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

Optimal real-valued coefficients for V:

p0 = -0.18117338915051454

q0 = 0.6331818230771687

CG = 8.60603

	p0	q0	s0	s1	CG
R=f	-11/64 -0.171875	36/64 0.5625	91/64 1.421875	85/64 1.328125	8.63473
R=t,D=f	-12/64 -0.1875	41/64 0.640625	92/64 1.4375	1093/768 1.423177	8.60486
R=t,D=t	-16/64 -0.25	41/64 0.640625	92/64 1.4375	93/64 1.453125	8.59886

8x16

Optimal real-valued coefficients for V:

p0 = -0.39460731547057293

p1 = -0.33002212811740816

p2 = -0.12391270981321137

q0 = 0.822154737511288

q1 = 0.632488694485779

q2 = 0.40214668677553894

CG = 9.56867

	p0	p1	p2	q0	q1	q2	s0	s1	s2	s3	CG
R=f
R=t,D=f	-26/64 -0.40625	-22/64 -0.34375	-8/64 -0.125	53/64 0.828125	41/64 0.640625	26/64 0.40625	11/8 1.375	879/768 1.14453125	1469/1280 1.14765625	275/224 1.2276785714285714	9.56627
R=t,D=t	-24/64 -0.375	-20/64 -0.3125	-4/64 -0.0625	53/64 0.828125	40/64 0.625	24/64 0.375	88/64 1.375	75/64 1.171875	76/64 1.1875	76/64 1.1875	9.56161