JP4191419B2 - Computer generated hologram production apparatus and method - Google Patents

Description

なる式で表される。ここで、ｉは虚数単位を表す。また、Δｈ_jkは、計算機ホログラムの画素（ｊ，ｋ）の複素振幅変化量を表し、
【００３０】
【数２】

なる式で表される。
【００３１】
上記(1)式から判るように、再生像上の各画素（ｍ，ｎ）の光の複素振幅変化量ΔＨ_mnは、計算機ホログラムの画素（ｊ，ｋ）の複素振幅変化量Δｈ_jkのみに依存している。このような変化の後の再生像上の画素（ｍ，ｎ）の光の複素振幅Ｈ'_mnは、変化前の複素振幅Ｈ_mnに対して、
【００３２】
【数３】

なる式で表される。したがって、再生像上の各画素（ｍ，ｎ）の光の複素振幅Ｈ_mnは、最初に１回だけ計算しておけば、後は変化量ΔＨ_mnを加算していくだけでよい。
【００３３】
ＳＡ法等の最適化法では、以上のようにして計算された再生像と目標再生像との誤差Ｃを計算し、その誤差Ｃを評価して、選択した画素の画素値の変化を受け入れるか否かを決定する。この誤差Ｃの計算に際しては、再生像のうちの関心領域Ω内においてのみ、
【００３４】
【数４】

なる二乗和を求めればよい。ここで、Ｆ_mnは目標再生像上の画素（ｍ，ｎ）の光の複素振幅である。なお、目標再生像のエネルギ（全ての画素の光の複素振幅の二乗和）を入力光のエネルギ（ｈ_jkの絶対値の二乗和）と等しくしておけば、上記(4)式のように計算範囲を関心領域Ω内に制限しても、最適化の結果には殆ど影響を与えない。
【００３５】
上記(1)式の右辺のexp関数は、実数部がcos関数を用いて計算され、虚数部がsin関数を用いて計算される。これらcos関数およびsin関数それぞれの計算に替えて、予め用意したルックアップテーブルを参照することにより、上記(1)式の右辺のexp関数の値を短時間に求めることができる。整数ＭおよびＮの最小公倍数Ｌ（Ｌ＝ｄＭ＝ｅＮ、ｄおよびｅそれぞれは整数。）を用いると、ここで用いられるルックアップテーブルＬＵＴ１は、
【００３６】
【数５】

なる式で表される。なお、Kirkらの文献では、(1)式右辺の指数関数部をルックアップテーブル化している。彼らの場合は、ホログラムが２値でありΔｈ_jkが２および−２の２つの値しかもたないので、これで十分に高速化の効果が現れた。しかし、多値の場合には、本実施形態のようにΔｈ_jkの部分も含めてルックアップテーブル化しないと大きな効果が期待できない。
【００３７】
計算機ホログラムでは、各画素（ｊ，ｋ）の複素振幅ｈ_jkは、強度および位相の双方または何れか一方が標本化され量子化される。したがって、上記(1)式の指数関数部だけでなく複素振幅変化量Δｈ_jkも、予め用意したルックアップテーブルを参照することにより求めることができる。
【００３８】
作成しようとする計算機ホログラムが強度ホログラムである場合には、計算機ホログラムの各画素（ｊ，ｋ）の複素振幅ｈ_jkおよびｈ'_jkそれぞれは量子化された実数である。したがって、この場合には、複素振幅変化量Δｈ_jkのルックアップテーブルを容易に作成することができる。また、上記(1)式の右辺の計算は、実数と複素数との積となるから、ルックアップテーブルを記憶したメモリの参照が３回であって、実数の乗算が２回であり、演算数が少ない。
【００３９】
作成しようとする計算機ホログラムが位相ホログラムである場合には、計算機ホログラムの各画素（ｊ，ｋ）の複素振幅ｈ_jkおよびｈ'_jkそれぞれは、振幅強度が一定の実数値であって、位相が量子化されていて、複素数である。計算機ホログラムが複素振幅ホログラムである場合には、計算機ホログラムの各画素（ｊ，ｋ）の複素振幅ｈ_jkおよびｈ'_jkそれぞれは、振幅強度および位相の双方が量子化されていて、複素数である。
【００４０】
計算機ホログラムが位相ホログラムまたは複素振幅ホログラムである場合に、計算機ホログラムの各画素（ｊ，ｋ）の複素振幅ｈ_jkの位相が０〜２πの範囲でＰ個のレベルで量子化されているものとする。計算機ホログラムの各画素（ｊ，ｋ）の複素振幅ｈ_jkの位相レベルがｐ（ただし、０≦ｐ＜Ｐ、ｐおよびＰそれぞれは整数。）であるとき、すなわち、位相値が２πｐ／Ｐであるとき、その複素振幅ｈ_jkは、
【００４１】
【数６】

なる式で表される。ここで、Ａ_jkは、実数値であり、計算機ホログラムの画素（ｊ，ｋ）からの再生光の振幅強度を表す。
【００４２】
最適化法の過程で計算機ホログラムの画素（ｊ，ｋ）の複素振幅の位相レベルがｐからｐ'へ変化したとすると、上記(2)式の計算機ホログラムの画素（ｊ，ｋ）の複素振幅変化量Δｈ_jkは、
【００４３】
【数７】

なる式で表される。この(7)式の右辺の括弧内はルックアップテーブル化が可能である。この場合のルックアップテーブルＬＵＴ２は、
【００４４】
【数８】

なる式で表される。そして、上記(1)式の再生像の画素（ｍ，ｎ）の複素振幅変化量ΔＨ_mnは、ルックアップテーブルＬＵＴ１およびＬＵＴ２を用いて、
【００４５】
【数９】

なる式で表される。
【００４６】
ルックアップテーブルＬＵＴ１およびＬＵＴ２それぞれの出力値は複素数であるので、この(9)式の右辺では、複素数と複素数との乗算を行うことになる。また、上記(9)式の右辺を計算するには、Ａ_jkのルックアップテーブルの参照、上記ルックアップテーブルＬＵＴ１の実数部および虚数部それぞれの参照、および、上記ルックアップテーブルＬＵＴ２の実数部および虚数部それぞれの参照が必要である。したがって、上記(9)式の右辺の計算では、ルックアップテーブルを記憶したメモリの参照が５回であって、実数の乗算が４回であり、実数の加減算が４回である。このように、上記(9)式に従って計算することで、再生像の画素（ｍ，ｎ）の複素振幅変化量ΔＨ_mnを短時間に求めることができる。
【００４７】
しかし、以下のようにルックアップテーブルを工夫することで、再生像の画素（ｍ，ｎ）の複素振幅変化量ΔＨ_mnを更に短時間に求めることができる。すなわち、上記(7)式を上記(1)式に代入すると、
【００４８】
【数１０】

なる式が得られる。そして、Ｐ，ＭおよびＮの最小公倍数をＱとして、
【００４９】
【数１１】

とする。ここで、ａ，ｂおよびｃそれぞれは整数である。このＱを用いると、上記(10)式は、
【００５０】
【数１２】

と表される。
【００５１】
この(12)式の右辺にある２つのexp関数それぞれは、その位相が０〜２πの範囲でＱ個のレベルで量子化されており、１つのルックアップテーブルに置き換えることができる。ここで用いられるルックアップテーブルＬＵＴ３は、
【００５２】
【数１３】

なる式で表される。ただし、０≦ｑ＜Ｑでり、ｑおよびＱそれぞれは整数である。このルックアップテーブルＬＵＴ３を用いれば、上記(12)式は、
【００５３】
【数１４】

と表される。
【００５４】
上記(14)式の右辺を計算するには、Ａ_jkのルックアップテーブルの参照、および、上記ルックアップテーブルＬＵＴ３の実数部および虚数部それぞれの参照が必要である。したがって、上記(14)式の右辺の計算では、ルックアップテーブルを記憶したメモリの参照が５回であって、実数の乗算が２回であり、実数の加減算が２回である。このように、上記(14)式に従って計算することで、上記(9)式に従う場合よりも、再生像の画素（ｍ，ｎ）の複素振幅変化量ΔＨ_mnを更に短時間に求めることができる。
【００５５】
また、計算機ホログラムの画素（ｊ，ｋ）の複素振幅ｈ_jkの位相レベルｐ，ｐ'それぞれの値の組み合わせによっては、計算機ホログラムの画素（ｊ，ｋ）の複素振幅ｈ_jkが実数または虚数になるので、この複素振幅ｈ_jkが実数、虚数およびその他の何れであるかによって場合分けして計算することで、再生像の画素（ｍ，ｎ）の複素振幅変化量ΔＨ_mnを更に短時間に求めることができる。
【００５６】
すなわち、計算機ホログラムの画素（ｊ，ｋ）の複素振幅ｈ_jkの位相レベルｐ，ｐ'それぞれの値は０以上Ｐ未満の整数であるから、
【００５７】
【数１５】

