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多角形のアンチエイリアス描画

アンチエイリアスのアルゴリズムというのはどっかに定番があるのだろうか。フォントだろうが図形だろうが、輪郭がギザギザにならないように、境界部分に中間色を配して滑らかに見せるテクニックは普通に用いられているが、ネットで検索してもなかなかその方法を解説した文書は見つからない。

そもそも多角形の内部を塗りつぶす方法は一般に走査線方式が用いられる。上から下まで水平線を走査し、多角形の辺とぶつかったら塗りつぶし開始、もう一度辺とぶつかったら塗りつぶし停止。要するに横に走査していって奇数番目の辺と交差したらピクセルを塗り、偶数版目の線と交差したらやめるというのが用いられる。このため、線が交差するような複雑な図形の場合、内部に塗られない島ができる。

問題となるのは、多角形の辺がピクセルを切断する場合だ。斜めの直線と水平の走査線の交点を整数に丸めてしまうと、1か0にきっちり分かれ、特定のピクセルを完全に塗りつぶすか、全く塗りつぶさないかの二通りになる。

アンチエイリアスを実現するためには、線が切断しているピクセルを、その切断部の大きさに応じた濃さのグレーで塗りつぶす必要がある。その方法として僕が考えたのは次の二通り。

1:サブピクセル法
ピクセルを内部をさらに分割して計算する。たとえば1ピクセルを3×3のサブピクセルに分割し、それぞれのサブピクセルが多角形の辺の内側か外側が判断する。この場合サブピクセルの総数は9になるが、その中で辺の内側に入るものが5、外側になるものが4であるならば、そのピクセルの濃度は5/9=約56%グレーということになる。

2:面積計算法
ピクセルが辺で切られる場合、辺の内部は必ず台形か三角形か長方形になる。従ってそれぞれの面積の公式で濃度が計算可能である。ついでに長方形は上底と下底が等しい台形と言えるし、三角形は上底か下底かどっちかが0の台形と言えるので、計算自体は統合できるかもしれない。

1の方法では、3×3で0~9の10階調、と微妙に階調が少ない。16×16にすれば、0~256で257階調の表現が可能だが、たかだか1ピクセルを256回評価することになり、単純計算でアンチエイリアス無しの場合の256倍時間がかかる計算になる。

2の方法では完全に浮動小数点演算をするので、それ自体結構重い処理になるかもしれないが、現代のCPUでは浮動小数点演算はかなり高速化しており、計算自体は台形であれば1ピクセルに1度「(上底+下底)×高さ÷2」を行えばいいだけである。また、ビット深度を十分に生かした階調を表現できる。ただし、計算すべき三角形もしくは台形を判別するために場合分けする必要があり、これが結構めんどくさい。また、ピクセルの途中に頂点がある場合、そこでピクセルを分割し、上半分と下半分、左半分と右半分(場合によってはもっと多くの部分)ごとに計算し、結果を合成しなければならない。ただし、一般に多角形全体の中で、検討するピクセル数に対して、頂点の数はかなり少ないので、ピクセル分割の処理はそう多くないと思う。

デジコミツールズの集中線、流線ツールでは2の方法を使ったが、まあまあ満足できる処理速度に収まったと思う。

なんかもっと上手い方法があるかもしれないが。僕には思いつかなかった。

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