福島県立医科大学総合科学教育研究センター紀要 Vol. 4, 1-10, 2015 原著論文 CT 2 ( ) CT 2 Received 2 October 2015, Accepted 16 October CT 2 f 0 (x, y) Radon f 0 2 f (x, y)
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1 福島県立医科大学総合科学教育研究センター紀要 Vol. 4, -, 5 原著論文 CT () CT Received October 5, Accepted 6 October 5 CT f (x, y) Radon f f (x, y) (FBP) Fourier Fourier (Bracewell & Riddle, 967 () ; Ramachangran & Lakshminarayanan, 97; Shepp & Logan, 974) f f ( ) f f ( 3) FBP ( 4) CT N M f (x i, y j ) f (x i, y j ) f f f. x θ s (u ) u s x cos θ + y sin θ s f (x, y) p(s, θ) p(s, θ) f (x, y)δ(x cos θ + y sin θ s)dxdy. () R δ(x) ( A. ) f (x, y) p(s, θ) Radon (x, y) b(x, y) p(x cos θ + y sin θ, θ)dθ. () () b(x, y) f (x x, y y dx dy ) (3) R (x ) + (y ) () Bracewell & Riddle FBP
2 /r( / x + y ) /r (3) b(x, y) (x, y). f (x, y) Fourier F (Q x, Q y ) (Q x, Q y ) (Q x Q cos θ, Q y Q sin θ) F ()(A.8) F (Q cos θ, Q sin θ) [ f (x, y)δ(x cos θ + y sin θ s)dxdy e iπqs ds R p(s, θ)e iπqs ds P(Q, θ) (4) () P(Q, θ) p(s, θ) s Fourier Q x θ Q x -Q y F (Q x, Q y ) x θ s f (x, y) (p(s, θ)) s Fourier F (Q x, Q y ) P(Q, θ) Fourier f (x, y) [ q (s, θ) h (s) Q P(Q, θ)e iπq(x cos θ+y sin θ) dq dθ. (5) Q P(Q, θ)e iπqs dq, (6) Q e iπqs dq (7) q h q (s, θ) p(s, θ)h (s s )ds (8) (5) q (s, θ) h Fourier H (Q)( Q ) (7) (8) p(s, θ) h (s) q (s, θ) h q f (x, y) h.3 H (Q).3. Ram Lak Ramachangran & Lakshminarayanan (97) H (Q) Ram Lak Q max H RL (Q) { Q ( Q < Qmax ) ( Q > Q max ) Fourier h RL (s) h RL (s) πq maxs sin (πq max s) cos (πq max s) π s π (s/ s) sin (πs/ s) cos (πs/ s). () π s s Q max / s s n s (n ) (n ) 4 s h RL,n h RL (n s) () ( )n (n ) π n s.3. Shepp Logan Ram Lak Shepp Logan (Shepp & Logan, 974) Q ( ) max πq H SL (Q) π sin ( Q Q max) Q max ( Q > Q max ) (9) () (9) H SL H RL sinc (3) H () Q x, Q y Q π θ < π p(s, θ) p(s, θ) < s <. θ < π Fourier P(Q, θ) < Q <, θ < π (3) sinc sinc(x) sin(πx) πx
3 sinc Q max Shepp-Logan h SL (s) π s [ (s/ s) sin (πs/ s) (s/ s) () : h SL,n h SL (n s).3.3 Kak Slaney Kak & Slaney (987). (3) π s ( 4n). (4) H KS (Q) Q e ɛ Q, (5) Ram-Lak Shepp-Logan H H (Q) h KS (s) h KS (s) ( ɛ 4π s ) ( ɛ + 4π s ). (6) ɛ r H H RL (Q) Q max ( s ) H (Q) H SL (Q) Q/Q max sinc(q/q max ) H SL (Q) H (Q) Kak- Slaney ɛ H KS (Q) H (Q) H(Q) H Fourier h H (Q) 3 p- f - H (Q),, p-,, h f - g (4) f f f (x, y) f (x, y )g(x x, y y )dx dy. (7) R p- f - 3. p- f - p- h(s) q(s, θ) q(s, θ) f f (x, y) ()(8) f (x, y) p(s, θ)h(s s )ds. (8) q(x cos θ + y cos θ, θ)dθ (9) R f (s cos θ u sin θ, s sin θ + u cos θ) h(x cos θ + y sin θ s )du ds dθ s (x x ) cos θ + (y y ) sin θ, u (x x ) sin θ + (y y ) cos θ f (x, y) f (x x, y y ) R [ h(x cos θ + y sin θ)dθ dx dy () (7) () [ f - p- f - g(x, y) h(x cos θ + y sin θ)dθ () (4) Fourier 3