および
【００５８】
【数１６】

の何れかの関係式が満たされる場合には、計算機ホログラムの画素（ｊ，ｋ）の複素振幅変化量Δｈ_jkは、実数であり、
【００５９】
【数１７】

なる式で表される。このとき、上記(12)式は、
【００６０】
【数１８】

と表される。なお、この(18)式においてRe[・]は実数部のみを参照する操作を表す。
【００６１】
一方、
【００６２】
【数１９】

なる関係式が満たされる場合には、計算機ホログラムの画素（ｊ，ｋ）の複素振幅変化量Δｈ_jkは、虚数であり、
【００６３】
【数２０】

なる式で表される。このとき、上記(12)式は、
【００６４】
【数２１】

と表される。なお、この(21)式においてIm[・]は虚数部のみを参照する操作を表す。
【００６５】
上記(18)式の右辺を計算するには、Ａ_jkのルックアップテーブルの参照、および、上記ルックアップテーブルＬＵＴ３の実数部の参照が必要である。また、上記(21)式の右辺を計算するには、Ａ_jkのルックアップテーブルの参照、および、上記ルックアップテーブルＬＵＴ３の虚数部の参照が必要である。したがって、上記(19)式または上記(21)式の右辺の計算では、ルックアップテーブルを記憶したメモリの参照が４回であって、実数の乗算が３回であり、実数の加減算が不要である。このように、上記(15)式または上記(16)式が成立するときには上記(18)式に従って計算し、上記(19)式が成立するときには上記(21)式に従って計算することで、再生像の画素（ｍ，ｎ）の複素振幅変化量ΔＨ_mnを更に短時間に求めることができる。なお、上記(15)式、上記(16)式および上記(19)式の何れも成立しないときには、再生像の画素（ｍ，ｎ）の複素振幅変化量ΔＨ_mnを上記(14)式に従って計算する。
【００６６】
さらに、最適化法（特にＳＡ法）において、計算機ホログラムの画素（ｊ，ｋ）の位相レベル変化量Δｐ（＝ｐ'−ｐ）の値が限定されている場合があり、この場合には、以下のようにルックアップテーブルを工夫することで、再生像の画素（ｍ，ｎ）の複素振幅変化量ΔＨ_mnを更に短時間に求めることができる。すなわち、この場合には、上記(12)式は、
【００６７】
【数２２】

なる式で表される。ここで、位相レベルｐおよびｐ'それぞれは０からＰ−１までの整数であるので、位相レベル変化量Δｐは、−Ｐ＋１からＰ−１までの整数である。ただし、複素指数関数の性質から exp[-ix]＝exp[i(2π-x)] であるので、上記(22)式中の [exp(２πｉΔｐ／Ｐ)−１] については、位相レベル変化量Δｐに関し０からＰ−１までの範囲でルックアップテーブルを用意すればよい。
【００６８】
したがって、ｑ＝ａｐ＋ｂｊｍ＋ｃｋｎと表し（０≦ｑ＜Ｑ）、
【００６９】
【数２３】

