A Bayesian Nonparametric Approach to Image Super-resolution
arxivからの論文
http://arxiv.org/pdf/1209.5019.pdf
A Bayesian Nonparametric Approach to Image
Super-resolution
Gungor Polatkan, Mingyuan Zhou, Lawrence Carin, David Blei, and Ingrid
Daubechies
ノンパラメトリックベイズでは有名な Bleiのグループとの共同研究といった感じでしょうか？
超解像技術（super-resolution）は低解像度画像から高解像度画像を作る技術．
全画素の組み合わせに対して，実際に観測される組み合わせは非常にスパースであることから，
パッチを組み合わせることで，高解像度画像を低解像度画像から復元することができます．
そのためには，辞書(Dictionary)を持つ必要があるのですが，
それをどのように作るかが問題となります．
筆者らは過去に
NIPSで
Non-Parametric Bayesian Dictionary Learning for
Sparse Image Representations
http://books.nips.cc/papers/files/nips22/NIPS2009_0190.pdf
を発表しており，ノンパラメトリックベイズを用いて，Dictionary Learning にノンパラメトリックベイズを
応用するということをやっています．
それをsuper resolutionに応用するというのが主な筋立てです．
基本的にスパースな表現を得る場合には，L1ノルムを用いて刈り込む事が多くて，
超解像でもこれがよく用いられます．
Image Super-Resolution via Sparse Representation
Jianchao Yang et al.
などが良くリファレンスされるらしいです．
これに対して，ノンパラメトリックベイズ業界（？）ではスパースな表現にする，
つまり用いない次元を作るような場合には，ベータ・ベルヌーイ分布を導入し，スイッチを作るのが定石です．
例えば，
Sharing Features among Dynamical Systems
with Beta Processes
Emily B. Fox et al.
http://videolectures.net/nips09_fox_sfa/
では，HDP-HMM の各隠れ状態に対してストリーム毎にベータ・ベルヌーイのスイッチを設けて，使わない隠れ状態をオフにします．
ちなみに， @k_ishiguro さんの，
Subset Infinite Relational Models
Katsuhiko Ishiguro et al.
でも，ベータ・ベルヌーイのスイッチをつくって，汎用的な出力分布を用いる（IRMの外に吐き出してしまう）か，
通常のIRMの側に入れるかをえらぶようにしていたりします．
というわけで，
「L1刈り込みの代わりを，ノンパラベイズでやるなら，やっぱベータ・ベルヌーイっしょ！」
という，結構ストレートフォワードな適用があるわけです．
グラフィカルモデルはこんな感じ．
l と h はlow resolution と high resolutionを表している．
xl と xh が観測．
Di と Dh が辞書．
で si が係数なんですが，
zi がいわゆるベータ・ベルヌーイのスイッチで，0,1 をとる．
これによって，使う基底，使わない基底が，0,1でオン・オフされることで，
スパース表現を得るわけである．
なんとも，ストレートフォワードな論理である．
ちなみに，これだけでは綺麗にならないみたいで，最後に平滑化処理っぽいことをやる．
実験の結果は
こんなかんじなのだが，正直，よくわからない・・・．
どうも既存手法に勝てているか微妙なのだが，
なるほどな，とおもったのは， Fig.8 で
こんな図がある．
これは，辞書の要素数（もとの次元数）を大きくしていった際，BPの場合は打ち切り最大数を大きくしていった時にどうなるか
を示しているのだと思うが，
その時に，ScSR（L1ノルムでのスパースコーディング）はピークを持ってしまう．
これに対しノンパラメトリックベイズのアプローチでは，十分な 要素数があれば，良い値を推定できるので，その良さが維持される．
これは，BPのアプローチがもともと無限の状態数を前提として組まれているのに対して，
L1の正則化項は 無次元量でもなく，要素数に影響を受けてしまうからだろう．
なるほどねー．
とは思うが，計算量とか考えても，実用的にはL1で行ったほうが，楽で実用的なのかなぁ，と思った次第でございます．
本内容は，
Xian-Hua Han (韓 先花） Ph.D にご紹介いただいて (Thank you very much
http://www.iipl.is.ritsumei.ac.jp/XHHan/index.html
それを，僕が勝手に理解したものを書いたものであり，
この記事の内容に誤りがあった場合は，僕を責めてくださいませ．
![ビブリオバトル[知的書評合戦]](img/bn_biblio.jpg)
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