Kernel Coupled Cross-Regression for Low-Resolution Face Recognition

Table 1

The KCCR method used for LR FR.

LR/HR training images: 𝐗^𝑡𝑙∈ℜ^𝑚×𝑁 and 𝐗^𝑡ℎ∈ℜ^𝑀×𝑁
HR gallery images: 𝐗^𝑔ℎ∈ℜ^𝑀×𝑁𝑔 with class labels: 𝐜^𝑔∈ℜ^𝐶
LR probe images: 𝐗^𝑝𝑙∈ℜ^𝑚×𝑁𝑝 with class labels: 𝐜^𝑝∈ℜ^𝐶
Offline training phase:
(1) Learning low embeddings: 𝐲𝑙∈ℜ^𝑁×𝑑 and 𝐲ℎ∈ℜ^𝑁×𝑑
(2) Computing kernel coupled cross projection matrices:
𝐀𝑙∈ℜ^𝑚×𝑑 and 𝐀ℎ∈ℜ^𝑀×𝑑
(3) Transforming HR gallery images:
𝐲^𝑔ℎ=Φ(𝐗^𝑔ℎ)^𝑇𝐀ℎ
=Φ(𝐗^𝑔ℎ)^𝑇Φ(𝐗ℎ)(𝐾ℎ+𝑁𝛽𝐈ℎ)^−1𝐲𝑙
=𝑘ℎ(𝐾ℎ+𝑁𝛽𝐈ℎ)^−1𝐲𝑙,
where 𝑘ℎ𝑖,𝑗=Φ(𝑥^𝑔ℎ𝑖)^𝑇Φ(𝑥ℎ𝑗)=𝑘(𝑥^𝑔ℎ𝑖,𝑥ℎ𝑗)
Online testing phase:
(1) Transforming LR probe images:
𝐲^𝑝𝑙=Φ(𝐗^𝑝𝑙)^𝑇𝐀𝑙
=Φ(𝐗^𝑝𝑙)^𝑇Φ(𝐗𝑙)(𝐾𝑙+𝑁𝛼𝐈𝑙)^−1𝐲ℎ
=𝑘𝑙(𝐾𝑙+𝑁𝛼𝐈𝑙)^−1𝐲ℎ,
where 𝑘𝑙𝑖,𝑗=Φ(𝑥^𝑝𝑙𝑖)^𝑇Φ(𝑥𝑙𝑗)=𝑘(𝑥^𝑝𝑙𝑖,𝑥𝑙𝑗)
(2) Performing feature matching: comparing 𝐲^𝑝𝑙 and 𝐲^𝑔ℎ by; nearest neighbour classifier and obtaining 𝐜^𝑝 from 𝐜^𝑔