4 g h h (5) g g(x, y) g(r) r r h(s)ds r s () g r( x + y ) (6) () () p- f - 3. f - p- g(x, y) h(s) g h g(x, y), h(s) Fourier G(Q x, Q y ) g(x, y)e iπ(q xx+q y y) dxdy R (3) H(Q) h(s)e iπqs ds (4) () Fourier H(Q) G(Q x, Q y ) ( B. ) H(Q) Q G(Q) ( < Q < ), (5) G(Q) G(Q cos θ, Q sin θ) g G(Q cos θ, Q sin θ) G(Q) G(Q, ) G(, Q) G( Q) (5) Fourier g h ( B.3 ) g() (s ) π h(s) [ d s r g(r)dr (s ). π ds s r (6) s () x y g() πh() (5) (6) f -p- 3.3 f -.3 p- f p- f - (3) b [ f (/r) D (7) f - /r Fourier / Q x + Q y / Q (5) g(r) G(Q)H(Q)h(s). p- f - g(x, y) G(Q) H(Q) h(s) r Q δ(s) p- q(s, θ) p(s, θ) /r 3.3. Ram-Lak Ram-Lak p- (9) G RL (Q) Θ(Q max Q ) (5) g RL (r) Qmax g RL (r) π QJ (πrq)dq Q m r J (πq m r). (7) (Bracewell & Riddle) J (x), J (x) Bessel ( C ) J (x) x cos(x + α)/ πx g RL (r) r 3/ (a) g RL (r) Shepp-Logan Shepp-Logan () f - G SL (Q x, Q y ) G SL (Q) (5) G SL (Q) ( sinc Q Q max ) ( Q Q max) ( Q > Q max ) (8) (5) (6) g(x, y) g(r) (7) [ D D ( A.. ) 4
5 (a) (b) (c) /Δ /Δ /Δ /Δ /Δ /Δ. f - f - g SL (r) g SL (r) 4Q max Qmax ( ) πq sin J (πrq)dq, (9) Q max (b) g RL (r) g SL g RL r Kak-Slaney Kak-Slaney (5) Fourier g KS (r) (C.6) g KS (r) πɛ { (πr) + ɛ } 3/. (3) (6) () g KS (r) /r (x, y) (x, y) Ram-Lak Shepp- Logan g RL (x, y) g SL (x, y)g KS (x, y) 3 Ram-Lak Shepp- Logan Q max Kak-Slaney ɛ p- h (s) f - f - Ram-Lak (7) Q max g RL (r) (7) Q max (C.9) g RL (x, y) δ () (x, y) Shepp-Logan (9) g SL (r) Qmax πq sinc (Q/Q max ) J (πrq)dq, Q max sinc(q/q max ) g SL (r) πqj (πrq)dq δ () (x, y), Kak-Slaney (3) ɛ { g KS (r) (πr) + ɛ } 3/ πr[(πr) +ɛ ɛ/π ε g KS (r) ε πr r + ε (A.3) g KS (x, y) g KS (r) δ(r)/πr δ () (x, y) 3 f - δ () (x, y) p- h (s) f - δ () (x, y) f - g(x, y) (5) (6) H(Q) h(s) (p- ) p- f - (5) (6) p- f - p- 4 p- 5
6 4. Gauss A. (A.) Gauss p- Gauss Gauss g Gs (x, y) ( πɛ exp x + y ) ɛ ) ( πɛ exp r, (3) ɛ ( (c) )ɛ x, y g Gs (x, y) point spread (PSF) (Yoshii, 993) g Gs (r) (6) Gauss p- h Gs (s) π ɛ [ s ɛ s exp ( s t ) dt. (3) ɛ ɛ s (33) g (x, y) Fourier G (Q cos θ, Q sin θ) G (Q) G (Q) 4π ɛ Q G(Q) (5) H (Q) 4π ɛ Q 3 G(Q) h (s) ɛ d h(s). (34) ds p- (3) (33) ɛ h Gs h Gs (s) ɛ d ds h Gs(s) π ɛ 4.3 Gauss (3 s ɛ ) h Gs (s). (35) Ram-LakShepp-Logan Gauss 4. / x + / y (Laplacian) f (x, y) p- h (s) h (s) f - g (x, y) ( ) ɛ x + f (x, y) y ( ) f (x, y )ɛ R x + g(x x, y y )dx dy y f (x, y )g (x x, y y )dx dy, R g ɛ g, (33) 4.3. Shepp & Logan (974) 3 3.