B-spline

\[ % 汎用 \newcommand{\ctext}[1]{\raise0.2ex\hbox{\textcircled{\scriptsize{#1}}}} % 数学 % 汎用 \DeclarePairedDelimiterX{\parens}[1]{\lparen}{\rparen}{#1} \DeclarePairedDelimiterX{\braces}[1]{\lbrace}{\rbrace}{#1} \DeclarePairedDelimiterX{\bracks}[1]{\lbrack}{\rbrack}{#1} \DeclarePairedDelimiterX{\verts}[1]{|}{|}{#1} \DeclarePairedDelimiterX{\Verts}[1]{\|}{\|}{#1} \newcommand{\as}{{\quad\textrm{as}\quad}} \newcommand{\st}{{\textrm{ s.t. }}} \DeclarePairedDelimiterX{\setComprehension}[2]{\lbrace}{\rbrace}{#1\,\delimsize\vert\,#2} \newcommand{\naturalNumbers}{\mathbb{N}} \newcommand{\integers}{\mathbb{Z}} \newcommand{\rationalNumbers}{\mathbb{Q}} \newcommand{\realNumbers}{\mathbb{R}} \newcommand{\complexNumbers}{\mathbb{C}} \newcommand{\field}{\mathbb{F}} \newcommand{\func}[2]{{#1}\parens*{#2}} \newcommand{\argmax}{\mathop{\textrm{arg~max}}} \newcommand{\argmin}{\mathop{\textrm{arg~min}}} % 集合論 \newcommand{\range}[2]{\braces*{#1,\dotsc,#2}} \providecommand{\complement}{}\renewcommand{\complement}{\mathrm{c}} \newcommand{\ind}[2]{\mathbbm{1}_{#1}\parens*{#2}} \newcommand{\indII}[1]{\mathbbm{1}\braces*{#1}} % 数論 \newcommand{\abs}[1]{\verts*{#1}} \newcommand{\combi}[2]{{_{#1}\mathrm{C}_{#2}}} \newcommand{\perm}[2]{{_{#1}\mathrm{P}_{#2}}} \newcommand{\GaloisField}[1]{\mathrm{GF}\parens*{#1}} % 解析学 \newcommand{\NapierE}{\mathrm{e}} \newcommand{\sgn}[1]{\operatorname{sgn}\parens*{#1}} \newcommand{\cl}[1]{\operatorname{cl}#1} \newcommand{\Img}[1]{\operatorname{Img}\parens*{#1}} \newcommand{\dom}[1]{\operatorname{dom}\parens*{#1}} \newcommand{\norm}[1]{\Verts*{#1}} \newcommand{\floor}[1]{\left\lfloor#1\right\rfloor} \newcommand{\ceil}[1]{\left\lceil#1\right\rceil} \newcommand{\expo}[1]{\exp\parens*{#1}} \newcommand{\sinc}{\mathop{\textrm{sinc}}} \newcommand{\GammaFunc}[1]{\Gamma\parens*{#1}} \newcommand{\erf}{\mathop{\mathrm{erf}}} % 逆三角関数 \newcommand{\asin}[1]{\operatorname{Sin}^{-1}{#1}} \newcommand{\acos}[1]{\operatorname{Cos}^{-1}{#1}} \newcommand{\atan}[1]{\operatorname{{Tan}^{-1}}{#1}} \newcommand{\atanEx}[2]{\atan{\parens*{#1,#2}}} % 畳み込み \newcommand{\cycConv}[2]{{#1}\underset{\text{cyc}}{*}{#2}} % 微分 \newcommand{\deriv}[3]{\frac{\operatorname{d}^{#3}#1}{\operatorname{d}{#2}^{#3}}} \newcommand{\derivLong}[3]{\frac{\operatorname{d}^{#3}}{\operatorname{d}{#2}^{#3}}#1} \newcommand{\partDeriv}[3]{\frac{\operatorname{\partial}^{#3}#1}{\operatorname{\partial}{#2}^{#3}}} \newcommand{\partDerivLong}[3]{\frac{\operatorname{\partial}^{#3}}{\operatorname{\partial}{#2}^{#3}}#1} \newcommand{\partDerivIIHetero}[3]{\frac{\operatorname{\partial}^2#1}{\partial#2\operatorname{\partial}#3}} \newcommand{\partDerivIIHeteroLong}[3]{{\frac{\operatorname{\partial}^2}{\partial#2\operatorname{\partial}#3}#1}} % 