なる式で表されるルックアップテーブルＬＵＴ４を用意することで、再生像の画素（ｍ，ｎ）の複素振幅変化量ΔＨ_mnは、
【００７０】
【数２４】

なる式で表される。ただし、ｓは、Δｐが正である場合はΔｐであり、Δｐが負である場合はＰ−Δｐで計算され、何れの場合にも０からＰ−１までの整数である。また、ＬＵＴ４'[・,・]は２次元配列を意味し、ＬＵＴ４[・]はこれを１次元配列で表したものである。
【００７１】
上記(24)式の右辺を計算するには、Ａ_jkのルックアップテーブルの参照、および、上記ルックアップテーブルＬＵＴ４の実数部および虚数部それぞれの参照が必要である。したがって、上記(24)式の右辺の計算では、ルックアップテーブルを記憶したメモリの参照が３回であって、実数の乗算が２回であり、実数の加減算が不要である。このように、計算機ホログラムの画素（ｊ，ｋ）の位相レベル変化量Δｐの値が限定されている場合には、上記(24)式に従って計算することで、再生像の画素（ｍ，ｎ）の複素振幅変化量ΔＨ_mnを更に短時間に求めることができる。
【００７２】
なお、各ルックアップテーブルは、実数部および虚数部を含むものであるのであって、実数部がcos関数を用いて計算される値であり、虚数部がsin関数を用いて計算される値である。このことを利用すれば、各ルックアップテーブルのサイズを小さくすることができる。例えば、cos(x)＝sin(x−π) なる関係を利用すれば、各ルックアップテーブルは、実数部と虚数部とを統合することができ、サイズが１／２になる。また、cos(x)＝−cos(x−π) および cos(x)＝cos(−x) なる関係を利用すれば、各ルックアップテーブルはサイズが１／４になる。これらの方法を同時に採用すれば、各ルックアップテーブルはサイズが１／８になる。ただし、この方法はルックアップテーブルＬＵＴ４には適用できない。
【００７３】
また、ＳＡ法では、内側のループ（ステップＳ２〜Ｓ７）における１回の探索操作において、計算機ホログラムにおいて画素値を変化させる画素は１つであるのが通常であるが、同時に複数の画素の画素値を変化させる場合がある。また、ＧＡ法では、１回の探索操作において同時に複数の画素の画素値を変化させる。このように１回の探索操作において同時に複数の画素の画素値を変化させる場合には留意を要する。すなわち、本実施形態による再生像の計算時間は、（再生像における関心領域Ωの画素数）×（計算機ホログラムにおいて画素値が変化する画素数）の値に比例する。計算機ホログラムにおいて画素値が変化する画素数が多い場合には、ＦＦＴによる計算時間と比べて、本実施形態のＤＦＴによる再生像の計算時間が長くなる場合があり得る。一方、ＦＦＴによる計算時間および本実施形態のＤＦＴによる計算時間それぞれは予め測定することができる。したがって、ＦＦＴによる計算時間と比べて本実施形態のＤＦＴによる計算時間が長くなると予想される場合には、ＦＦＴを用いて再生像を計算し、そうでない場合には、本実施形態のＤＦＴを用いて再生像を計算するようにすればよい。
【００７４】
以上では、本実施形態に係る計算機ホログラム作成装置および方法における再生像計算処理の理論面を中心に説明した。次に、本実施形態に係る計算機ホログラム作成装置の構成および動作について説明するとともに、本実施形態に係る計算機ホログラム作成方法について説明する。図１は、本実施形態に係る計算機ホログラム作成装置１の構成図である。
【００７５】
本実施形態に係る計算機ホログラム作成装置１は、上述したルックアップテーブルＬＵＴ３またはＬＵＴ４を参照してＤＦＴにより再生像を計算し、この計算した再生像と目標再生像との誤差を評価し、この誤差に基づいて計算機ホログラムを最適化していって、目標再生像を再生し得る計算機ホログラムを作成するものである。計算機ホログラム作成装置１は、最適化処理部１０、再生像計算部２０、ルックアップテーブル格納メモリ３０、再生領域格納メモリ４０および再生像格納メモリ５０を備えている。
【００７６】
最適化処理部１０は、上記ステップＳ１〜Ｓ３，Ｓ５〜Ｓ９の処理を実行するとともに、再生像計算部２０に対して再生像計算（上記ステップＳ４の処理）を実行するよう指示する。最適化処理部１０は、再生像計算部２０による再生像計算に必要なルックアップテーブルを、ルックアップテーブル格納メモリ３０に格納する。最適化処理部１０は、再生像計算部２０による再生像計算の際に計算すべき再生像のうちの関心領域Ωに含まれる各画素（ｍ，ｎ）を示す位置情報ｍ，ｎを、目標再生像に基づいて求めて、再生領域格納メモリ４０に格納する。最適化処理部１０は、計算機ホログラムの初期値を決定して再生像格納メモリ５０に格納する。また、最適化処理部１０は、再生像計算の処理が終了した旨を示す終了信号を再生像計算部２０から受け取り、再生像格納メモリ５０に格納されている更新された計算機ホログラムを受け取って、最適化処理を行う。
【００７７】
再生像計算部２０は、最適化処理部１０からの指示に基づいて上記ステップＳ４の処理を実行する。このとき、再生像計算部２０は、最適化処理部１０、ルックアップテーブル格納メモリ３０、再生領域格納メモリ４０および再生像格納メモリ５０それぞれから再生像計算に必要な情報を獲得して、これらの情報に基づいて、再生像の画素（ｍ，ｎ）の複素振幅変化量ΔＨ_mnを計算し、変化後の再生像の画素（ｍ，ｎ）の複素振幅Ｈ'_mnを求める。この再生像計算部２０における処理の内容の詳細は後述する。
【００７８】
ルックアップテーブル格納メモリ３０は、最適化処理部１０により計算されたルックアップテーブルＬＵＴ３（上記(13)式）を、最適化処理開始の当初より格納している。再生領域格納メモリ４０は、再生像のうちの関心領域Ωに含まれる各画素（ｍ，ｎ）を示す位置情報ｍ，ｎの集合を、最適化処理開始の当初より格納している。なお、再生像のうちの関心領域Ωは、一般には目標再生像のうち画素値が０でない領域であり、目標再生像に基づいて最適化処理部１０により求められたものである。再生像格納メモリ５０は、最適化処理部１０により決定された計算機ホログラムの初期値を最適化処理開始の当初に格納し、また、再生像計算部２０により計算された計算機ホログラムを更新記憶する。
【００７９】
図２は、本実施形態に係る計算機ホログラム作成装置および方法における再生像計算処理を説明するフローチャートである。図２に示したフローチャートは、上記(14)式を用いて再生像計算部２０において行われる再生像計算処理（上記ステップＳ４の処理）の詳細なフローを示すものである。
【００８０】
(Ｓ１１) 先ず、再生像計算部２０は、再生像を計算すべき旨の指示を最適化処理部１０から受け取り、さらに、計算機ホログラムに含まれる多数の画素のうち位相レベルが変化する画素（ｊ，ｋ）を示す位置情報ｊ，ｋ、および、その画素（ｊ，ｋ）の複素振幅ｈ_jkの変化前後の位相レベルｐ，ｐ'を、最適化処理部１０から受け取る。(Ｓ１２) また、再生像計算部２０は、再生像のうちの関心領域Ωに含まれる各画素（ｍ，ｎ）を示す位置情報ｍ，ｎを、再生領域格納メモリ４０から受け取る。
【００８１】
(Ｓ１３) 再生像計算部２０は、これらのパラメータｐ，ｐ'，ｍ，ｎ，ｊおよびｋに基づいて、ルックアップテーブル格納メモリ３０に格納されているデータを読み出す為のアドレス値ａｄｒ１（＝ａｐ＋ｂｊｍ＋ｃｋｎ）およびａｄｒ２（＝ａｐ'＋ｂｊｍ＋ｃｋｎ）を作成する。(Ｓ１４) 再生像計算部２０は、当該アドレスに格納されているデータＬＵＴ３[ａｄｒ１]およびＬＵＴ３[ａｄｒ２]をルックアップテーブル格納メモリ３０より読み出す。(Ｓ１５) そして、再生像計算部２０は、再生像の画素（ｍ，ｎ）の複素振幅変化量ΔＨ_mnを上記(14)式に従って計算する。
【００８２】
(Ｓ１６) 再生像計算部２０は、パラメータｍおよびｎに基づいて、再生像格納メモリ５０に格納されている再生像の画素（ｍ，ｎ）の複素振幅Ｈ_mnを読み出す。(Ｓ１７) 再生像計算部２０は、この複素振幅Ｈ_mnに上記複素振幅変化量ΔＨ_mnを加算することで、変化後の再生像の画素（ｍ，ｎ）の複素振幅Ｈ'_mnを求めて、この複素振幅Ｈ'_mnを再生像格納メモリ５０に戻す。(Ｓ１８) 再生領域格納メモリ４０に格納されている関心領域Ω内の全て画素について処理を行ったか否かを判断し、もし「否」であればステップＳ１２に戻る。(Ｓ１９) 再生領域格納メモリ４０に格納されている関心領域Ω内の全て画素について処理が終了すると、再生像計算部２０は、最適化処理部１０に対して終了信号を出力する。その後、最適化処理部１０は、再生像格納メモリ５０に格納されている更新された計算機ホログラムを受け取って、最適化処理を行う。
【００８３】
位相レベルｐとｐ'との和について上記(15)式、上記(16)式または上記(19)式が成立するか否かを判断して、この判断結果に基づいて、再生像の画素（ｍ，ｎ）の複素振幅変化量ΔＨ_mnを、異なる式（上記(18)式または(21)式）で計算する場合には、上記ステップＳ１１，Ｓ１２およびＳ１６〜Ｓ１９それぞれにおける処理内容は上記と同様であるが、ステップＳ１３〜Ｓ１５それぞれにおける処理内容は異なる。
【００８４】
位相レベルｐとｐ'との和について上記(15)式または上記(16)式が成立するときには、ステップＳ１３〜Ｓ１５それぞれにおける処理内容は、以下のステップＳ２３〜Ｓ２５のように置き換えられる。(Ｓ２３) 再生像計算部２０は、これらのパラメータｐ'，ｍ，ｎ，ｊおよびｋに基づいて、ルックアップテーブル格納メモリ３０に格納されているデータを読み出す為のアドレス値ａｄｒ３（＝ａｐ'）およびａｄｒ４（＝ｂｊｍ＋ｃｋｎ）を作成する。(Ｓ２４) 再生像計算部２０は、当該アドレスに格納されているデータＬＵＴ３[ａｄｒ３]およびＬＵＴ３[ａｄｒ４]をルックアップテーブル格納メモリ３０より読み出す。(Ｓ２５) そして、再生像計算部２０は、再生像の画素（ｍ，ｎ）の複素振幅変化量ΔＨ_mnを上記(18)式に従って計算する。
【００８５】
一方、位相レベルｐとｐ'との和について上記(19)式が成立するときには、ステップＳ１３〜Ｓ１５それぞれにおける処理内容は、以下のステップＳ３３〜Ｓ３５のように置き換えられる。(Ｓ３３) 再生像計算部２０は、これらのパラメータｐ'，ｍ，ｎ，ｊおよびｋに基づいて、ルックアップテーブル格納メモリ３０に格納されているデータを読み出す為のアドレス値ａｄｒ３（＝ａｐ'）およびａｄｒ４（＝ｂｊｍ＋ｃｋｎ）を作成する。(Ｓ３４) 再生像計算部２０は、当該アドレスに格納されているデータＬＵＴ３[ａｄｒ３]およびＬＵＴ３[ａｄｒ４]をルックアップテーブル格納メモリ３０より読み出す。(Ｓ３５) そして、再生像計算部２０は、再生像の画素（ｍ，ｎ）の複素振幅変化量ΔＨ_mnを上記(21)式に従って計算する。
【００８６】
また、計算機ホログラムの画素（ｊ，ｋ）の位相レベル変化量Δｐの値が限定されている場合に、再生像の画素（ｍ，ｎ）の複素振幅変化量ΔＨ_mnを上記(24)式に従って計算するときには、上記(23)式で表されるルックアップテーブルＬＵＴ４がルックアップテーブル格納メモリ３０に格納される。この場合には、上記ステップＳ１１，Ｓ１２およびＳ１６〜Ｓ１９それぞれにおける処理内容は上記と同様であるが、ステップＳ１３〜Ｓ１５それぞれにおける処理内容は、以下のステップＳ４３〜Ｓ４５のように置き換えられる。(Ｓ４３) 再生像計算部２０は、これらのパラメータｐ，ｐ'，ｍ，ｎ，ｊおよびｋに基づいて、ルックアップテーブル格納メモリ３０に格納されているデータを読み出す為のアドレス値ａｄｒ５（＝Ｑ(ｐ'−ｐ)＋ａｐ＋ｂｊｍ＋ｃｋｎ）を作成する。(Ｓ４４) 再生像計算部２０は、当該アドレスに格納されているデータＬＵＴ４[ａｄｒ５]をルックアップテーブル格納メモリ３０より読み出す。(Ｓ４５) そして、再生像計算部２０は、再生像の画素（ｍ，ｎ）の複素振幅変化量ΔＨ_mnを上記(24)式に従って計算する。
【００８７】
図３は、再生像の画素（ｍ，ｎ）の複素振幅変化量ΔＨ_mnを計算する際の計算量を纏めた図表である。ルックアップテーブルＬＵＴ１およびＬＵＴ２を用いる上記(9)式と比較して、ルックアップテーブルＬＵＴ３を用いる上記(14)式は、メモリ参照回数が同じであるものの、実数乗算回数および実数加減算回数それぞれが少ないので、計算時間が短い。また、位相レベルｐとｐ'との和が所定の関係を満たす場合に用いられる上記(18)式および上記(21)式それぞれは、メモリ参照回数、実数乗算回数および実数加減算回数の何れも少ないので、計算時間が更に短い。さらに、計算機ホログラムの画素（ｊ，ｋ）の位相レベル変化量Δｐの値が限定されている場合に用いられる上記(24)式は、メモリ参照回数、実数乗算回数および実数加減算回数の何れも更に少ないので、計算時間が更に短い。
【００８８】
【発明の効果】
以上、詳細に説明したとおり、本発明によれば、入力値ｑ（ただし、０≦ｑ＜Ｑ、ｑおよびＱそれぞれは整数。）の入力に対して関数値 exp(２πｉｑ／Ｑ) が量子化された出力値ＬＵＴ[ｑ]を出力するルックアップテーブルを用いることにより、或いは、入力値ｓＱ＋ｑ（ただし、０≦ｑ＜Ｑ、０≦ｓ＜Ｐ、ｑ，Ｑ，ｓおよびＰそれぞれは整数。）の入力に対して関数値 [exp(２πｉｓ／Ｐ)−１]・exp(２πｉｑ／Ｑ) が量子化された出力値ＬＵＴ[ｓＱ＋ｑ]を出力するルックアップテーブルを用いることにより、再生像を計算する際の演算量が削減され、計算機ホログラムが短時間に作成される。
【図面の簡単な説明】
【図１】本実施形態に係る計算機ホログラム作成装置の構成図である。
【図２】本実施形態に係る計算機ホログラム作成装置および方法における再生像計算処理を説明するフローチャートである。
【図３】再生像の画素の複素振幅変化量ΔＨ_mnを計算する際の計算量を纏めた図表である。
【符号の説明】
１…計算機ホログラム作成装置、１０…最適化処理部、２０…再生像計算部、３０…ルックアップテーブル格納メモリ、４０…再生領域格納メモリ、５０…再生像格納メモリ。[0001]
BACKGROUND OF THE INVENTION
The present invention relates to an apparatus and a method for creating a computer generated hologram (phase hologram or complex amplitude hologram) for reproducing a target reproduction image by generating reproduction light whose phase is modulated by incidence of illumination light by an optimization method. Is.
[0002]
[Prior art]
In fields such as laser marking, laser processing, optical modeling, and optical tweezers, there is a demand for an image pattern forming technique for forming laser light into a desired shape. One of the image pattern forming techniques is a technique using a computer generated hologram. In this technique, a computer generated hologram capable of generating a target reconstructed image having a desired shape (target reconstructed image) is obtained by calculation, and illumination light is applied to the computer generated hologram to form a target reconstructed image. Since this technique using a computer generated hologram uses a holographic technique, the degree of freedom is high. In addition, if a phase hologram is used as a computer generated hologram, the loss of light is small, and the utilization efficiency of light energy is high.
[0003]
This computer generated hologram cannot be analytically performed, but is performed by an optimization method based on iterative calculation. The creation of a computer generated hologram by optimization starts with a certain initial computer generated hologram and performs iterative calculations. The computer generated holograms are sequentially reconstructed so that the reconstructed image reproduced from the computer generated by this calculation approaches the target reconstructed image. Update. With such an optimization method, good results are obtained.
[0004]
For example, a computer described in the literature “AG Kirk and TJ Hall,“ Design of binary computer generated holograms by simulated annealing: coding density and reconstruction error, ”Opt. Comm. Vol. 94, pp. 491-496 (1992)”. In the hologram creation method, a simulated annealing (SA) method is used as an optimization method. Kirk et al. Applied the SA method to the creation of a binary phase computer hologram with hologram values limited to binary. In addition, there are various documents in which the SA method is applied to the creation of a computer generated hologram. Among them, the above-mentioned document by Kirk et al. Clearly mentions a device for shortening the creation time.
[0005]
First, a general procedure of a computer generated hologram method using the SA method configured based on this document will be described. In this procedure, (S1) the initial value of the computer generated hologram is determined, and initial values of parameters used in the SA method are determined. (S2) one or more pixels among a number of pixels included in the computer generated hologram (S3) change the pixel value of the selected pixel, (S4) calculate a reproduction image reproduced by the computer generated hologram after changing the pixel value, and (S5) calculate the reproduction image And (S6) evaluate the error and determine whether or not to accept the change of the pixel value of the selected pixel (step S3 above), and (S7) included in the computer generated hologram It is determined whether or not the processing in steps S2 to S6 has been performed for all the pixels to be processed. If “No”, the process returns to step S2, and (S8) updates parameters used in the SA method. ) Calculate It is determined whether or not may be completion, the flow returns to step S2 if if "No".
[0006]
This SA calculation procedure has a double loop. In the inner loop (steps S2 to S7), pixels included in the computer generated hologram are scanned and the pixel values are optimized while the parameters used in the SA method are constant. In the outer loop (steps S2 to S9), optimization is repeated while updating parameters used in the SA method, and the degree of optimization is increased.
[0007]
[Problems to be solved by the invention]
However, the computer hologram creation method based on the SA method described in the above document has a problem that it takes a very long time because both the number of repetitions and the calculation amount are large.
[0008]
As for the number of repetitions, the loop inside the SA method is performed for all the pixels included in the computer generated hologram, and therefore the number of repetitions is the same as the number of pixels. In general, the number of pixels is very large. For example, considering the creation of a computer generated hologram for a VGA standard liquid crystal monitor, the number of repetitions of the inner loop is 300,000 times or more. Furthermore, in order to obtain a good solution, it is necessary to repeat the outer loop of steps S2 to S9 many times. Thus, since the number of repetitions of both the outer and inner loops is large, the creation time becomes longer.
[0009]
As the number of pixels increases, the creation time becomes longer. In recent years, with the development of process technology, the number of pixels of a device that can be used for hologram display is increasing at a rapid pitch. As the number of pixels in the hologram increases, the number of iterations for pixel scanning in the inner loop also increases. Also, the amount of calculation of the reproduced image in step S4 increases. Furthermore, since the degree of freedom becomes high, the number of iterations for optimization in the outer loop increases. Thus, when the number of pixels of a computer generated hologram increases, the hologram creation time increases rapidly.
[0010]
Even if the hologram value becomes multi-level, the calculation time increases. In the above-mentioned document of Kirk et al., A binary hologram is created by using the SA method. However, in the case of a binary hologram, since the amount of calculation in step 4 is small, the total amount of calculation is originally relatively small. Further, since the degree of freedom is low, the number of repetitions can be relatively small. On the other hand, when the hologram value is multi-valued, the calculation amount in step S4 is twice or more that when the hologram value is binary. Furthermore, since the degree of freedom is increased, the number of times the outer loop is repeated increases. From these two effects, as the number of hologram value levels increases, the hologram production time increases rapidly.
