Medical Imaging. Transmission Measurement

Transcription

1 Medical Imaging Image Recontruction from Projection Prof Ed X. Wu Tranmiion Meaurement ( meaurable unknown attenuation, aborption cro-ection hadowgram ) I(ϕ,ξ) µ(x,) I = I o exp A(x, )dl L X-ra ource

2 X-Ra Tomograph Ue X-ra to generate everal hadowgram I(ϕ,ξ). I(ϕ,ξ) X-ra ource X-Ra Tomograph I(ϕ,ξ) X-ra ource

3 X-Ra Tomograph I(ϕ,ξ) X-ra ource X-Ra Tomograph unknown aborption cro-ection µ(x,) I(ϕ,ξ) X-ra ource To obtain image from projection data ue invere radon tranform.

4 Overview Image Recontruction from Projection: Radon Tranform Projection Slice Theorem Image Recontruction from Projection Data - Method I: Fourier Recontruction - Method II: Backprojection Filtering - Method III: Fourier Filtered Backprojection - Method IV: Convolution Filtered Backprojection Radon Tranform g(,) µ(x,) unknown aborption µ(x,) x x X-ra ource g(,) I = I o exp µ(x, )dl L g ln I o (Signal) I g(,) = µ(x, )dl L The Radon tranform g(,) of a function µ(x,) i defined a it line integral along a line inclined at an angle from the -axi and at a ditance from the origin.

5 Radon Tranform The Radon tranform g(,) of a function µ(x,) i the one-dimenional projection of µ(x,) at an angle. g(,) = L µ(x, )dl The Radon tranform map the patial domain (x,) to the domain (,). µ(x,) g(,) g(,) µ(x,) unknown aborption x x Each point in the (,) pace correpond to a line in the patial domain (x,). / in co Radon Tranform g (, ) = µ (, xdl ) = µ (, xdl ) L xco+ in = = µ ( x, ) δ( xco + in dxd ) in x Equation of line (red) given and ; (x) = m x + c m = - co / in c = () = /co(9- ) L = /in => = - co /in x + /in = - in - co x + = x co + in -

6 More Radon Tranform g (, ) = µ (, x) δ( xco + in dxd ) = µ ( x = x 'co 'in, = x 'in + 'co ) δ( x ' ) dx ' d ' = µ ( x= x'co 'in, = x'in + 'co ) d' = µ ( co 'in, in + 'co ) d' Achieved b rotating coordinate tem o that the integration line i along -axe: x x = x'co 'in = x'in + 'co x' = x' = xco + in ' = xin + co x Radon Tranform: Example 1 A(x,) = δ(x, ) (point at center) x g (, ) = δ(, x) δ( xco + in dxd ) [ ] x=, = g (, ) = δ( xco + in ) = δ( ) = δ() x Sinogram δ(x,) Line in Radon pace S= (,)

7 Radon Tranform: Example 2 A(x,) = δ(x-1,) (point on x-axi) g (, ) = δ( x 1, ) δ( xco + in dxd ) [ ] x= 1, = g (, ) = δ( xco + in ) = δ(co ) Non zero => co = 1 Sinogram x δ(x-1, ) x Radon Tranform: Example 3 Example II: A(x,) = δ(r,φ) (arbitrar point) Sinogram 1.5 [ ] [ ] g(, ) = A(, x ) δ( x co + in ) dxd g (, ) = δ( xco + in ) = δ rco( φ) Non zero => r co( φ) = x= rco φ, = rin φ φ r x δ(r,φ) φ x 1.5 r

8 Radon Tranform: Example 4 Example III: A(x,) = δ(r,φ) + δ(x-1,) x φ φ x r δ(x-1,) δ(r,φ) Radon Tranform 2D Real Space 2D Radon Space ( to 18 degree)

9 Radon Tranform 2D Real Space 2D Radon Space Where thee tructure are in Radon pace? ( to 18 degree) Propertie of Radon Tranform

10 Overview Image Recontruction from Projection: Radon Tranform Projection Slice Theorem Image Recontruction from Projection Data - Method I: Fourier Recontruction - Method II: Backprojection Filtering - Method III: Fourier Filtered Backprojection - Method IV: Convolution Filtered Backprojection Projection Theorem ( alo Central Slice Theorem or Projection Slice Theorem) If g(,) i the Radon tranform of a function f(x,), then the one-dimenional Fourier tranform G(ω,) with repect to of the projection g(,) i equal to the central lice, at angle, of the two dimenional Fourier tranform F(ω x, ω ) of the function f(x,).