積分 \newcommand{\integrate}[5]{\int_{#1}^{#2}{#3}{\mathrm{d}^{#4}}#5} \newcommand{\LebInteg}[4]{\int_{#1} {#2} {#3}\parens*{\mathrm{d}#4}} % 複素解析 \newcommand{\conj}[1]{\overline{#1}} \providecommand{\Re}{}\renewcommand{\Re}[1]{{\operatorname{Re}{\parens*{#1}}}} \providecommand{\Im}{}\renewcommand{\Im}[1]{{\operatorname{Im}{\parens*{#1}}}} \newcommand{\Arg}[1]{\operatorname{Arg}{\parens*{#1}}} \newcommand{\Log}[1]{\operatorname{Log}{#1}} % ラプラス変換 \newcommand{\LPLC}[1]{\operatorname{\mathcal{L}}\parens*{#1}} \newcommand{\ILPLC}[1]{\operatorname{\mathcal{L}}^{-1}\parens*{#1}} % 離散Fourier変換 \newcommand{\DFT}[1]{\mathrm{DFT}\parens*{#1}} % Z変換 \newcommand{\ZTrans}[1]{\operatorname{\mathcal{Z}}\parens*{#1}} \newcommand{\IZTrans}[1]{\operatorname{\mathcal{Z}}^{-1}\parens*{#1}} % 線形代数 \newcommand{\bm}[1]{{\boldsymbol{#1}}} \newcommand{\matEntry}[3]{#1\bracks*{#2}\bracks*{#3}} \newcommand{\matPart}[5]{\matEntry{#1}{#2:#3}{#4:#5}} \newcommand{\diag}[1]{\operatorname{diag}\parens*{#1}} \newcommand{\tr}[1]{\operatorname{tr}{\parens*{#1}}} \newcommand{\inprod}[2]{\left\langle#1,#2\right\rangle} \newcommand{\HadamardProd}{\odot} \newcommand{\HadamardDiv}{\oslash} \newcommand{\Span}[1]{\operatorname{span}\bracks*{#1}} \newcommand{\Ker}[1]{\operatorname{Ker}\parens*{#1}} \newcommand{\rank}[1]{\operatorname{rank}\parens*{#1}} % ベクトル % 単位ベクトル \newcommand{\vix}{\bm{i}_x} \newcommand{\viy}{\bm{i}_y} \newcommand{\viz}{\bm{i}_z} % 確率論 \newcommand{\PDF}[2]{\operatorname{PDF}\bracks*{#1,\;#2}} \newcommand{\Ber}[1]{\operatorname{Ber}\parens*{#1}} \newcommand{\Beta}[2]{\operatorname{Beta}\parens*{#1,#2}} \newcommand{\ExpDist}[1]{\operatorname{ExpDist}\parens*{#1}} \newcommand{\ErlangDist}[2]{\operatorname{ErlangDist}\parens*{#1,#2}} \newcommand{\PoissonDist}[1]{\operatorname{PoissonDist}\parens*{#1}} \newcommand{\GammaDist}[2]{\operatorname{Gamma}\parens*{#1,#2}} \newcommand{\cind}[2]{\ind{#1\left| #2\right.}} %条件付き指示関数 \providecommand{\Pr}{}\renewcommand{\Pr}[1]{\operatorname{Pr}\parens*{#1}} \newcommand{\cPr}[2]{\operatorname{Pr}\cPrParens{#1}{#2}} \newcommand{\E}[2]{\operatorname{E}_{#1}\bracks*{#2}} \newcommand{\cE}[3]{\E{#1}{\left.#2\right|#3}} \newcommand{\Var}[2]{\operatorname{Var}_{#1}\bracks*{#2}} \newcommand{\Cov}[2]{\operatorname{Cov}\bracks*{#1,#2}} \newcommand{\CovMat}[1]{\operatorname{Cov}\bracks*{#1}} % グラフ理論 \newcommand{\neighborhood}{\mathcal{N}} % プログラミング \newcommand{\plpl}{\mathrel{++}} \newcommand{\pleq}{\mathrel{+}=} \newcommand{\asteq}{\mathrel{*}=} \]