[0011]
In the above-mentioned document of Kirk et al., The calculation of the reproduced image is speeded up by eliminating the unnecessary calculation and using the lookup table in the calculation of the reproduced image in step S4. The former can also be applied to multi-value holograms. Unfortunately, however, their lookup tables are designed for binary holograms, so the creation time cannot be significantly reduced even when applied to multi-value holograms.
[0012]
As described above, the SA method is originally a method that takes time to calculate, and the time required for creation increases rapidly as the number of pixels and the number of levels of hologram values increase. A long preparation time is a major obstacle to practical use. In the case of a multi-value hologram, even if a lookup table described in Kirk et al. Is used, sufficient speedup cannot be expected.
[0013]
The present invention has been made to solve the above problems, and an object of the present invention is to provide a computer generated hologram apparatus and method capable of generating a computer generated hologram in a short time.
[0014]
[Means for Solving the Problems]
The computer generated hologram generating apparatus according to the present invention generates reproduction light having at least a phase modulated by incident illumination light to reproduce a target reproduction image. Fourier transform type (1) A function value exp (2πiq / Q with respect to an input of an input value q (where 0 ≦ q <Q, q and Q are integers). ) Is an output value LUT [q] quantized, and (2) a phase value before the change according to a change in the phase value of any pixel included in the computer generated hologram The output value LUT [q] is obtained from the lookup table storage means based on the input value q corresponding to the pixel position, and the lookup is performed based on the phase value after the change and the input value q ′ corresponding to the pixel position. Reproduction image calculation means for obtaining an output value LUT [q ′] from the table storage means and calculating a reproduction image reproduced by the computer generated hologram based on these output values LUT [q] and LUT [q ′]; 3) Included in computer generated hologram The phase value of each pixel is changed, and the computer generated hologram for reproducing the target reproduction image is optimized by the optimization method based on the reproduction image and the target reproduction image calculated by the reproduction image calculation means according to the change. And an optimization processing means to be created.
[0015]
Further, the computer generated hologram method according to the present invention reproduces a target reproduction image by generating reproduction light whose phase is modulated at least by the incidence of illumination light. Fourier transform type (1) A function value exp (2πiq / Q for an input value q (where 0 ≦ q <Q, q and Q are integers), respectively. ) Prepares a look-up table for outputting the quantized output value LUT [q], and (2) according to a change in the phase value of any pixel included in the computer generated hologram, The output value LUT [q] is obtained from the lookup table based on the input value q corresponding to the pixel position, and from the lookup table based on the phase value after the change and the input value q ′ corresponding to the pixel position. An output value LUT [q ′] is obtained, and a reproduced image reproduced by a computer generated hologram is calculated based on the output values LUT [q] and LUT [q ′]. (3) Each pixel included in the computer generated hologram Change the phase value of A computer generated hologram for reproducing the target reproduction image is created by an optimization method based on the reproduction image calculated in accordance with the change and the target reproduction image.
[0016]
The computer generated hologram creating apparatus according to the present invention and the computer generated hologram creating method according to the present invention are based on the same technical idea. In the present invention, by preparing a lookup table as described above, the amount of calculation when calculating a reproduced image is reduced, and a computer generated hologram is created in a short time.
[0017]
In the computer generated hologram creating apparatus according to the present invention, the reproduced image calculating means is (1) the sum of the phase value 2πp / P before the change of the pixel and the phase value 2πp ′ / P after the change (2π (p + p ′) / P ) Is π or 3π, the real part of the output value is obtained from the look-up table storage means based on the input value corresponding to the phase value 2πp ′ / P after the change, and the input corresponding to the position of the pixel. Based on the values, output values are obtained from the lookup table storage means, and based on these values, a reproduction image reproduced by the computer generated hologram is calculated. (2) When the sum value is 2π, An imaginary part of the output value is obtained from the lookup table storage means based on the input value corresponding to the phase value 2πp ′ / P, and an output value is obtained from the lookup table storage means based on the input value corresponding to the pixel position. To these values Then, a reproduced image reproduced by a computer generated hologram is calculated. (3) When the sum value is not any of π, 2π, and 3π, the phase value 2πp / P before the change and the position of the pixel are determined. The output value LUT [q] is obtained from the lookup table storage means based on the input value q, and the lookup table is stored based on the phase value 2πp ′ / P after the change and the input value q ′ corresponding to the pixel position. An output value LUT [q ′] is obtained from the means, and a reproduced image reproduced by a computer generated hologram is calculated based on the output values LUT [q] and LUT [q ′].
[0018]
In addition, when the computer generated hologram method according to the present invention calculates a reconstructed image reproduced by a computer generated hologram, (1) the phase value 2πp / P before the change of the pixel and the phase value 2πp ′ / P after the change When the value of the sum (2π (p + p ′) / P) is π or 3π, the real part of the output value is obtained from the lookup table based on the input value corresponding to the phase value 2πp ′ / P after the change. The output value is obtained from the lookup table based on the input value corresponding to the pixel position, and the reproduced image reproduced by the computer generated hologram is calculated based on these values. (2) The sum value is 2π. In some cases, an imaginary part of the output value is obtained from the lookup table based on the input value corresponding to the changed phase value 2πp ′ / P, and the output value from the lookup table is obtained based on the input value corresponding to the pixel position. Get this Based on these values, a reconstructed image reproduced by a computer generated hologram is calculated. (3) When the sum value is not any of π, 2π, and 3π, the phase value 2πp / P before the change and the pixel value The output value LUT [q] is obtained from the lookup table based on the input value q corresponding to the position, and the lookup is performed based on the phase value 2πp ′ / P after the change and the input value q ′ corresponding to the pixel position. An output value LUT [q ′] is obtained from the table, and a reproduction image reproduced by a computer generated hologram is calculated based on the output values LUT [q] and LUT [q ′].
[0019]
As described above, when the sum of the phase value 2πp / P before the change of the pixel and the phase value 2πp ′ / P after the change (2π (p + p ′) / P) is any one of π, 2π, and 3π. In this case, the amount of calculation for calculating the reproduced image is further reduced, and a computer generated hologram is created in a shorter time.
[0023]
DETAILED DESCRIPTION OF THE INVENTION
Hereinafter, embodiments of the present invention will be described in detail with reference to the accompanying drawings. In the description of the drawings, the same elements are denoted by the same reference numerals, and redundant description is omitted.
[0024]
First, a reproduction image calculation process (the process of step S4) in the computer generated hologram creating apparatus and method according to the present embodiment will be described. When the computer generated hologram is a Fourier transform hologram, a reproduced image reproduced by the computer generated hologram can be obtained by Fourier transforming the computer generated hologram. For the calculation by Fourier transform, Fast Fourier Transform (FFT) or Discrete Fourier Transform (DFT) is used. In general, when the number of data is V, the calculation amount when calculating a reproduced image by FFT is proportional to V · logV, whereas the calculation amount when calculating a reproduced image by DFT is V ² Is faster, the FFT is faster.
[0025]
However, in the optimization method such as the SA method, as described above, (S2) one or a plurality of pixels are selected from many pixels included in the computer generated hologram, and (S3) the pixel value of the selected pixel. (S4) Since the reconstructed image reproduced by the computer generated hologram after changing the pixel value is calculated by Fourier transform, it is only necessary to calculate the pixel whose pixel value has changed. Therefore, in this case, since the amount of calculation when calculating a reproduced image by DFT is proportional to V, DFT is faster. In addition, by using a look-up table for calculation by DFT, a reproduced image can be calculated in a short time.
[0026]
Furthermore, in FFT, all pixels included in the reproduced image on the reproduction surface are calculated simultaneously, so even if only a part of the reproduced image is required, all pixels included in the reproduced image are calculated. Must. On the other hand, in the DFT, each pixel included in the reconstructed image is calculated independently. Therefore, when only a part of the reconstructed image is necessary, only the necessary pixels need be calculated. That is, when calculating the reproduced image by DFT, it is only necessary to calculate the pixels included in the region of interest in the reproduced image. If the total number of pixels in the reproduced image is V and the number of pixels in the region of interest is U, the amount of calculation of the region of interest by DFT is proportional to U, and the amount of calculation in the inner loop (steps S2 to S7) is V · U. Is proportional to Note that the region of interest in the reconstructed image is a region where the pixel value is not 0, although it varies depending on the application.
[0027]
The computer generated hologram creation method according to the present embodiment is based on the above-described idea, and calculates a reproduced image by DFT using a lookup table, and based on the calculated reproduced image and the target reproduced image. Thus, a computer generated hologram for reproducing the target reproduction image is created by the optimization method. In addition to the SA method described above, a genetic algorithm (GA) method or a Fourier transform iterative method is used as the optimization method. In the following, a lookup table that is referred to when calculating a reproduced image by DFT will be mainly described.
[0028]
Let the number of pixels of the computer generated hologram to be created be M × N. The complex amplitude of a pixel located in the j-th row and k-th column (1 ≦ j ≦ M, 1 ≦ k ≦ N) in the computer generated hologram (hereinafter, this pixel is referred to as “pixel (j, k)”). h _jk It describes as follows. In addition, a pixel located in the m-th row and n-th column (1 ≦ m ≦ M, 1 ≦ n ≦ N) in the reproduced image on the reproduction surface (hereinafter, this pixel is referred to as “pixel (m, n)”). )) For the complex amplitude of light _mn It describes as follows. The complex amplitude of pixel (j, k) of the computer generated hologram is h _jk To h ' _jk Change to the complex amplitude change amount ΔH of the light of the pixel (m, n) on the reproduced image. _mn Is
[0029]
[Expression 1]