11 Projection Theorem ( alo Central Slice Theorem or Projection Slice Theorem) 2D-pace domain of µ(x,) 2D-frequenc domain of µ(x,) g(,) 1D-Fourier tranform F(g()) ω µ(x,) x ω x If g(,) i the Radon tranform of a function f(x,), then the one-dimenional Fourier tranform G(ω,) with repect to of the projection g(,) i equal to the central lice, at angle, of the two dimenional Fourier tranform F(ω x, ω ) of the function f(x,). Projection Theorem ( alo Central Slice Theorem or Projection Slice Theorem) Proof: Aignment g(,) 1D-Fourier tranform F(g()) ω µ(x,) x ω x Clue?

12 Projection Theorem ( alo Central Slice Theorem or Projection Slice Theorem) 2D-pace domain of µ(x,) 2D-frequenc domain of µ(x,) 2D-FT F(µ(x,)) ω ω x under viewing angle = Projection Theorem g(, ) 1 1D-FT F(g()) ξ [cm] ω ω x Projection Theorem ( alo Central Slice Theorem or Projection Slice Theorem) 2D-pace domain of µ(x,) 2D-frequenc domain of µ(x,) µ(x,) 2D-FT F(µ(x,)) ω ω x under viewing angle = & = 9 g(, or 9 ) 1 1D-FT F(g()) ξ [cm] ω ω x

13 Preparing for Cla Project Computer kill (MatLab or C from ELEC221)? Skill in numerical computation? Overview Image Recontruction from Projection: Radon Tranform Projection Slice Theorem Image Recontruction from Projection Data - Method I: Fourier Recontruction - Method II: Backprojection Filtering - Method III: Fourier Filtered Backprojection - Method IV: Convolution Filtered Backprojection

14 Projection Theorem How can we ue Projection lice theorem to recontruction patial ditribution of aborption profile µ(x,)? I. Fourier Recontruction Object Space µ(x,) 2-D Invere Fourier Tranform 2 Dimenional Fourier-Object Space F(ω x,ω ) = FT(µ(x,)) X-ra aborption meaurement ield Radon Tranform Man lice at different fill 2D Fourier-object pace Radon Space g(,) 1-D Fourier Tranform FT(g()) 1 Dimenional Fourier-Radon Space G(ω, ) = FT(g())

15 Fourier Image Recontruction with Projection Theorem Fourier Recontruction g(,) 1D-Fourier tranform F(g()) ω µ(x,) x ω x

16 Fourier Recontruction 1D-Fourier tranform F(g(, 1 )) of one projection 2D-Invere Fourier tranform Fourier Recontruction g(, 1, 2 ) 1D-Fourier tranform F(g()) ω µ(x,) x ω x

17 Fourier Recontruction 1D-Fourier tranform F(g(, 1,2 )) of two projection 2D-Invere Fourier tranform Fourier Recontruction g(, 1,2,3,4 ) 1D-Fourier tranform F(g()) ω x ω x µ(x,)

18 Fourier Recontruction 1D-Fourier tranform F(g(, 1,2,3,4 )) of 4 projection 2D-Invere Fourier tranform Fourier Recontruction g(, 1,2,...8 ) 1D-Fourier tranform F(g()) ω µ(x,) x ω x

19 Fourier Recontruction 1D-Fourier tranform F(g(, 1,2,...,8 )) of 8 projection 2D-Invere Fourier tranform Fourier Recontruction 1D-Fourier tranform F(g(, 1,2,..., 16 )) of 16 projection 2D-Invere Fourier tranform

20 Fourier Recontruction 1D-Fourier tranform F(g(, 1,2,..., 64 )) of 64 projection 2D-Invere Fourier tranform Influence of # of Projection 1 projection 2 projection 4 projection 8 projection 16 projection 64 projection

