Image analysis I C9940 3-Dimensional Transmission Electron Microscopy S1007 Doing structural biology with the electron microscope March 2, 2015 Syllabus, original Week Date Instructor Topic 1 02/16 D. Nemecek & T. Shaikh Introduction/Tour/History 2 02/23 D. Nemecek & T. Shaikh Electron Optics 3 03/02 D. Nemecek Specimen preparation 4 03/09 T. Shaikh Image analysis I 5 03/16 T. Shaikh Image analysis II 6 03/23 T. Shaikh 3D reconstruction 7 03/30 T. Shaikh Single-particle reconstruction (Easter) 8 04/13 D. Nemecek Tomography I 9 04/20 D. Nemecek Tomography II 10 04/27 D. Nemecek Visualization/Segmentation 11 05/04 D. Nemecek Hybrid methods Syllabus, updated Week Date Instructor Topic 1 02/16 D. Nemecek & T. Shaikh Introduction/Tour/History 2 02/23 D. Nemecek & T. Shaikh Electron Optics 3 03/02 T. Shaikh Image analysis I 4 03/09 D. Nemecek Specimen preparation 5 03/16 T. Shaikh Image analysis II 6 03/23 T. Shaikh 3D reconstruction 7 03/30 T. Shaikh Single-particle reconstruction (Easter) 8 04/13 D. Nemecek Tomography I 9 04/20 D. Nemecek Tomography II 10 04/27 D. Nemecek Visualization/Segmentation 11 05/04 D. Nemecek Hybrid methods Correction/clarification from Feb. 16 http://ernst.ruska.de First Siemens microscope, 1939 http://emu.msim.org.uk First commercial EM, 1937 Metropolitan-Vickers EM1 (EM2 shown) The first commercial electron microscope was actually by the British company Metropolitan-Vickers in 1937. However, the magnification was worse than for the light microscope, so the Siemens is considered “first.” Correction/clarification Metropolitan Vickers eventually became AEI, which built a 1.2 million volt EM-7. http://www.wadsworth.org Outline Correction/clarification form Feb. 16 Intro to image analysis Relationship between imaging and diffraction Fourier transforms Theory Examples in 1D Examples in 2D Digitization Fourier filtration Contrast transfer function Resolution Outline Correction/clarification form Feb. 16 Intro to image analysis Relationship between imaging and diffraction Fourier transforms Theory Examples in 1D Examples in 2D Digitization Fourier filtration Contrast transfer function Resolution http://www.microscopy.ethz.ch http://electron6.phys.utk.edu Relationship between imaging and diffraction http://electron6.phys.utk.edu 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 Outline Correction/clarification form Feb. 16 Intro to image analysis Relationship between imaging and diffraction Fourier transforms Theory Examples in 1D Examples in 2D Digitization Fourier filtration Contrast transfer function Resolution Relevance of Fourier transforms to EM Fourier transform ~ diffraction pattern see John Rodenburg's site, http://rodenburg.org 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: Definition f: function which we are transforming (1D) x: axis coordinate i: √-1 k: spatial frequency F(k): Fourier coefficient at frequency k Fourier transforms: Definition 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 Some simple 1D transforms: a Gaussian 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 2D lattice Some simple 2D Fourier transforms: a sharp disc Some simple 2D Fourier transforms: a series of lines 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. 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 Outline Correction/clarification form Feb. 16 Intro to image analysis Relationship between imaging and diffraction Fourier transforms Theory Examples in 1D Examples in 2D Digitization Fourier filtration Contrast transfer function Resolution How do we evaluate the quality of a reconstruction? Now, how do we compare the two half-set reconstructions? images “odd” reconstruction “even” reconstruction We split the data set into halves and compare them. Fourier Shell Correlation (FSC) Properties: - Fourier terms have amplitude + phase. - Correlation values range from -1 to +1. - Noise should give an average of 0. - The comparison is done as a function of spatial frequency (or “resolution”) Reconstruction1 Reconstruction2 term 1 term 2 Fourier Shell Correlation; A better example It is controversial what single number to use to describe this curve, but a common practice is to report the value where the FSC=0.5 as the nominal resolution.