It is expressed by the following formula. Here, i represents an imaginary unit. Δh _jk Represents the amount of complex amplitude change of pixel (j, k) of the computer generated hologram,
[0030]
[Expression 2]

It is expressed by the following formula.
[0031]
As can be seen from the above equation (1), the complex amplitude change amount ΔH of the light of each pixel (m, n) on the reproduced image. _mn Is the complex amplitude variation Δh of the pixel (j, k) of the computer generated hologram _jk Rely only on. The complex amplitude H ′ of the light of the pixel (m, n) on the reproduced image after such a change _mn Is the complex amplitude H before the change _mn Against
[0032]
[Equation 3]

It is expressed by the following formula. Therefore, the complex amplitude H of the light of each pixel (m, n) on the reproduced image _mn Is calculated only once, then the amount of change ΔH _mn You just have to add.
[0033]
In the optimization method such as the SA method, an error C between the reproduced image calculated as described above and the target reproduced image is calculated, and the error C is evaluated to accept a change in the pixel value of the selected pixel. Decide whether or not. In calculating this error C, only within the region of interest Ω of the reproduced image,
[0034]
[Expression 4]

What is necessary is just to obtain | require the square sum. Where F _mn Is the complex amplitude of the light of the pixel (m, n) on the target reproduced image. Note that the energy of the target reproduced image (the sum of squares of the complex amplitudes of the light of all the pixels) is the energy of the input light (h _jk If the calculation range is limited to the region of interest Ω as in the above equation (4), the optimization result is hardly affected.
[0035]
In the exp function on the right side of the above equation (1), the real part is calculated using the cos function, and the imaginary part is calculated using the sin function. By referring to a lookup table prepared in advance instead of calculating the cos function and the sin function, the value of the exp function on the right side of the above equation (1) can be obtained in a short time. Using the least common multiple L of integers M and N (L = dM = eN, where d and e are integers), the lookup table LUT1 used here is
[0036]
[Equation 5]