21 Fourier Image Recontruction with Projection Theorem Problem: Point in 2D Fourier Space are not on rectangular grid. => Invere Fourier tranform not trivial. ω ω x Fourier Image Recontruction with Projection Theorem A practical algorithm:

22 Fourier Image Recontruction with Projection Theorem How i the Fourier recontruction method connected with the backprojection method? II. Backprojection Filtering Backprojection operator π bx (, ) fˆ ( x, ) Bg = g ( = xco + in, ) d The value of the backprojection Bg i evaluated b integrating g(,) over for all line that pa through that point. Example: Backproject 2 projection (g1 and g2) onl 2 projection repreented in Randon pace g (, ) = g( ) δ ( φ) + g( ) δ ( φ) Backproject 2 Projection b(x,) = g 1 ( 1 ) + g 2 ( 2 ) = b(r,φ) = rco( φ φ), = rco( φ φ)

23 Backprojection & Radon Tranform Real Space Radon Space, g(,) (r, φ ) = r co( φ) -1 x The backprojection at (r, φ ) i the integration of g(,) along the inuoid = r co( φ) WHY? (Optional) Thi i wh we aw

24 Backprojection & Radon Tranform Object Simple Backprojection (BP) Sinogram Backprojection Meaurement Difference between Object & BP image ( to π) Backprojection Operator: Mathematic It can be hown from π f ˆ( x, ) Bg = g( = x co + in, ) d g(, ) = = L f ( x, ) dl f ( x, ) δ ( x co + in ) dxd that the backprojected Radon tranform data g or Rf (i.e., the imple backprojection image) fˆ( x, ) Bg = BRf 1 = f( x, ) 2 2 x + Proof? Aignment Therefore backprojection of radon tranform give the original image convolved with 1/qrt(x ). Thi reult in blurred image. What could ou do to get back f(x,)?

25 Image Recontruction b Backprojection Filtering Ue a filter! - But what filter? fˆ( x, ) Bg = BRf Ue convolution theorem: ( ) ( ) 2 2 ( ) 1/2 = f( x, ) x / /2 ( ) ( ) (( ) ) F fˆ( x, ) = F f ( x, ) x + = F f ( x, ) F x+ F 2 2 ( (, ))( x ) = F f x ω + ω 1/2 2 2 ( (, ))( ) 1/ f x ω + ω = F( f ( x, ) )( ω + ω ) 1/ ( ω + ω ) ˆ 1/ 2 x x x = F ( f ( x, ) ) 2 2 1/2 ( ( ) ( x + ) ) ( ) ( ) IF F fˆ( x, ) ω ω = IF F f ( x, ) = f ( x, ) SQRT - Filter ω 1 2 = ω x + ω 2 ( ) 1/2 Filter (LP) 2 ω = ω x + ω 2 ( ) 1/2 Filter (HP) ω ω x ω ω x

26 Image Recontruction b Backprojection Filtering Backprojection Filtering Algorithm: (1) Get Radon tranform g(,) of f(x,) b performing tomographic X-tra imaging. (2) Backproject the Radon tranform data. (3) Take Fourier tranform of backprojected data. (4) Multipl with filter qrt(ω x2 + ω 2 ) (5) Perform invere Fourier Tranform to obtain f(x,) ( ) ( ( ) f (x, ) = IF 2 ω F 2 BRf Backprojection Filtering Recontruction & Fourier Recontruction Backprojection Filtering Method: ( ) ( ( ) f (x, ) = IF 2 ω F 2 BRf Fourier Recontruction Method: ( ) f (x, ) = IF 2 ( PST Rf ) N = IF 2 F 1, g(, n ) n=1 There are alo other technique!!!! PST := Projection Slice Theorem or Central Slice Theorem ( ) Note that re-griding i required

27 Radon Tranform 2D Real Space 2D Radon Space Wh uch 2D Radon Tranform i important? - Formulation of x-ra attenuation meaurement in CT - Mathematic for later ue in image recontruction - Invere Radon Tranform poible? π f ( x, ) = gˆ( = x co + in, ) d with gˆ (, ) = ω G( ω, )exp( iω ) dω with G(ω, ) = F 1, g(,) ( ) III. Fourier Filtered Backprojection Invere Radon Tranform Theorem: π f ( x, ) = gˆ( = x co + in, ) d with gˆ(, ) = ω F ( ω, ) exp( iω ) dω with 1 F ( ω, ) = F 1 1, ( g(, )) The invere Radon tranform i obtained in two tep: (1) Each projection i filtered b a one dimenional filter whoe frequenc repone i ω. (2) The reult of tep (1) i backprojected to ield f(x,). Proof?

