Image analysis I C9940 3-Dimensional Transmission Electron Microscopy S1007 Doing structural biology with the electron microscope March 6, 2017 SyllabusWeek Date Instructor Topic 1 20.2 T. Shaikh Introduction/History/Optics 2 27.2 J. Novacek Specimen preparation 3 6.3 J. Novacek Instrumentation 4 13.3 T. Shaikh Image analysis I 5 20.3 T. Shaikh Image analysis II 6 27.3 J. Novacek Tomography I 7 3.4 J. Novacek Tomography I 8 10.4 T. Shaikh Image analysis III 17.4 (Easter) 9 24.4 T. Shaikh Image analysis IV 1.5 (May Day) 8.5 (Liberation Day) 10 15.5 J. Novacek Visualization/Hybrid methods 11 22.5 J. Novacek & T. Shaikh Journal club Syllabus Week Date Instructor Topic 1 20.2 T. Shaikh Introduction/History/Optics 2 27.2 J. Novacek Specimen preparation 3 6.3 T. Shaikh Image analysis I 4 13.3 J. Novacek Instrumentation 5 20.3 T. Shaikh Image analysis II (Room 1S102) 6 27.3 J. Novacek Tomography I 7 3.4 J. Novacek Tomography I 8 10.4 T. Shaikh Image analysis III 17.4 (Easter) 9 24.4 T. Shaikh Image analysis IV 1.5 (May Day) 8.5 (Liberation Day) 10 15.5 J. Novacek Visualization/Hybrid methods 11 22.5 J. Novacek & T. Shaikh Journal club Outline Image analysis I Fourier transforms Why do we care? Theory Examples in 1D Examples in 2D Digitization Fourier filtration Contrast transfer function Resolution Fourier transforms Outline Image analysis I Fourier transforms Why do we care? Theory Examples in 1D Examples in 2D Digitization Fourier filtration Contrast transfer function A quiz http://www.microscopy.ethz.ch A quiz http://electron6.phys.utk.edu http://www.microscopy.ethz.ch http://electron6.phys.utk.edu Relationship between imaging and diffraction The only difference between microscopy and diffraction is that, in microscopy, you can (easily) focus the scattered radiation into an image. How do X-ray microscopes work? http://ssrl.slac.stanford.eduhttp://www.microscopy.ethz.ch How do X-ray microscopes work? http://ssrl.slac.stanford.edu Fabrizio...Barrett, 1999, Nature Best resolution: ~20nm Outline Image analysis I Fourier transforms Relationship between imaging and diffraction Theory Examples in 1D Examples in 2D Digitization Fourier filtration Contrast transfer function Relevance of Fourier transforms to EM Fourier transform ~ diffraction pattern see John Rodenburg's site, http://rodenburg.org NOTE: ν=α/λ Fourier series A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines Fourier transforms: Exponential form f: function which we are transforming (1D) x: axis coordinate i: √-1 k: spatial frequency F(k): Fourier coefficient at frequency k Fourier transforms: Exponential form Euler's Formula: ba +i Fourier transforms: Sines + cosines (NOTE: This isn't the same a & b from the previous slide.) Fourier transforms: Definition ba +i a b Amplitude, A: Phase, Ф: Ф A Fourier coefficients, discrete functions sine and cosine are orthogonal. The inner product of the two is zero. A function with the same frequency but with an offset will have some components of both sine and cosine. That is, a and b will be non-zero. A function with a different frequency will have coefficients of zero. http://cnx.org The higher the spatial frequencies (i.e., higher resolution) that are included, the more faithful the representation of the original function will be. Some properties  As n increases, so does the spatial frequency, i.e., the “resolution.” − For example, sin(2x) oscillates faster than sin(x)  Computation of a Fourier transform is a completely reversible operation. − There is no loss of information.  Fourier terms (or coefficients) have amplitude and phase.  The diffraction pattern is the physical manifestation of the Fourier transform – Phase information is lost in a diffraction pattern. – An image contains both phase and amplitude information. Some simple 1D transforms: a 1D lattice Some simple 1D transforms: a box http://cnx.org Later, you will learn that multiplying a step function is bad, because of these ripples in Fourier space. Fourier transforms: plot of a Gaussian Xx f(x) F(X) Some simple 1D transforms: a sharp point (Dirac delta function) http://en.labs.wikimedia.org/wiki/Basic_Physics_of_Nuclear_Medicine/Fourier_Methods Some simple 2D Fourier transforms: a row of points Some simple 2D Fourier transforms: a series of lines Some simple 2D Fourier