# 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 |