(a) p(s, θ)(b) Ram-Lak q RL (s, θ)(c) Shepp- Logan q SL (s, θ)(d) Gauss q Gs (s, θ)(e) Gauss (35) q Gs (s, θ)(f) Gauss q Gs+Gs (s, θ) Gauss Ram-Lak Shepp-Logan f - Gauss Gauss (a) (b) (c) (d) (e) (f) 3. (a) p(s, θ), (b) q RL (s, θ), (c) q SL (s, θ), (d) q Gs (s, θ), (e) q Gs (s, θ), (f) q Gs Gs (s, θ) 6
7 (a) 原画像 (b) Ram-Lak (c) Shepp-Logan (d) Gauss 型 (e) エッジフィルタ エッジフィルタ 図 4. ファントムヘッドを用いた各フィルタによる再構成像 がってしまうのを記述していることに対応している エッ 5 ジフィルタを通した投影 (e) では (d) の変化がうまく表現さ れている 更に (f) では (d) と (e) を合わせると (b) (c) と同様なシャープさが回復することがわかる (f) Gauss 型 5 図 4 はそれぞれのフィルタを用いて再構成したファント 5 ムヘッドである (a) はファントムヘッドの原画像である (b) (f) は図 3 の (b) (f) に対応している (d) では (b) (c) に 5 5 比べやや縁が曖昧な画像になっているが (f) では Ram-Lak (a) 再構成像 や Shepp-Logan より鮮やかな画像になっている 図 6. Ram-Lak フィルタによる再構成像 5 (b) f f の相関 デジタル画像によるフィルタの評価 ファントムヘッドは幾何学的な図形の重ね合わせででき ているので 高々 種類程度の色 f (x, y) の値は高々 種類程度 でしか f (x, y) の値を評価することができない 5 より多くの f (x, y) の値でフィルタの性能を評価するため 5 (8) に 図 5 の白黒の写真 (ビットマップファイル ) を使って 再構成像を作成し f (xi, y j ) と f (xi, y j ) i, j,..., (a) 再構成像 (b) f f の相関 図 7. Shepp-Logan フィルタによる再構成像 を比較した その手順は次のとおりである (a) 再構成像 (b) f f の相関 図 8. Gauss 型フィルタによる再構成像 図 5. 原画像 まず 図 5 の写真に対して ごとに角度を変えて投影 p(si, θ j ) をとった ここで 写真のサイズを スケー 5 ルは任意 とし s 軸方向の分解能 s を とし た これは各角度で約 6 本の平行ビームを放射したこと に相当する 次にこの投影に 4 種類のフィルタ (Ram-Lak, 5 Shepp-Logan, Gauss 型 Gauss 型 + エッジ) をあて 再構成 5 像 f (xi, y j ) を求めた 図 6(a) は Ram-Lak フィルタを使っ た再構成像 frl 図 7(a) は Shepp-Logan フィルタを使った (a) 再構成像 (b) f f の相関 図 9. Gauss 型 + エッジフィルタによる再構成像 再構成像 fsl 図 8(a) は Gauss 型フィルタを使った再構成 像 fgs 図 9(a) は Gauss 型 + エッジフィルタを使った再構 (8) 白黒のビットマップファイルでは各ピクセルに から 55 の整数値が割り当てられている また 本レポートで使用した画像のサイズは 4 4[pxl である 7
8 f Gs f (x i, y j ) f (x i, y j ) 6 9 (b) f (x i, y j ) 55 (x i, y j ) f (x i, y j ) f f f ( f ) f f f a + b f. a b f f Gauss + 4. f f f a + b f a b Ram-Lak 8.5 ±.4.87 ±.7 Shepp-Logan 3.3 ±.5.84 ±.8 Gauss 8.5 ± ±.5 Gauss ±..9 ±.8 5 CT (p-) f f p- Gauss Gauss + ()(5)(6) h g(x, y) Bracewell, R.N. and Riddle, A.C,, Astrophys. J, 5, 47, 967 Gradshteyn, I.S. and Ryzhil, I.M, Table of Integrals, Series and Products, Academic Press, 996 Kak, A.C. and Slaney, M., Principles of Computerized Tomographic Imaging, IEEE Press (New York), 987 Ramachandran, G.N. and Lakshminarayanan, A.V., Proc. Not, Acod. Sci, 68, 36, 97 Shepp, L.A. and Logan, B.F., IEEE. Trans. Nucl. Sci., NS-, 974 Yoshii, Y. Astrophys. J, 43, 55, 993 A Fourier A. Dirac b a δ(x a) f (x)δ(x c)dx { (x a) (x a) { f (c) (a c b) (a > c b < c) ( x a ɛ/) δ(x a) lim ɛ (A.) ɛ + ( x a > ɛ/) (x a) lim exp [, (A.) ɛ + πɛ ɛ ɛ lim ɛ + π (x a) + ɛ. (A.3) δ(ax) a δ(x), (A.4) n δ(g(x)) g (x i ) δ(x x i). (A.5) i x i g(x i ) i,..., n δ () (x, y) A. Fourier A.. Fourier δ () (x, y) πr δ(r), (A.6) Fourier 8
