程序代写代做代考 graph 5.3 Image Reconstruction from Projections

5.3 Image Reconstruction from Projections
51

Agenda
Image reconstruction from projections:
• The reconstruction problem
• Principles of Computed Tomography (CT) • The Radon transform
• The Fourier-slice theorem
• Reconstruction by filtered back-projections
52

The Reconstruction Problem
• Consider a single object on a uniform background (suppose that this is a cross section of 3D region of a human body).
• Background represents soft, uniform tissue and the object is also uniform but with higher absorption characteristics.
53

The Reconstruction Problem
• A beam of X-rays is emitted and part of it is absorbed by the object.
• The energy of absorption is detected by a set of detectors.
• The collected information is the absorption signal.
54

The Reconstruction Problem
• A simple way to recover the object is to back-project the 1D signal across the direction in which the beam came.
• This simply means duplicating the signal across the 1D beam.
55

The Reconstruction Problem
• Werotatethe position of the source-detector pair and obtain another 1D signal.
• Werepeatthe procedure and add the signals from the previous back- projections.
• Wecannowtellthat the object of interest is located at the central square.
56

The Reconstruction Problem
• After a large number of views have been back- projected in this manner, the original point is reconstructed as a diffuse “blob”.
• We only consider projections from 0 to 180 degrees as projections differing 180 degrees are mirror images of each other.
57

The Reconstruction Problem
58

Principles of X-ray CT
• ThegoalofCTistoobtaina3D representation of the internal structure of an object by X-raying it from many different directions.
• Back-projecting the image would result in slices through the body.
• A 3D representation is then obtained by stacking the slices.
59