28 Proof The invere Fourier tranform i given b: f ( x, ) F( ω x, ω )exp[ i( ωxx + ω )] dωxdω = Rewriting in polar coordinate reult in: f ( x, ) 2π = Fp ( ω, )exp[ iω ( x co + Changing the limit of integration we get: π in )] ω dω d f ( x, ) = ω F ( ω, )exp[ iω ( x co + in )] dω d Since the Projection Slice Theorem p (1D Fourier tranform with repect to of Radon tranform equal lice through 2D Fourier tranform at angle of the object function f) f ( x, ) = f ( x, ) = π π gˆ( x co + in, ) d = ω G( ω, )exp[ iω] dω d π gˆ(, ) d Projection Theorem ( alo Central Slice Theorem or Projection Slice Theorem) = g(,) 1D-Fourier tranform F(g()) ω µ(x,) x ω x

29 Fourier Filtered Backprojection Recontruction & Backprojection Filtering Recontruction π f ( x, ) = gˆ( = x co + in, ) d with gˆ(, ) = ω F ( ω, )exp( iω ) dω with 1 F ( ω, ) = F 1 1, ( g(, )) Fourier Filtered Backprojection Method ( ( ( ) ) f (x, ) = BIF 1, ω F 1, Rf Backprojection Filtering Method ( ) ( ( ) f (x, ) = IF 2 ω F 2 BRf Fourier Filtered Backprojection Recontruction baic concept: dicrete implementation:

30 ω - Filter RAM - LAK Shepp-Logan Hamming Lowpa Coine IV. Convolution Filtered Backprojection π f ( x, ) = gˆ( = x co + in, ) d with gˆ (, ) ω G( ω, )exp( iω ) dω = with G(ω,) = F 1, g(,) = ω G(ω,) gn(ω ) exp(iω )dω = [ IF 1 ω G(ω,) ] IF 1 gn(ω ) 1 = g(,) i2π 1 iπ 1 g(t,) 1 = 2π 2 t t dt page 446 { } [ { }] convolution theorem ( ) Hilbert Tranform

31 Convolution Filtered Backprojection The invere Radon tranform i obtained in three tep: (1) Each projection i differentiated with repect to. (2) A Hilbert tranformation i performed with repect to. (3) The reult of tep (2) i backprojected to ield f(x,). ( ( )) f (x, ) = ( 1/2π)BH D Rf ( ) convolution

32 Summar Fourier Recontruction - Method I: N f (x, ) = IF 2 F 1, g(, n ) n=1 Backprojection Filtering - Method II: ( ) ( ( ) ( ) f (x, ) = IF 2 ω F 2 BRf Fourier Filtered Backprojection - Method III ( ( ( ) ) f (x, ) = BIF 1, ω F 1, Rf Convolution Filtered Backprojection Method IV: ( ( )) f (x, ) = ( 1/2π)BH D Rf Project ( ) Group of work on the ame problem but with different approache. Conult each other and divide work whenever poible. Preentation in cla Will be graded a ~6 homework ( or ~1% Grade).

33 Group Project 25 tudent => 5 group of 5 Each group will write recontruction program in Matlab: A. Fourier Recontruction - Method I: N f (x, ) = IF 2 F 1, ( g(, n )) n=1 B. Backprojection Filtering - Method II f (x, ) = IF 2 ω F 2 BRf D. Convolution Filtered Backprojection - Method IV: ( ( ( ) ) f (x, ) = ( 1/2π)BH D Rf ( ( ) ( ) C. Fourier Filtered Backprojection - Method III: ( ( )) ( ) f (x, ) = BIF 1, ω F 1, Rf E. Iterative Recontruction