TDLT
This page holds the results of Time Domain Lapped Transform (TDLT) optimization problems looking for integer transform coefficients that provide optimal coding gain. Wherever possible the assumptions are stated. Later we should include testing against actual image data to verify the results (see test data here).
The coding gain objective used as the objective is taken from slide 13 of Tim's presentation An Introduction to Video Coding
<need figure with block matrix diagrams>
The free parameters are initially just the coefficients p_0,...,p_m,q_0,...,q_m where m=(n/2)-1. We limit these to being dyadic rationals, e.g., x/2^d with d=6, between [-1,1].
Given p's and q's and assuming a linear ramp constrains the s's.
4x8
s0 = 4*(1-q0) s1 = 4*(1-p0*(1-q0))/3
Optimal real-valued coefficients for V:
p0 = -0.18117338915051454
q0 = 0.6331818230771687
CG = 8.60603
Optimal integer-valued coefficients (d=6) for V:
p0 = -12/64 = -0.1875
q0 = 41/64 = 0.640625
CG = 8.60486
Optimal integer-valued coefficients (d=6) were (1-p0*(1-q0)) is divisible by 3:
p0 = -13/64 = -0.203125
q0 = 41/64 = 0.640625
CG = 8.60446
8x16
Optimal real-valued coefficients for V:
p0 = -0.3625885819608689
p1 = -0.41911703104506826
p2 = -0.06962862896922385
q0 = 0.8423686986993569
q1 = 0.6393969893363041
q2 = 0.3894929113709107
CG = 9.52441
Optimal [maybe] integer-valued coefficients (d=6) for V:
p0 = -26/64 = -0.40625
p1 = -22/64 = -0.34375
p2 = -8/64 = -0.125
q0 = 53/64 = 0.828125
q1 = 41/64 = 0.640625
q2 = 26/64 = 0.40625
9.56627