First-generation CT scanner
• Single source, single detector
• Veryslow
– e.g., need 785,000 measurements
– 1 ms per measurement – 785 s (13 min!)
• Modern CT scanners collect data in < 1s. 60 First-generation CT scanner https://www.youtube.com/watch?v=fNaCxhhhZTE 61 2nd generation CT scanner • Similar to first generation, but multiple detectors • Translation still required • faster scan than 1st generation scanner 62 2nd generation CT scanner www.youtube.com/watch?v=Ni4Hsi3GhXo 63 3rd generation CT scanner • Multiple detectors (500- 1000) cover entire fan beam through patient. • Both detectors and x-ray source rotate in synchrony • much faster (<1s per scan) 64 3rd generation CT scanner http://www.youtube.com/watch?v=bdf0kXn5Eeg 65 Ring artifact in 3rd generation scanner • error in detector results in a bad measurement at each projection view. • artifact is a ring of erroneous μ • error in 0.1% can result in observable rings 66 Ring artifact in 3rd generation scanner Fig. 1 Sinogram illustrating the effect of one detector with sensitivity reduced to 90% Fig. 2 Sinogram illustrating the effect of one detector with sensitivity reduced to 95% Reconstruction of the data in Fig. 1 (10% detector error) Reconstruction of the data in Fig. 2 (5% detector error) 67 4th generation CT scanner • Onlyx-raysource rotates, detectors remain fixed. • Avoidsringartifact 68 4th generation CT scanner http://www.youtube.com/watch?v=AWVz3yke_bY 69 3D image reconstructed from slices ..... 70 Spiral (Helical) CT • Componentsrotatecontinuouslytobuildupa3D image. • Could be 3rd or 4th generation designs • Rotationcanbeasfastastwopersecond. 71 Modern CT scanner instrumentation Main components: • Gantry: Contains x- ray source, detector arrays and electronic controlling the collection of data. • Sliding Table: Slides into the gantry. • Computer – Controls the sliding table, which moves to the table so that the desired body part is scanned – Reconstruct the image 72 Radon Transform Think of every point 􏴁, 􏴂 on the line, projected onto the unit vector (􏵄􏴒􏵂􏵃, 􏵂􏴊􏴏􏵃) have a length of 􏵑. That is: 􏴁,􏴂 ⋅ 􏵄􏴒􏵂􏵃,􏵂􏴊􏴏􏵃 = 􏵑 73 Radon Transform 74 Radon Transform 75 Radon Transform 76 Radon Transform and Sinogram • The representation of the Radon transform 􏴄(􏵑, 􏵃) as an image with 􏵑 and 􏵃 as coordinates is called a sinogram. 􏵑 􏵃 􏵃 􏵑 77 Why is it called “sinogram”? 􏵑=􏴎cos 􏵒􏴢􏵃 =􏴎cos 􏵃􏴢􏵒 􏵃 􏵑 􏵃 􏵑 www.youtube.com/watch?v=XA7GXhPbRT0 78 Backprojection Mathematical Description of Backprojection: For a fixed rotation angle 􏵃􏴰, and a fixed distance 􏵑􏵓, backprojecting the value of the projection 􏴄(􏵑􏵓,􏵃􏴰) is equivalent to copying the value 􏴄(􏵑􏵓,􏵃􏴰) to the image pixels belonging to the line 􏴁􏵄􏴒􏵂􏵃􏴰 + 􏴂􏵂􏴊􏴏􏵃􏴰 = 􏵑􏵓. 79 Backprojection • Projecting the entire profile for a fixed angle (i.e., 􏴄(􏵑, 􏵃􏴰 )) yield: 􏳿􏵔􏵕 􏴁,􏴂 =􏴄􏵑,􏵃􏴰 =􏴄(􏴁cos􏵃􏴰+􏴂sin􏵃􏴰,􏵃􏴰) 􏵑􏵓 = 􏴁 cos 􏵃􏴰 + 􏴂 sin 􏵃􏴰 (􏴁, 􏴂) 80 Backprojection • This equation holds for every angle 􏵃: 􏳿􏵔 􏴁,􏴂 =􏴄(􏴁cos􏵃+􏴂sin􏵃,􏵃) • The final image is formed by integrating over all the backprojected images: 􏳿􏴁,􏴂 =􏵖􏵗􏳿􏵔 􏴁,􏴂􏴑􏵃 􏴜 Backprojection results in blurred images. 