Biophysical Techniques (BPHS 4090/PHYS 5800)

Transcription

1 Biophysical Techniques (BPHS 4090/PHYS 5800) Instructors: Prof. Christopher Bergevin Schedule: MWF 1:30-2:30 (CB 122) Website: York University Winter 2017 Lec.3

2 Example of possible Journal Club paper

3 à Close connection to BPHS 4080 topics... Weiss (1996)

4

5 Hoppe et al (1983)

6 à Close connection to BPHS 4090 topics...

7 à Close connection to BPHS 4080 topics... Weiss (1996)

8 Wikipedia

9 à Technique used here is two-photon microscopy (not CT; we ll aim to come back to this later in the semester) Denk et al. (Neuron, 1997)

10

11 Observation: Stacks of 2-D images are sliced from a 3-D object Idea: (Re-)Build up 3-D object from series of 2-D images* * MRI, CT imaging, two photon imaging, confocal microscopy, etc... all allow for 3-D imaging, but work under very different principles Wikipedia

12 à Goal is to obtain detailed 3-D information (e.g., for medical imaging, this may reveal a crucial aspect of patient s condition to radiologist) Herman (ch.2)

13 Focal point: CT Scanner Ø Uses x-rays (i.e., ionizing radiation) Ø Different tissue types have different attenuation coefficients Ø Detector signal depends upon effective attenuation coefficient of what is in path between it and source Ø Source and detector rotate around object, thereby tracing out a series of projected images Franklin et al.

14 Tomography Derived from the Greek tomē ("cut") or tomos ("part" or "section") and graphein ("to write ) à Think of 2-D projections as the sum of slices of a 3-D object à From the 2-D projections, goal is to reconstruct 3-D object Note: We can only (directly) measure P, not S 1 or S 2 [we can only know the slices from the reconstruction] Wikipedia

15 à We ll focus here on a 2-D object projected onto 1-D axes, but general idea scales back up to higher dimensions Herman (ch.2)

16 Projections Note: Other projections are possible beyond just those shown here Darvey (Phys. Teach. 2013)

17 Ex. à Are these all the projections? Dobrovolny

18 Projections Delaney & Rodriguez (Am. J. Phys. 2002)

19 Reconstruction?? Dobrovolny

20 Backprojection à Simple means to reconstruct N dimensional objects from series of projected N-1 dimensional images Delaney & Rodriguez (Am. J. Phys. 2002)

21 Backprojection à Potential improvement using more refined methodology (e.g., filtered backprojection ) Delaney & Rodriguez (Am. J. Phys. 2002)

22 Ex. IL tz1 3 1l, dlrejrr' o{ p*jojr,^ Dobrovolny

23 Ex. (cont.) à Reconstructed object is not necessarily same as original Dobrovolny

24 Radon Transforms Transformation coord. system Hobbie & Roth (ch.12.5)

25 Aside: Basic notion of a transform transformer pic Intuitive connection back to Taylor series: Taylor series à Expand as a (infinite) sum of polynomials Different Idea: Fourier series à Expand as a (infinite) sum of sinusoids à Different types of series expansions allow us to transform a function into other bases that might be easier to work with

26 Aside: Basic notion of a transform e.g., consider the conversion between Cartesian and polar coordinates for complex #s Cartesian Form Polar Form à In many instances, polar form is much easier to work with

27 Aside: Basic notion of a transform Fourier transforms are very useful in image processing (as we will see), but not that intuitive Somewhat more straightforward in hearing (as we will see later, your ear is a hydrodynamic Fourier analyzer)

28 Radon Transforms Transformation coord. system à Radon transform using straight lines as basis function à Inverse radon transform is the backprojection Hobbie & Roth (ch.12.5)

29 Figure 2.2: ID Projection go(x') of a 2D Function f (x,y1 obtained by integrating f (x, y1 along the y' direction. Nishimura

30 Radon Transforms à Note role of attenuation coefficient here (we ll come back to this soon) Herman (ch.2)

31 Sinogram Dobrovolny

32 Radon Transforms - Practical considerations Ø Finite measurement space (i.e. resolution) Ø Propagation of error do to uncertainties and noise à Limit of reconstruction quality? Ø Efficient means to compute integral? à Projection-slice theorem (& Fourier transforms)

33 Radon Transforms - Practical considerations Note: While we will outline the math here, we acknowledge that the necessary tools are not yet in our pocket. Put another way, let this serve to motivate us to (subsequently) develop Fourier analysis Ø Projection à Radon transform [i.e., projection of f (x,y) onto a straight line] Ø Inverse Radon transform à Reconstruction [i.e., mathematical basis for tomography] Ø Radon transform (RT) ßà Fourier transform (FT) à 2-D FT (of object) is related to 1-D FT of the RT (i.e., projection) Fourier slice theorem (aka Projection-slice theorem) Build back up image by taking FT of various projections, then inverse FT of array you built up for different slices, thereby to reconstruct object

34 Fourier-Slice Theorem (Proof) Ø Projection of f (x,y) onto x-axis Ø Fourier transform of f (x,y) Spatial domain Frequency domain Wave number (à spatial frequency) Ø Single slice of Fourier transform (by definition) à Just the Fourier transform of the projection Wikipedia (Projection-slice theorem)

35 Summary Ø One common form of modern medical imaging uses Computerized Tomography (CT scan) Ø Connection between Radon and Fourier transforms allows efficient (although not exact) reconstruction Ø Idea is to build up projections based upon absorption of EM radiation Ø Provides valuable tool(s) for radiologists

36 Summary We have encountered an interesting problem (tomography), but clearly there are some key concepts we need to develop further to really understand what is going on here: Basic signal processing considerations re an image (e.g., convolution) EM-matter interactions (e.g., attenuation coefficients ) Notion of a transform Fourier analysis (i.e., spectral decomposition) à We ll take a detour to explore these topics in some detail...

37 Looking Back/Ahead Question: Was this type of image acquired using tomography? Short answer: Not directly. Very different principle using nuclear magnetic resonance (NMR), where there is no incident (ionizing) EM radiation and subsequent attenuation