It is expressed by the following formula. In Kirk et al., A look-up table is used for the exponential function part on the right side of equation (1). In their case, the hologram is binary and Δh _jk Since there are only two values of 2 and -2, the effect of speeding up sufficiently appeared. However, in the case of multiple values, Δh as in this embodiment. _jk A large effect cannot be expected unless a look-up table is created including this part.
[0037]
In a computer generated hologram, the complex amplitude h of each pixel (j, k) _jk Are sampled and / or quantized in intensity and / or phase. Therefore, not only the exponential function part of the above equation (1) but also the complex amplitude change amount Δh _jk Can also be obtained by referring to a lookup table prepared in advance.
[0038]
When the computer generated hologram to be created is an intensity hologram, the complex amplitude h of each pixel (j, k) of the computer generated hologram _jk And h ' _jk Each is a quantized real number. Therefore, in this case, the complex amplitude change amount Δh _jk Can easily create a lookup table. In addition, since the calculation of the right side of the above equation (1) is the product of the real number and the complex number, the reference to the memory storing the lookup table is three times, the real number multiplication is two times, and the number of operations Less is.
[0039]
When the computer generated hologram to be created is a phase hologram, the complex amplitude h of each pixel (j, k) of the computer generated hologram _jk And h ' _jk Each is a real value with a constant amplitude intensity, a phase is quantized, and a complex number. When the computer generated hologram is a complex amplitude hologram, the complex amplitude h of each pixel (j, k) of the computer generated hologram _jk And h ' _jk Each is complex, with both amplitude intensity and phase quantized.
[0040]
When the computer generated hologram is a phase hologram or a complex amplitude hologram, the complex amplitude h of each pixel (j, k) of the computer generated hologram _jk Are phase-quantized at P levels in the range of 0 to 2π. Complex amplitude h of each pixel (j, k) of the computer generated hologram _jk When the phase level of p is p (where 0 ≦ p <P, p and P are integers), that is, when the phase value is 2πp / P, the complex amplitude h _jk Is
[0041]
[Formula 6]

It is expressed by the following formula. Where A _jk Is a real value and represents the amplitude intensity of the reproduction light from the pixel (j, k) of the computer generated hologram.
[0042]
Assuming that the phase level of the complex amplitude of the pixel (j, k) of the computer generated hologram changes from p to p ′ during the optimization process, the complex amplitude of the pixel (j, k) of the computer hologram of the above formula (2) Change amount Δh _jk Is
[0043]
[Expression 7]

It is expressed by the following formula. A look-up table can be formed in parentheses on the right side of the equation (7). The lookup table LUT2 in this case is
[0044]
[Equation 8]

It is expressed by the following formula. Then, the complex amplitude change amount ΔH of the pixel (m, n) of the reproduced image of the above expression (1) _mn Using the lookup tables LUT1 and LUT2,
[0045]
[Equation 9]

It is expressed by the following formula.
[0046]
Since the output values of the lookup tables LUT1 and LUT2 are complex numbers, multiplication of the complex number and the complex number is performed on the right side of the equation (9). In order to calculate the right side of the above equation (9), A _jk Lookup table lookup, lookup of the real part and imaginary part of the lookup table LUT1, and lookup of the real part and imaginary part of the lookup table LUT2. Therefore, in the calculation of the right side of the above equation (9), the memory in which the lookup table is stored is referenced five times, the real number multiplication is four times, and the real number addition / subtraction is four times. Thus, by calculating according to the above equation (9), the complex amplitude change amount ΔH of the pixel (m, n) of the reproduced image is obtained. _mn Can be obtained in a short time.
[0047]
However, by devising the look-up table as follows, the complex amplitude change amount ΔH of the pixel (m, n) of the reproduced image _mn Can be obtained in a shorter time. That is, substituting the above equation (7) into the above equation (1),
[0048]
[Expression 10]

The following formula is obtained. And the least common multiple of P, M and N is Q,
[0049]
[Expression 11]

And Here, each of a, b, and c is an integer. Using this Q, the above equation (10) becomes
[0050]
[Expression 12]

It is expressed.
[0051]
Each of the two exp functions on the right side of the equation (12) is quantized with Q levels in the range of 0 to 2π, and can be replaced with one lookup table. The lookup table LUT3 used here is:
[0052]
[Formula 13]

It is expressed by the following formula. However, 0 ≦ q <Q, and q and Q are integers. Using this lookup table LUT3, the above equation (12) is
[0053]
[Expression 14]

It is expressed.
[0054]
To calculate the right side of equation (14) above, _jk Need to be referred to, and the real part and imaginary part of the lookup table LUT3. Therefore, in the calculation of the right side of the above equation (14), the memory in which the lookup table is stored is referenced five times, the real number multiplication is performed twice, and the real number addition / subtraction is performed twice. Thus, by calculating according to the above equation (14), the complex amplitude change amount ΔH of the pixel (m, n) of the reproduced image is larger than that according to the above equation (9). _mn Can be obtained in a shorter time.
[0055]
Also, the complex amplitude h of the pixel (j, k) of the computer generated hologram _jk Depending on the combination of the respective values of the phase levels p and p ′, the complex amplitude h of the pixel (j, k) of the computer generated hologram _jk Since this becomes a real number or an imaginary number, this complex amplitude h _jk Is calculated depending on whether it is a real number, an imaginary number, or the like, so that the complex amplitude change amount ΔH of the pixel (m, n) of the reproduced image is calculated. _mn Can be obtained in a shorter time.
[0056]
That is, the complex amplitude h of the pixel (j, k) of the computer generated hologram _jk Since the values of the phase levels p and p ′ are integers of 0 or more and less than P,
[0057]
[Expression 15]

and
[0058]
[Expression 16]

Is satisfied, the complex amplitude change amount Δh of the pixel (j, k) of the computer generated hologram is satisfied. _jk Is a real number,
[0059]
[Expression 17]

It is expressed by the following formula. At this time, the above equation (12) is
[0060]
[Expression 18]

It is expressed. In this equation (18), Re [•] represents an operation that refers to only the real part.
[0061]
on the other hand,
[0062]
[Equation 19]

If the following relational expression is satisfied, the complex amplitude change amount Δh of the pixel (j, k) of the computer generated hologram _jk Is an imaginary number,
[0063]
[Expression 20]

It is expressed by the following formula. At this time, the above equation (12) is
[0064]
[Expression 21]

It is expressed. In this equation (21), Im [•] represents an operation that refers to only the imaginary part.
[0065]
To calculate the right side of equation (18) above, _jk Need to be referred to, and the real part of the lookup table LUT3. In order to calculate the right side of the above equation (21), A _jk Of the lookup table and the imaginary part of the lookup table LUT3 are required. Therefore, in the calculation of the right side of the above equation (19) or (21), the memory storing the lookup table is referenced four times, the real number multiplication is three times, and the real number addition / subtraction is not required. is there. Thus, when the above formula (15) or the above formula (16) is established, the calculation is performed according to the above formula (18), and when the above formula (19) is established, the calculation is performed according to the above formula (21). Complex amplitude change ΔH of pixel (m, n) _mn Can be obtained in a shorter time. When none of the above formulas (15), (16), and (19) holds, the complex amplitude change amount ΔH of the pixel (m, n) of the reproduced image _mn Is calculated according to the above equation (14).
[0066]
Further, in the optimization method (particularly the SA method), the value of the phase level change amount Δp (= p′−p) of the pixel (j, k) of the computer generated hologram may be limited. In this case, By devising the look-up table as follows, the complex amplitude change amount ΔH of the pixel (m, n) of the reproduced image _mn Can be obtained in a shorter time. That is, in this case, the above equation (12) is
[0067]
[Expression 22]

It is expressed by the following formula. Here, since each of the phase levels p and p ′ is an integer from 0 to P−1, the phase level change amount Δp is an integer from −P + 1 to P−1. However, since exp [-ix] = exp [i (2π-x)] due to the nature of the complex exponential function, [exp (2πiΔp / P) -1] in the above equation (22) has a phase level change. A lookup table may be prepared in the range from 0 to P-1 with respect to the amount Δp.
[0068]
Therefore, q = a p + bjm + ckn (0 ≦ q <Q),
[0069]
[Expression 23]

By preparing a look-up table LUT4 represented by the following formula, the complex amplitude change amount ΔH of the pixel (m, n) of the reproduced image _mn Is
[0070]
[Expression 24]