81 The Fourier-Slice Theorem • Let the 1D FT of a projection with respect to 􏵑 (at a given angle) be: 􏵛 􏴘􏵓􏴩􏵗􏵙􏵚 􏵘􏳿,􏵃 =􏵖 􏴄􏵑,􏵃􏴍 􏴑􏵑 􏴘􏵛 • Substituting the projection 􏴄 􏵑, 􏵃 : 􏵘 􏳿,􏵃 =􏵖􏵛 􏵖􏵛 􏵖􏵛􏳿 􏴁,􏴂 􏵜 􏴁cos􏵃+􏴂sin􏵃−􏵑 􏴑􏴁􏴑􏴂 􏴍􏴘􏵓􏴩􏵗􏵙􏵚􏴑􏵑 􏴘􏵛 􏴘􏵛 􏴘􏵛 =􏵖􏵛 􏵖􏵛 􏳿 􏴁,􏴂 􏵖􏵛􏵜 􏴁cos􏵃+􏴂sin􏵃−􏵑 􏴍􏴘􏵓􏴩􏵗􏵙􏵚􏴑􏵑 􏴑􏴁􏴑􏴂 􏴘􏵛 􏴘􏵛 􏴘􏵛 =􏵥􏵛 􏵥􏵛 􏳿 􏴁,􏴂 􏴍􏴘􏵓􏴩􏵗􏵙 􏴤􏵝􏵞􏵟􏵔􏵠􏴾􏵟􏵡􏵢􏵔 􏴑􏴁􏴑􏴂 􏴘􏵛 􏴘􏵛 (by sifting property) 82 The Fourier-Slice Theorem 􏵘 􏳿,􏵃 =􏵖􏵛 􏵖􏵛􏳿 􏴁,􏴂 􏴍􏴘􏵓􏴩􏵗􏵙 􏴤􏵝􏵞􏵟􏵔􏵠􏴾􏵟􏵡􏵢􏵔 􏴑􏴁􏴑􏴂 􏴘􏵛 􏴘􏵛 Let 􏴖 = 􏳿 cos 􏵃 and v = 􏳿 sin 􏵃 􏵘 􏳿,􏵃 =􏵖􏵛 􏵖􏵛􏳿 􏴁,􏴂 􏴍􏴘􏵓􏴩􏵗 􏴚􏴤􏵠􏴧􏴾 􏴑􏴁􏴑􏴂􏵦􏴚􏴳􏵙􏵝􏵞􏵟􏵔, 􏴘􏵛 􏴘􏵛 􏴧􏴳􏵙􏵟􏵡􏵢􏵔 which is the 2D FT of the image 􏳿(􏴁, 􏴂) evaluated at the indicated frequencies 􏴖, 􏴦: 􏵘 􏳿,􏵃 = 􏴠(􏳿cos􏵃,􏳿sin􏵃) 83 The Fourier-Slice Theorem • The resulting equation 􏵘 􏳿, 􏵃 = 􏴠(􏳿 cos 􏵃 , 􏳿 sin 􏵃) is known as the Fourier- slice theorem. • It states that the 1D FT of a projection (at a given angle θ) is a slice of the 2D FT of the image. 84 The Fourier-Slice Theorem • We could obtain 􏳿(􏴁, 􏴂) by evaluating the FT of every projection and inverting them. • However, this procedure needs irregular interpolation which introduces inaccuracies. 85 Filtered Backprojection • The 2D inverse Fourier transform of 􏴠(􏴖, 􏴦) is 􏳿 􏴁,􏴂 =􏵖􏵛 􏵖􏵛􏴠 􏴖,􏴦 􏴍􏴩􏵗 􏴚􏴤􏵠􏴧􏴾 􏴑􏴖􏴑􏴦 􏴘􏵛 􏴘􏵛 • Express the integral in polar coordinate with 􏴖 = 􏴎cos􏵃,􏴦 = 􏴎sin􏵃 • The differential 􏴑􏴖􏴑􏴦 is equal to 􏴎􏴑􏴎􏴑􏵃 in polar coordinate • The integral in polar coordinate becomes 􏳿 􏴁,􏴂 =􏵖􏴩􏵗􏵖􏵛􏴠 􏴎cos􏵃,􏴎sin􏵃 􏴍􏴩􏵗􏵣 􏴤􏵝􏵞􏵟􏵔􏵠􏴾􏵟􏵡􏵢􏵔 􏴎􏴑􏴎􏴑􏵃 􏴜􏴜 86 Filtered Backprojection • Using the Fourier-slice theorem, 􏳿 􏴁,􏴂 =􏵖􏴩􏵗􏵖􏵛􏵘 􏴎,􏵃 􏴍􏴩􏵗􏵣 􏴤􏵝􏵞􏵟􏵔􏵠􏴾􏵟􏵡􏵢􏵔 􏴎􏴑􏴎􏴑􏵃 􏴜􏴜 • With some manipulation: 􏳿 􏴁,􏴂 =􏵖􏵗􏵖􏵛 􏴎􏵘 􏴎,􏵃 􏴍􏴩􏵗􏵣 􏴤􏵝􏵞􏵟􏵔􏵠􏴾􏵟􏵡􏵢􏵔 􏴑􏴎􏴑􏵃 􏴜 􏴘􏵛 • Theterm􏴁cos􏵃 +􏴂sin􏵃=􏵑andis independent of 􏴎: 􏳿􏴁,􏴂 =􏵖􏵗􏵖􏵛 􏴎􏵘􏴎,􏵃􏴍􏴩􏵗􏵣􏵚􏴑􏴎􏴑􏵃 􏴜 􏴘􏵛 87 Filtered Backprojection 􏳿(􏴁,􏴂)=􏵖􏵗 􏵖􏵛􏴎􏵘􏴎,􏵃􏴍􏴩􏵗􏵣􏵚􏴑􏴎􏴑􏵃 􏴜 􏴘􏵛 • For a given angle 􏵃, the inner expression is the 1-D Fourier transform of the projection multiplied by a ramp filter |􏵤|. • This is equivalent in filtering the projection with a high-pass filter with Fourier Transform |􏵤| before backprojection. 88 Filtered Backprojection 􏳿(􏴁,􏴂)=􏵖􏵗 􏵖􏵛􏴎􏵘􏴎,􏵃􏴍􏴩􏵗􏵣􏵚􏴑􏴎􏴑􏵃 􏴜 􏴘􏵛 Problem: the filter 􏴔(􏵤) = |􏵤| is not integrable in the inverse Fourier transform as it extends to infinity in both directions. • It should be truncated in the frequency domain. • The simplest approach is to multiply it by a box filter in the frequency domain, but ringing will be noticeable. • Windows with smoother transitions are used. 89 Filtered Backprojection • Ringing decreases with these windows. 90 Filtered Backprojection 91 Filtered Backprojection Filtered backprojection is obtained by the following steps: 1. Compute the 1-D Fourier transform of each projection (with padding to mitigate the effect of wraparound error) 2. Multiply each Fourier transform by the filter function |􏵤| (multiplied by a suitable window, e.g., Hamming). 3. Obtain the inverse 1-D Fourier transform of each resulting filtered transform. 4. Backproject and integrate all the 1-D inverse transforms from step 3. 92 Filtered Backprojection Backprojection Box-windowed FBP Hamming FBP 93 Filtered Backprojection • Ringing is more Box-Windowed FBP Hamming FBP pronounced in box-windowed FBP • Ringing is reduced in Hamming FBP, at the expense of slight blurring 94 Filtered Backprojection Box-Windowed FBP Hamming FBP 95