix, defined by
Thus, the entire transformation process is truly invertible, with proper
selection of the boundary matrices, 
to give
entire matrices
The transformation relationships then become