It is expressed by the following formula. However, s is Δp when Δp is positive, and is calculated as P−Δp when Δp is negative, and in any case, s is an integer from 0 to P−1. Also, LUT4 '[•, ·] means a two-dimensional array, and LUT4 [•] represents this in a one-dimensional array.
[0071]
To calculate the right side of equation (24) above, _jk Need to be referred to, and the real part and the imaginary part of the lookup table LUT4. Therefore, in the calculation of the right side of the above equation (24), the memory storing the lookup table is referenced three times, the real number is multiplied twice, and the real number addition / subtraction is not required. Thus, when the value of the phase level variation Δp of the pixel (j, k) of the computer generated hologram is limited, the pixel (m, n) of the reproduced image is calculated by calculating according to the above equation (24). Complex amplitude change ΔH _mn Can be obtained in a shorter time.
[0072]
Each lookup table includes a real part and an imaginary part, and the real part is a value calculated using a cos function, and the imaginary part is a value calculated using a sin function. If this is utilized, the size of each lookup table can be reduced. For example, if a relationship of cos (x) = sin (x−π) is used, each lookup table can integrate the real part and the imaginary part, and the size becomes 1/2. Further, if the relations cos (x) = − cos (x−π) and cos (x) = cos (−x) are used, the size of each lookup table becomes ¼. If these methods are adopted at the same time, the size of each lookup table becomes 1/8. However, this method cannot be applied to the lookup table LUT4.
[0073]
In the SA method, in one search operation in the inner loop (steps S2 to S7), it is normal that there is one pixel whose pixel value is changed in the computer generated hologram. The value may be changed. In the GA method, the pixel values of a plurality of pixels are changed simultaneously in one search operation. Thus, care must be taken when changing the pixel values of a plurality of pixels simultaneously in one search operation. In other words, the calculation time of the reconstructed image according to the present embodiment is proportional to the value of (number of pixels of the region of interest Ω in the reconstructed image) × (number of pixels whose pixel value changes in the computer generated hologram). When the number of pixels whose pixel values change in a computer generated hologram is large, the calculation time of the reproduced image by the DFT of this embodiment may be longer than the calculation time by the FFT. On the other hand, the calculation time by FFT and the calculation time by DFT of this embodiment can each be measured in advance. Therefore, when the calculation time by the DFT of the present embodiment is expected to be longer than the calculation time by the FFT, the reproduced image is calculated by using the FFT. Otherwise, the DFT of the present embodiment is used. Thus, the reproduced image may be calculated.
[0074]
The above description has focused on the theoretical aspect of the reproduction image calculation processing in the computer generated hologram creating apparatus and method according to the present embodiment. Next, the configuration and operation of the computer generated hologram creating apparatus according to this embodiment will be described, and the computer generated hologram creating method according to this embodiment will be described. FIG. 1 is a configuration diagram of a computer generated hologram creating apparatus 1 according to the present embodiment.
[0075]
The computer generated hologram creating apparatus 1 according to the present embodiment calculates a reproduced image by DFT with reference to the lookup table LUT3 or LUT4 described above, evaluates an error between the calculated reproduced image and the target reproduced image, and calculates the error. Based on the above, the computer generated hologram is optimized, and a computer generated hologram capable of reproducing the target reproduced image is created. The computer generated hologram production apparatus 1 includes an optimization processing unit 10, a reproduction image calculation unit 20, a lookup table storage memory 30, a reproduction area storage memory 40, and a reproduction image storage memory 50.
[0076]
The optimization processing unit 10 executes the processes of steps S1 to S3 and S5 to S9 and instructs the reproduction image calculation unit 20 to execute reproduction image calculation (the process of step S4). The optimization processing unit 10 stores a lookup table necessary for the reproduction image calculation by the reproduction image calculation unit 20 in the lookup table storage memory 30. The optimization processing unit 10 uses the position information m, n indicating each pixel (m, n) included in the region of interest Ω in the reproduced image to be calculated when the reproduced image calculation unit 20 calculates the reproduced image as target information. It is obtained based on the reproduced image and stored in the reproduction area storage memory 40. The optimization processing unit 10 determines an initial value of the computer generated hologram and stores it in the reproduced image storage memory 50. Further, the optimization processing unit 10 receives an end signal indicating that the processing of the reproduction image calculation is completed from the reproduction image calculation unit 20, receives the updated computer generated hologram stored in the reproduction image storage memory 50, Perform optimization processing.
[0077]
The reproduced image calculation unit 20 executes the process of step S4 based on an instruction from the optimization processing unit 10. At this time, the reconstructed image calculation unit 20 obtains information necessary for reconstructed image calculation from the optimization processing unit 10, the look-up table storage memory 30, the reproduction area storage memory 40, and the reconstructed image storage memory 50, respectively. Based on the information, the complex amplitude change amount ΔH of the pixel (m, n) of the reproduced image _mn And the complex amplitude H ′ of the pixel (m, n) of the reproduced image after the change _mn Ask for. Details of processing in the reproduction image calculation unit 20 will be described later.
[0078]
The look-up table storage memory 30 stores the look-up table LUT3 (the above equation (13)) calculated by the optimization processing unit 10 from the beginning of the optimization process. The reproduction area storage memory 40 stores a set of position information m and n indicating each pixel (m, n) included in the region of interest Ω of the reproduction image from the beginning of the optimization process. Note that the region of interest Ω in the reproduced image is generally a region having a pixel value other than 0 in the target reproduced image, and is obtained by the optimization processing unit 10 based on the target reproduced image. The reproduced image storage memory 50 stores the initial value of the computer generated hologram determined by the optimization processing unit 10 at the beginning of the optimization process, and updates and stores the computer generated hologram calculated by the reproduced image calculating unit 20.
[0079]
FIG. 2 is a flowchart for explaining a reproduction image calculation process in the computer generated hologram creating apparatus and method according to the present embodiment. The flowchart shown in FIG. 2 shows a detailed flow of the reproduction image calculation process (the process of step S4) performed in the reproduction image calculation unit 20 using the above equation (14).
[0080]
(S11) First, the reconstructed image calculation unit 20 receives an instruction to calculate a reconstructed image from the optimization processing unit 10, and further, a pixel (j whose phase level changes among a large number of pixels included in the computer generated hologram (j , K) and position information j, k indicating the complex amplitude h of the pixel (j, k) _jk Are received from the optimization processing unit 10. (S12) Further, the reproduced image calculation unit 20 receives position information m, n indicating each pixel (m, n) included in the region of interest Ω of the reproduced image from the reproduction region storage memory 40.
[0081]
(S13) Based on these parameters p, p ′, m, n, j, and k, the reproduced image calculation unit 20 reads an address value adr1 (=) for reading data stored in the lookup table storage memory 30. ap + bjm + ckn) and adr2 (= ap ′ + bjm + ckn). (S14) The reproduced image calculation unit 20 reads the data LUT3 [adr1] and LUT3 [adr2] stored at the address from the lookup table storage memory 30. (S15) Then, the reproduced image calculation unit 20 calculates the complex amplitude change amount ΔH of the pixel (m, n) of the reproduced image. _mn Is calculated according to the above equation (14).
[0082]
(S16) Based on the parameters m and n, the reproduction image calculation unit 20 calculates the complex amplitude H of the pixel (m, n) of the reproduction image stored in the reproduction image storage memory 50. _mn Is read. (S17) The reproduced image calculation unit 20 calculates the complex amplitude H _mn And the complex amplitude variation ΔH _mn Is added to the complex amplitude H ′ of the pixel (m, n) of the reproduced image after the change. _mn For this complex amplitude H ′ _mn Is returned to the reproduced image storage memory 50. (S18) It is determined whether or not all the pixels in the region of interest Ω stored in the reproduction region storage memory 40 have been processed. If “No”, the process returns to step S12. (S19) When the processing is completed for all the pixels in the region of interest Ω stored in the reproduction region storage memory 40, the reproduction image calculation unit 20 outputs an end signal to the optimization processing unit 10. Thereafter, the optimization processing unit 10 receives the updated computer generated hologram stored in the reproduced image storage memory 50 and performs an optimization process.
[0083]
It is determined whether the above equation (15), the above equation (16), or the above equation (19) is established for the sum of the phase levels p and p ′, and based on the determination result, a pixel ( m, n) complex amplitude change ΔH _mn Is calculated by a different formula (the above formula (18) or the formula (21)), the processing contents in the above steps S11, S12 and S16 to S19 are the same as the above, but in each of the steps S13 to S15. Processing contents are different.
[0084]
When the above expression (15) or the above expression (16) is established for the sum of the phase levels p and p ′, the processing contents in steps S13 to S15 are replaced as in the following steps S23 to S25. (S23) Based on these parameters p ′, m, n, j and k, the reproduced image calculation unit 20 reads out the address value adr3 (= ap ′) for reading the data stored in the lookup table storage memory 30. ) And adr4 (= bjm + ckn). (S24) The reproduced image calculation unit 20 reads the data LUT3 [adr3] and LUT3 [adr4] stored at the address from the lookup table storage memory 30. (S25) Then, the reproduced image calculation unit 20 calculates the complex amplitude change amount ΔH of the pixel (m, n) of the reproduced image. _mn Is calculated according to the above equation (18).
[0085]
On the other hand, when the above equation (19) is established for the sum of the phase levels p and p ′, the processing contents in steps S13 to S15 are replaced as in the following steps S33 to S35. (S33) Based on these parameters p ′, m, n, j and k, the reconstructed image calculation unit 20 reads out the address value adr3 (= ap ′) for reading the data stored in the lookup table storage memory 30. ) And adr4 (= bjm + ckn). (S34) The reproduced image calculation unit 20 reads the data LUT3 [adr3] and LUT3 [adr4] stored in the address from the lookup table storage memory 30. (S35) Then, the reproduced image calculation unit 20 calculates the complex amplitude change amount ΔH of the pixel (m, n) of the reproduced image. _mn Is calculated according to the above equation (21).
[0086]
Further, when the value of the phase level change Δp of the pixel (j, k) of the computer generated hologram is limited, the complex amplitude change ΔH of the pixel (m, n) of the reproduced image is limited. _mn Is calculated according to the above equation (24), the lookup table LUT4 represented by the above equation (23) is stored in the lookup table storage memory 30. In this case, the processing contents in steps S11, S12 and S16 to S19 are the same as described above, but the processing contents in steps S13 to S15 are replaced as in steps S43 to S45 below. (S43) Based on these parameters p, p ′, m, n, j, and k, the reproduced image calculation unit 20 reads an address value adr5 (=) for reading data stored in the lookup table storage memory 30. Q (p′−p) + ap + bjm + ckn) is created. (S44) The reproduced image calculation unit 20 reads the data LUT4 [adr5] stored at the address from the lookup table storage memory 30. (S45) Then, the reproduced image calculation unit 20 calculates the complex amplitude change amount ΔH of the pixel (m, n) of the reproduced image. _mn Is calculated according to the above equation (24).
[0087]
FIG. 3 shows the complex amplitude change amount ΔH of the pixel (m, n) of the reproduced image. _mn It is the table | surface which put together the calculation amount at the time of calculating. Compared with the above equation (9) using the look-up tables LUT1 and LUT2, the above equation (14) using the look-up table LUT3 has the same number of times of memory reference, but the number of times of real number multiplication and the number of real number addition / subtraction are small. So the calculation time is short. Further, each of the above formula (18) and the above formula (21) used when the sum of the phase levels p and p ′ satisfies a predetermined relationship has a small number of memory reference times, real number multiplication times, and real number addition / subtraction times. Therefore, the calculation time is even shorter. Further, the above equation (24) used when the value of the phase level change amount Δp of the pixel (j, k) of the computer generated hologram is limited is any of the memory reference count, the real multiplication count, and the real addition / subtraction count. Since there are few, calculation time is still shorter.
[0088]
【The invention's effect】
As described above in detail, according to the present invention, the function value exp (2πiq / Q) is quantized with respect to the input of the input value q (where 0 ≦ q <Q, q and Q are integers). By using a look-up table that outputs the output value LUT [q], or the input value sQ + q (where 0 ≦ q <Q, 0 ≦ s <P, q, Q, s, and P are integers). ) Is used to output an output value LUT [sQ + q] in which a function value [exp (2πis / P) −1] · exp (2πiq / Q) is quantized. The amount of calculation at the time of calculation is reduced, and a computer generated hologram is created in a short time.
[Brief description of the drawings]
FIG. 1 is a configuration diagram of a computer generated hologram creating apparatus according to the present embodiment.
FIG. 2 is a flowchart for explaining a reproduction image calculation process in the computer generated hologram generation apparatus and method according to the present embodiment.
FIG. 3 shows a complex amplitude change amount ΔH of a pixel of a reproduced image. _mn It is the table | surface which put together the calculation amount at the time of calculating.
[Explanation of symbols]
DESCRIPTION OF SYMBOLS 1 ... Computer hologram production apparatus, 10 ... Optimization process part, 20 ... Reproduction image calculation part, 30 ... Look-up table storage memory, 40 ... Reproduction area storage memory, 50 ... Reproduction image storage memory.

