# 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 |

## 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 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

R=f | |||||||||||||||||||||||

R=t,D=f | -26/64 -0.40625 |
-23/64 -0.359375 |
-20/64 -0.3125 |
-18/64 -0.28125 |
-14/64 -0.21875 |
-14/64 -0.21875 |
-2/64 -0.03125 |
58/64 0.90625 |
52/64 0.8125 |
45/64 0.703125 |
36/64 0.5625 |
31/64 0.484375 |
23/64 0.359375 |
16/64 0.25 |
3/2 1.5 |
77/64 1.20313 |
373/320 1.16563 |
543/448 1.21205 |
109/96 1.13542 |
1543/1408 1.09588 |
1823/1664 1.09555 |
131/120 1.09167 |
9.79008 |

R=t,D=t |