9 Fourier x Q f (x) Fourier F(Q) F(Q) f (x) f (x)e iπqx dx, F(Q)e iπqx dq, (A.7) (A.8) Fourier Fourier Fourier x, y Q x, Q y f (x, y) Fourier F(Q x, Q y ) f (x, y)e iπ(q xx+q y y) dxdy, (A.9) F(Q x, Q y ) f (x, y) R F(Q x, Q y )e iπ(q xx+q y y) dqx dq y, R (A.) (A.) a Fourier ɛ/ sin(πqɛ) lim ɛ ɛ/ ɛ eiπqx dx lim ɛ πɛq, Fourier A.. δ(x) Fourier e iπxq dq. (A.) Fourier f (x) g(x) [ f g D (x) [ f g D (x) f (y)g(x y)dy. (A.) [ D D f g Fourier F(Q)G(Q) (A.) [ [ f g D (x) F(Q)e iπqy dq G(Q )e iπq (x y) dq dy [ { } F(Q) e iπ(q Q )y dy G(Q )e iπq x dq dq [ F(Q) δ(q Q )G(Q )e iπq x dq dq F(Q)G(Q)e iπqx dq (A.3) [ f g Fourier F(Q)G(Q) B B. () h(s) (h( s) h(s)) [ h(x cos θ + y sin θ)dθ h(x cos θ + y sin θ)dθ π π r r r r h(s) r s ds. [ h(x cos θ + y sin θ) + h( x cos θ y sin θ) dθ [ h(x cos θ + y sin θ) +h(x cos(θ π) + y sin(θ π)) dθ h(x cos θ + y sin θ)dθ [ h(s)δ(x cos θ + y sin θ s)ds dθ [ h(s) δ(x cos θ + y sin θ s)dθ ds π [ h(s) δ(r cos(θ ϕ) s)dθ ds π h(s) Θ(r s )ds r s h(s) r s ds (B.) x + y r Θ(x) (A.5) B. (5) (3) g () h (4) G(Q x, Q y ) R { [ } H(ρ)e iπρ(x cos θ+y sin θ) dρ dθ e iπ(q xx+q y y) dxdy [ H(ρ) { } e iπ(ρ cos θ Qx) e iπ(ρ sin θ Qy) dxdy dρ dθ R π [ H(ρ)δ(ρ cos θ Q x )δ(ρ sin θ Q y )dρ dθ H ( ) / σ Q x + Q y Q x + Q y, σ Q x < σ Q x σ + Q σ Q x + Q y H(Q) Q x + Q yg(q x, Q y ) Q G(Q x, Q y ). (5) (B.) 9
10 B.3 (6) g Fourier G(Q x, Q y ) (5) g Fourier (3) [ G(Q cos θ, Q sin θ) e iπrq cos(θ ϕ) dϕ r g(r)dr π π J (πqr)r g(r)dr J (x) Q G(Q) (5) Fourier h(s) π 4π Q G(Q)e iπqs dq [ Q J (πqr) g(r)rdr e iπqs dq [ g(r)r QJ (πrq) cos(πsq)dq dr. (B.3) h s s s cos(πsq) (A.6)(C.9) h() [ h() g(r)r πr δ(r) dr π g() s > h(s) r g(r) d [ J (πrq) sin(πsq)dq dr, ds d π ds s r g(r)dr s r. (C.7) C Bessel J ν (z) ν Bessel ( d z d ) ) J ν (z) + ( ν J ν (z) z dz dz z (B.4) (B.5) (C.) ( z ν ( ) J ν (z) ) n (z/) n n!γ(ν + n + ) n J (x) π e ix cos θ dθ J (x), (C.) (C.3) J (x) (Gradshteyn & Ryzhil, 996) Q J (kt)dt k, QJ (aq)dq Q a J (aq ), e aq J (kt)dt J (ax) sin(bx)dx J (ax) cos(bx)dx (C.4) (C.5) a + k (a > ), (C.6) b a ( < a < b), (C.7) (b a) ( < b < a) a b (a b) ( < a < b). (C.8) Bessel δ () (x, y) e iπ(q xx+q y y) dqx dq y R [ Q e iπqr cos(θ ϕ) dϕ dq π π QJ (πrq)dq. (C.9)
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