MTF Definition Kyle R Bryant Tutorial Presentation OPTI

MTF Definition Kyle R. Bryant Tutorial Presentation: OPTI 521 Distance 25 November 2020 • MTF is a measure of intensity contrast transfer per unit resolution of an image or signal. • It is used in optics, electronics, and related signal processing applications. 1

Imaging Task Kyle R. Bryant Tutorial Presentation: OPTI 521 Distance As spatial separation decreases, the “good” system maintains clear separation of point source images, while the “poor” system eventually can no longer distinguish them. MTF quantifies this phenomenon in terms of contrast between the center peak intensities versus intensity at their midpoint across a scale of separation distances. Good 25 November 2020 Poor At large separations, even a poor system can completely resolve the two images. As separation decreases, only the good systems can still recognize separate sources. 2

Contrast Modulation: A Basic MTF Kyle R. Bryant Tutorial Presentation: OPTI 521 Distance Contrast Modulation is defined simply by averaging the difference of maximum and minimum transmitted intensities: 25 November 2020 Output Signal Imax Imin Original Signal “Spatial Frequency” typically implies an array of sine or bar targets at a given spacing, expressed in line-pairs-permillimeter (lp/mm) or cylces - One line pair per-milliradian (cy/mrad) 3

Nyquist Sampling Theory Tutorial Presentation: OPTI 521 Distance Kyle R. Bryant 25 November 2020 Input Waveform Nyquist Theory: In order to achieve perfect reconstruction of an input signal which has a maximum spatial frequency "f" (the cutoff), sampling must occur at a rate of at least "2 f". (Note: Phase is still an issue!) Samples Out of Phase Sampled Output Sample Intervals in Phase 4

Optical MTF Kyle R. Bryant Tutorial Presentation: OPTI 521 Distance Imaging optical systems perform sampling, with the maximum sample frequency determined by the “spot size” image of a perfect point source object (e. g. , “Impulse Response”). A “perfect” optical system is limited in resolution by wavelength (l) dependent diffraction effects. Lens aberrations can only worsen performance. The MTF of an optical system is found by Fourier operations on the “spot size”, or Point Spread Function. 25 November 2020 Image Object For an Object source at infinity distance: For a Diffraction Limited Circular Aperture: Frequency Scale Conversion: 5

System MTF Kyle R. Bryant Tutorial Presentation: OPTI 521 Distance The MTF of cascaded optical assemblies is NOT equal to the product of component MTF’s! Why? Lenses transmit not just intensity, but wavefront phase as well, and hence aberrations in one lens cancel those in another. MTF of cascaded objective lenses, detector, and displays may be multiplied for composite “System MTF”, with a component MTF measured at each intensity transfer point. 25 November 2020 Focal Plane Image MTFsystem ¹ MTFLens 1*MTFLens 2 MTFsystem = MTFobjective*MTFdetector*MTFdisplay*MTFeyepiece*MTFeye 6

System MTF Tutorial Presentation: OPTI 521 Distance Kyle R. Bryant 25 November 2020 Example System MTF calculation: Freq. Obj. Lens FPA Display Eyepiece System cy/mrad MTF MTF MTF 1. 0 0. 9999 0. 99 2. 0 0. 99 0. 90 0. 79 3. 0 0. 85 0. 98 0. 92 0. 88 0. 67 For average human observers, MTF values around 0. 05 are considered barely resolvable. If the above system MTF reached 0. 05 at 10 cy/mrad, for example, then you can predict that a human observer could identify (6 cylce criteria) a 2. 4 meter taget through this sensor at a maximum range (this is a coarse estimate!) of about 7

Measurement: Mathematical Functions Kyle R. Bryant Tutorial Presentation: OPTI 521 Distance 1. Measure the instensity profile ("spot size") Point Spread Function, or in 1 D case, the Line Spread Function, with an optical instrument (not as easy as it sounds sometimes) 2. Calculate the modulus (absolute value) of the Fourier Transform of the PSF or LSF, cut out the foldover reflection, normalize to 1. 0, and set the frequency scale based on cutoff calculation for n 0 as shown earlier. 1 2 2 1 1 1 2 2 2 1 1 1 3 1 1 1 25 52 25 1 3 52 100 52 3 1 25 52 25 1 1 1 3 1 1 1 2 2 2 1 1 25 November 2020 1 2 2 2 1 1 Discrete Fourier Transform: A typical MTF plot is really one-dimensional (e. g. , derived from the Line Spread Function. Hence, MTF's are typically plotted with curves for orthogonal Radial and Tangential orientations. 8

Measurement: Knife Edge Tutorial Presentation: OPTI 521 Distance Kyle R. Bryant 25 November 2020 Knife Edge Measure of the Line Spread Function (LSF): 1. Drag a knife-edge across the focal plane of the optic to be tested and record the intensity 2. Calculating the derivative of this data gives us the LSF we are looking for so we can continue with MTF. 3. The discrete derivative sequence di of a sequence of numbers xi is easy: Differentiate di = ||xi - xi+1|| (we want the absolute values here) Edge Response Line Spread 9