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	SBA	Filterbank
R=f	-11/64 -0.171875	36/64 0.5625	91/64 1.421875	85/64 1.328125	8.63473	22.0331
R=t,D=f	-12/64 -0.1875	41/64 0.640625	92/64 1.4375	1093/768 1.423177	8.60486	20.0573
R=t,D=t max CG	-16/64 -0.25	41/64 0.640625	92/64 1.4375	93/64 1.453125	8.59886	18.9411
R=t,D=t max SBA	-8/64 -0.125	30/64 0.46875	136/64 2.125	91/64 1.421875	8.23230	25.1934	File:4x8rd sba.png

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	Filterbank
R=f	-23/64 -0.359375	-18/64 -0.28125	-6/64 -0.09375	48/64 0.75	34/64 0.53125	20/64 0.3125	90/64 1.40625	73/64 1.140625	72/64 1.125	75/64 1.171875	9.60021
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

16x32

Best-known real-valued coefficients for V (R=t):

p0 = -0.42111473798940136

p1 = -0.4121736499899753

p2 = -0.3350240707669929

p3 = -0.3224547931861314

p4 = -0.25883387978005545

p5 = -0.20951913473498104

p6 = -0.0598657149803332

q0 = 0.9107782439906195

q1 = 0.8109855829278226

q2 = 0.715846584586721

q3 = 0.6135951570714172

q4 = 0.49846644853347627

q5 = 0.3945215834922529

q6 = 0.21822275136248082

CG = 9.81157

	p0	p1	p2	p3	p4	p5	p6	q0	q1	q2	q3	q4	q5	q6	s0	s1	s2	s3	s4	s5	s6	s7	CG	Filterbank
R=f	-23/64 -0.359375	-23/64 -0.359375	-17/64 -0.265625	-12/64 -0.1875	-14/64 -0.21875	-15/64 -0.234375	-12/64 -0.1875	50/64 0.78125	41/64 0.640625	30/64 0.46875	20/64 0.3125	16/64 0.25	15/64 0.234375	11/64 0.171875	90/64 1.40625	72/64 1.125	74/64 1.15625	72/64 1.125	67/64 1.046875	66/64 1.03125	68/64 1.0625	74/64 1.15625	9.87871
R=t,D=f	-26/64 -0.40625	-27/64 -0.421875	-22/64 -0.34375	-18/64 -0.28125	-16/64 -0.25	-14/64 -0.21875	-5/64 -0.078125	58/64 0.90625	52/64 0.8125	45/64 0.703125	36/64 0.5625	31/64 0.484375	23/64 0.359375	13/64 0.203125	3/2 1.5	77/64 1.203125	77/64 1.203125	1105/896 1.23326	218/192 1.135417	197/176 1.119318	1919/1664 1.153245	4351/3840 1.133073	9.79398
R=t,D=t	-16/64 -0.25	-16/64 -0.25	-8/64 -0.125	-17/64 -0.265625	-27/64 -0.421875	-13/64 -0.203125	-2/64 -0.03125	57/64 0.890625	50/64 0.78125	40/64 0.625	32/64 0.5	32/64 0.5	28/64 0.4375	12/64 0.1875	112/64 1.75	84/64 1.3125	88/64 1.375	80/64 1.25	72/64 1.125	72/64 1.125	73/64 1.140625	70/64 1.09375	9.65884