Hi, I was wondering if you can help in the attached Super Resolution Imaging assignment? we should plot the answers on Matlab. attached some slide to help, and some equationsHW 2 Super Resolution Imaging: SIM Use the Air Force Target 1951 as an object for the SIM image reconstruction. The image of the Air Force Target can be found online. Step 1: Construct your baseline Image (let’s call it image 1): Image 1 is the result of the limited bandwidth of your optical system. Simulate this by taking the Fourier transform of the object and let your optical transfer function (OTF) to be about 30-50% of your object’s max frequency. IFFT the resultant spectrum to obtain the baseline image that shows degradation. Plot image 1, the original object, and the corresponding spectra (that of image 1 and object). Step 2: Devise an appropriate structured illumination pattern (i.e. frequency of the pattern) for SIM processing. Multiple the object with 9 different variations of the pattern to obtain 9 different images (3 rotation angles (0. 60 and 120 degrees) and three phases for each rotation angle (0,120 and 240)). Show the corresponding images and spectra using the same OTF as in step 1. Step3: Extract high frequency components from the resultant spectra and rearrange them in correct k-space so that overall spectrum is larger than OTF. Show the resultant spectrum. Step 4: Take IFFT of the combined spectrum to obtain SIM image. Describe your finding Below are some additional information on the SIM recontrruction to help you. Figure 5 illustrates the general phenomenon of how moiré fringes are generated by two overlapping patterns of patterned lines (Figure 5(a)). The observable region of reciprocal space produced by a microscope objective (which is analogous to its diffraction pattern) is limited at the edges by the highest spatial frequencies that the objective can transmit (2NA / λ), where the central spot represents the zeroth order component. The zeroth (yellow dot) and first order (red dots) diffraction components representing a pattern of parallel lines are presented in Figure 5(c). If the pattern spacings lie at the limits of resolution, the first order spots occur at the edge of the observable field (shown here as red dots on the k0 boundary). Due to frequency mixing, the observable regions also contains, in addition to the normal image of spatial frequencies (center circle), two new offset frequency images (Figure 5(d)) that are centered on the edge of the original field. These offset images contain higher spatial frequencies that are not observed using traditional microscope optical systems. Finally, in Figure 5(e), a set of images is shown that were prepared from three phases at 120 degree orientations, which ultimately after processing, yield a real image that contains twice the spatial resolution as would be observed in widefield fluorescence microscopy. SR-SIM in its original form utilizes a sinusoidal pattern of parallel stripes given by the equation: I(r) = I0[1 + cos(k0 · r + ϕ)] (9) where ϕ specifies the phase of the illumination pattern. The effect of the sinusoidal illumination structure becomes apparent when comparing the Fourier transforms of images both without (Figure 7(a)) and with (Figure 7(b)) structured illumination. Displaced high frequency information becomes apparent due to moiré effects and is indicated by the arrowheads in Figure 7(b). The two dimensional Fourier transform of the illumination function consists of three delta function components that, when convolved with D̃(k) yields: Ẽ(k) = I0[D̃(k) + 0.5D̃(k + k0)eiϕ + 0.5D̃(k - k0)e-iϕ] (10) The observed emission light at each point in frequency space thus has three Fourier components corresponding to the three (yellow and red) points seen in Figure 5(c). If the periodicity of illumination structure is maximized, that is if the value of k0 is equivalent to the highest theoretical value calculated using Formula (6), then the center of each calculated transform will be located exactly along the outside edge of the original transform at a distance of magnitude k0 from the origin. In practice, this is accomplished by ensuring that the first diffraction order spots just enter the objective at opposite sides of the rear aperture. Acquiring a set of three images while adjusting the illumination phase for each yields three independent combinations of D̃(k), D̃(k + k0), and D̃(k - k0). The illumination phase translation increment should be equal to 1/3 of the grating period, which is required for proper separation of the three components summed in each image (Figure 7(c)). Three images thus result: the central normal image defined by D̃(k) and the two offset by D̃(k ± k0) that are separated by 180 degrees and are located on either side of the central transform. Since the illumination structure is known, the resulting 3 x 3 system of linear equations can then be solved algebraically to find the values of the constituent terms D̃(k), D̃(k + k0), and D̃(k - k0) in order to restore high frequency information to its correct place in frequency space (Figure 7(d)). However doing this only extends resolution in a single direction, as shown by Figure 5(d). In order to approximate an OTF support of radius 2k0 one must repeat this process using several different orientations of the grid pattern, as illustrated by Figure 5(e), to circumscribe the extended OTF support. The offset Fourier transforms obtained using 3 grid rotations is illustrated by Figure 7(c) and their computationally restored locations by Figure 7(d). Typically, one will use 3 orientations of the grid pattern to maximize acquisition speed or 5 orientations to increase the image quality of the superresolution reconstruction. The set of raw images corresponding to a single SIM reconstruction is referred to as a "SIM frame". When the specimen is illuminated by structured light generated by the grid pattern, moiré fringes will appear that represent information that has changed position in reciprocal space. Figure 7 illustrates the generalized procedure for reconstructing high frequency information in reciprocal space as described in detail above. A typical Fourier transform of a microscope image with uniform illumination is presented in Figure 7(a) and with structured illumination in Figure 7(b). The arrows in Figure 7(b) denote high frequency information that has undergone frequency mixing with the illumination structure and as a result has been displaced into the observable region in the form of moiré fringes. Acquiring three images at different phases of the grid pattern for each of three azimuthal rotations of the grid allows for the computation of the 7 components displayed in Figure 7(c). Subsequent computational restoration of each component to its proper place results in an image transform (Figure 7(d)) with approximately twice the radius of the conventional image transform area as given by Figure 7(a), ultimately yielding a doubling of lateral resolution. The Process of Image Formation in Fluorescence Microscopy Light Emission ¥ (x,y) Image Space ® (x,y) i) a ZI WE 3 AN RE Convolution [15 EE] Multiplication ——- Or) = D(r)- I(r) Ok) = Dk) ® I (k) I(r)=1, [1+ cos(2r 1, +4) I(k)=1I,[5(k) +0 Tk )e? + 0k =e] Ok) = I,[D(k) + SD Lk)e + > Dk —k Yet] Illumination Patterned Object in Reciprocal Space Angle= 120 Phase= 240 50 100 150 200 250 300 350 400 450 500 100 200 300 400 500 - tion = Or Orientation = 60° Orientation = 120° H,(0)H, (k) Dbar(k) 1 1 =|1 -0.5+0.86 1 -0.5-0.86 ~~ & = + ~~ TT = 3S Sz Sf 2 8 2 xX x, ~~ «1 3 x = - o <5 ~. - = z =i ie eee ay | ~.="" -="z" =i="" ie="" eee="" ay="">