transforms: a sharp disc Some simple 2D Fourier transforms: a 2D lattice Some simple 2D Fourier transforms: a 2D lattice Some simple 2D Fourier transforms: a helix Outline Image analysis I Fourier transforms Relationship between imaging and diffraction Theory Examples in 1D Examples in 2D Digitization Fourier filtration Contrast transfer function Digitization in 2D Digitization in 1D: Sampling Digitization: Is our sampling good enough? Here, our sampling is good enough. Digitization in 1D: Bad sampling What's the best resolution we can get from a given sampling rate? A 4-pixel “image” 1 2 3 4 In other words, what is the most rapid oscillation we can detect? What's the best resolution we can get from a given sampling rate? In other words, what is the most rapid oscillation we can detect? ANSWER: Alternating light and dark pixels. A 4-pixel “image” The period of this finest oscillation is 2 pixels. The spatial frequency of this oscillation is 0.5 px-1 . The finest detectable oscillation is what is known as “Nyquist frequency.” The edge of the Fourier transform corresponds to Nyquist frequency. origin spatial frequencyspatial frequency Nyquist frequency Nyquist frequency The period of this finest oscillation is 2 pixels. The spatial frequency of this oscillation is 0.5 px-1 . The finest detectable oscillation is what is known as “Nyquist frequency.” The edge of the Fourier transform corresponds to Nyquist frequency. What do we mean by pixel size? http://www.en.wikipedia.org Typical magnification: 50,000X Typical detector element: 15μm (pixel size on the camera scale) Pixel size on the specimen scale: 15 x 10-6 m/px / 50000 = 3.0 x 10-10 m/px = 3.0 Å/px In other words, the best resolution we can achieve (or, the finest oscillation we can detect) at 3.0 Å/px is 6.0 Å. It will be worse due to interpolation, so to be safe, a pixel should be 3X smaller than your target resolution. What happens if you're not oversampled enough? Aliasing https://www.youtube.com/watch?v=6LzaPARy3uA What do we mean by spatial frequency? origin spatial frequency From Wikipedia Fourier filtration From Wikipedia PowerspectrumProfile A “high-pass” filter A “low-pass” filter Contrast transfer function Why do we defocus? 1.0 0.1 Typical amplitude contrast is estimated a 0.08-0.12 (minus noise) water macromolecule Instead of amplitude contrast, we'll use phase contrast. Phase contrast in light microscopy Bright-field image Phase-contrast image http://www.microbehunter.com In EM, even with defocus, the contrast is poor. Signal-to-noise ratio for cryoEM typically given to be between 0.07 and 0.10. E. coli 70S ribosomes, field width ~1440Å. unfiltered filtered Optical path Specimen Back focal plane Image plane At focus, all we would see is amplitude contrast. Specimen Back focal plane Image plane Optical path with defocus Perfect focus Image plane Focal plane O A B OA path of unscattered beam OB path of scattered beam The length OA is also the amount of defocus Δf Optical path with defocus What is the path difference between the scattered and unscattered beams? Path difference as a function of Δf OB = OA/cos(a) Expressed in the number of wavelengths λ Phase difference is the sine O A B a Some typical values OA = Δf = 10,000 Å λ = 0.02 Å a < 0.01 A more precise formulation of the CTF can be found in Erickson & Klug A (1970). Philosophical Transactions of the Royal Society B. 261:105. QUICK QUIZ: What other example did we discuss where rays scattered at different angles experienced different path lengths? photographylife.com Proper form the CTF where: Cs: spherical aberration k: spatial frequency (resolution) How does the CTF affect an image? original original combined Still a zero present QUICK QUIZ: What would happen if you collected all of your images at the same defocus? Syllabus Week Date Instructor Topic 1 20.2 T. Shaikh Introduction/History/Optics 2 27.2 J. Novacek Specimen preparation 3 6.3 T. Shaikh Image analysis I 4 13.3 J. Novacek Instrumentation 5 20.3 T. Shaikh Image analysis II (Room 1S102) 6 27.3 J. Novacek Tomography I 7 3.4 J. Novacek Tomography I 8 10.4 T. Shaikh Image analysis III 17.4 (Easter) 9 24.4 T. Shaikh Image analysis IV 1.5 (May Day) 8.5 (Liberation Day) 10 15.5 J. Novacek Visualization/Hybrid methods 11 22.5 J. Novacek & T. Shaikh Journal club