はじめに

3次splineを勉強していたらB-splineも見つけたので寄り道してみた。Wikipediaの記事で紹介されている性質の証明を与える。

定義

$\dotsb < t_{-1} < t_0 < t_1 < t_2 < \dotsb$に対して次の漸化式で定められる関数$B_{i,n}$を$n$次のB-spline関数と呼ぶ。

\[ B_{i,0}(t) \coloneqq \begin{cases} 1 & (t_i \leq t < t_{i+1}) \\ 0 & (\text{otherwise}) \end{cases} \] \[ B_{i,n+1}(t) \coloneqq \omega_{i,n}(t)B_{i,n}(t) + (1-\omega_{i+1,n}(t))B_{i+1,n}(t),\quad \omega_{i,n}(t) \coloneqq \frac{t – t_i}{t_{i+n+1} – t_i} \]

この定義から、$B_{i,n}$の台は$[t_i,t_{i+n+1})$であることが判る。

性質

次式が成り立つ。

\[\forall t \in [t_j,t_{j+1}), \sum_{i=j-n}^j B_{i,n}(t) = 1\]

$\textit{Proof}$

$n = 0$のときは明らかに成り立つ。$n = m \in \integers$のときに成り立つと仮定して$n = m+1$のときに成り立つことを示す。

\begin{align*} \sum_{i=j-m-1}^j B_{i,m+1}(t) &= \sum_{i=j-m-1}^j \left[ \omega_{i,m}(t)B_{i,m}(t) + (1-\omega_{i+1,m}(t))B_{i+1,m}(t) \right] \\ &= \omega_{j-m-1,m}(t)B_{j-m-1,m}(t) + \sum_{i=j-m-1}^{j-1} B_{i+1,m}(t) + (1-\omega_{j+1,m}(t))B_{j+1,m}(t) \end{align*}

$B_{i,n}$の台が$[t_i,t_{i+n+1})$であることから、上式の$B_{j-m-1,m}(t)$と$B_{j+1,m}(t)$は$t \in [t_j,t_{j+1})$に対して0となるので

\[ \sum_{i=j-m-1}^j B_{i,m+1}(t) = \sum_{i=j-m-1}^{j-1} B_{i+1,m}(t) = \sum_{k=j-m}^j B_{k,m}(t) = 0 \]

$\square$

微分

次式が成り立つ。

\[\derivLong{B_{i,k}(t)}{t}{} = k\left(\frac{B_{i,k-1}(t)}{t_{i+k}-t_i} – \frac{B_{i+1,k-1}(t)}{t_{i+1+k} – t_{i+1}}\right)\]

$\textit{Proof}$

$k=1$のときに成り立つことは容易に解る。$k=l \in \naturalNumbers$のときに成り立つと仮定して$k=l+1$のときに成り立つことを示す。

\begin{align*} \derivLong{B_{i,l+1}(t)}{t}{} &= \derivLong{\left(\frac{t-t_i}{t_{i+l+1}-t_i}B_{i,l}(t) + \frac{t_{i+l+2}-t}{t_{i+l+2}-t_{i+1}}B_{i+1,l}(t)\right)}{t}{} \\ &= \frac{B_{i,l}(t)}{t_{i+l+1}-t_i} + \frac{t-t_i}{t_{i+l+1}-t_i}\derivLong{B_{i,l}(t)}{t}{} – \frac{B_{i+1,l}(t)}{t_{i+l+2} – t_{i+1}} + \frac{t_{i+l+2}-t}{t_{i+l+2} – t_{i+1}}\derivLong{B_{i+1,l}(t)}{t}{} \\ &= \frac{B_{i,l}(t)}{t_{i+l+1}-t_i} – \frac{B_{i+1,l}(t)}{t_{i+l+2} – t_{i+1}} + \underbrace{\frac{t-t_i}{t_{i+l+1}-t_i}l\left(\frac{B_{i,l-1}(t)}{t_{i+l}-t_i} – \frac{B_{i+1,l-1}(t)}{t_{i+1+l} – t_{i+1}}\right)}_{(1)} \\ &\phantom{=} + \underbrace{\frac{t_{i+l+2}-t}{t_{i+l+2} – t_{i+1}}l\left(\frac{B_{i+1,l-1}(t)}{t_{i+1+l}-t_{i+1}} – \frac{B_{i+2,l-1}(t)}{t_{i+2+l} – t_{i+2}}\right)}_{(2)} \end{align*}

ここで

\begin{align*} (1) &= \frac{l}{t_{i+l+1}-t_i}\left[\frac{t-t_i}{t_{i+l}-t_i}B_{i,l-1}(t) + \frac{t_{i+1+l} – t}{t_{i+1+l} – t_{i+1}}B_{i+1,l-1}(t) + \frac{t_i – t_{i+1+l}}{t_{i+1+l} – t_{i+1}}B_{i+1,l-1}(t)\right] \\ &= \frac{l}{t_{i+l+1}-t_i}B_{i,l}(t) – \frac{l}{t_{i+1+l} – t_{i+1}}B_{i+1,l-1}(t) \end{align*}

また

\begin{align*} (2) &= -\frac{l}{t_{i+l+2} – t_{i+1}}\left[\frac{t – t_{i+1}}{t_{i+1+l}-t_{i+1}}B_{i+1,l-1}(t) + \frac{t_{i+l+2}-t}{t_{i+2+l} – t_{i+2}}B_{i+2,l-1}(t) + \frac{t_{i+1} – t_{i+l+2}}{t_{i+1+l}-t_{i+1}}B_{i+1,l-1}(t)\right] \\ &= -\frac{l}{t_{i+l+2}-t_{i+1}}B_{i+1,l}(t) + \frac{l}{t_{i+1+l} – t_{i+1}}B_{i+1,l-1}(t) \end{align*}

以上より

\[ \derivLong{B_{i,l+1}(t)}{t}{} = (l+1)\left[\frac{B_{i,l}(t)}{t_{i+l+1}-t_i} – \frac{B_{i+1,l}(t)}{t_{i+l+2} – t_{i+1}}\right] \]

$\square$

投稿者: motchy

An embedded software and FPGA engineer for measuring instrument.

コメントを残す