Claims (2)

An apparatus for creating a Fourier transform type computer generated hologram by an optimization method for reproducing a target reproduction image by generating reproduction light whose phase is modulated by incident illumination light,
A lookup table that outputs an output value LUT [q] in which a function value exp (2πiq / Q) is quantized with respect to an input of an input value q (where 0 ≦ q <Q, q and Q are integers). Storage means;
In response to a change in the phase value of any of the pixels included in the computer generated hologram, an output value LUT [from the lookup table storage means based on the phase value before the change and the input value q corresponding to the position of the pixel. q], and an output value LUT [q ′] from the lookup table storage means based on the phase value after the change and the input value q ′ corresponding to the position of the pixel, and these output values LUT [ reconstructed image calculation means for calculating a reconstructed image reconstructed by the computer generated hologram based on q] and LUT [q ′];
The phase value of each pixel included in the computer generated hologram is changed, and the target reproduced image is reproduced based on the reproduced image calculated by the reproduced image calculating means and the target reproduced image according to the change. An optimization processing means for creating the computer generated hologram by an optimization method;
With
The reproduced image calculation means includes:
When the sum of the phase value 2πp / P before the change of the pixel and the phase value 2πp ′ / P after the change (2π (p + p ′) / P) is π or 3π, the phase value 2πp ′ after the change The real part of the output value is obtained from the lookup table storage means based on the input value corresponding to / P, and the output value is obtained from the lookup table storage means based on the input value corresponding to the position of the pixel. Based on these values, the reproduced image reproduced by the computer generated hologram is calculated,
When the sum value is 2π, an imaginary part of the output value is obtained from the lookup table storage means based on the input value corresponding to the phase value 2πp ′ / P after the change, and the sum value is determined according to the pixel position. Obtaining an output value from the lookup table storage means based on the input value, and calculating a reproduction image reproduced by the computer generated hologram based on these values;
When the sum value is neither π, 2π, or 3π, the output value LUT is output from the lookup table storage means based on the phase value 2πp / P before the change and the input value q corresponding to the position of the pixel. obtaining [q], obtaining the output value LUT [q ′] from the lookup table storage means based on the phase value 2πp ′ / P after the change and the input value q ′ corresponding to the position of the pixel, Based on these output values LUT [q] and LUT [q ′], a reproduced image reproduced by the computer generated hologram is calculated.
A computer generated hologram device. A method of creating a Fourier transform type computer generated hologram by an optimization method for reproducing a target reproduction image by generating reproduction light whose phase is modulated at least by incidence of illumination light,
Lookup for outputting an output value LUT [q] in which a function value exp (2πiq / Q) is quantized with respect to an input of an input value q (where 0 ≦ q <Q, q and Q are integers). Prepare a table
In response to a change in the phase value of any pixel included in the computer generated hologram, an output value LUT [q] from the lookup table based on the phase value before the change and the input value q corresponding to the position of the pixel. And the output value LUT [q ′] is obtained from the lookup table based on the phase value after the change and the input value q ′ corresponding to the position of the pixel, and these output values LUT [q] and LUT Based on [q ′], a reproduced image reproduced by the computer generated hologram is calculated,
The phase value of each pixel included in the computer generated hologram is changed, and the computer generated hologram for reproducing the target reproduced image is optimized based on the reproduced image calculated according to the change and the target reproduced image. Create by law,
When calculating a reproduced image reproduced by the computer generated hologram,
When the sum of the phase value 2πp / P before the change of the pixel and the phase value 2πp ′ / P after the change (2π (p + p ′) / P) is π or 3π, the phase value 2πp ′ after the change The real part of the output value is obtained from the look-up table based on the input value according to / P, and the output value is obtained from the look-up table based on the input value according to the position of the pixel. Based on the above, the reproduced image reproduced by the computer generated hologram is calculated,
When the sum value is 2π, an imaginary part of the output value is obtained from the lookup table based on the input value corresponding to the phase value 2πp ′ / P after the change, and the input value corresponding to the position of the pixel To obtain an output value from the lookup table, and based on these values, calculate a reproduced image reproduced by the computer generated hologram,
When the value of the sum is neither π, 2π, or 3π, the output value LUT [q from the lookup table based on the phase value 2πp / P before the change and the input value q corresponding to the position of the pixel. ], And an output value LUT [q ′] is obtained from the lookup table based on the phase value 2πp ′ / P after the change and the input value q ′ corresponding to the position of the pixel, and these output values LUT Based on [q] and LUT [q ′], a reproduced image reproduced by the computer generated hologram is calculated.
A method for creating a computer generated hologram.

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