1 Statistics, Data Analysis and Image Processing Lectures Vlad Stolojan Advanced Technology Institute University of Surrey

2 Learning Outcomes - how to calculate experimental errors - image processing - filtering - thresholding - particle analysis. - data analysis - fitting peaks - filtering - displaying

3 The labbook Date, title Aim of the experiment Description of the apparatus (brief), arrangement (sketch) Experimental method (main steps, precautions) Measurements (if recorded), environmental parameters Graphs + Calculations Conclusions (brief). Number pages.

4 Experimental results Repeatability Calculations = levels of confidence (eg. Organic solar cell with 8% efficiency ± ?). Uncertainties in measurements = significant figures: 3.14 ± ± ± SI UNITS – always check your formula. (particularly in exam).

5 True value, Accuracy, Precision True value: when measuring we estimate the exact, true value. The more measurements, the closer the mean of these is to the true value. Accurate: close to the true value. Precise: small uncertainty bu not necessarily close to the true value. Accurate and Precise

6 Error Propagation If a has uncertainty Δa and b has uncertainty Δb, then the uncertainty of a function of a and b, may not be just Δa+Δb – Δa may be positive and Δb negative. What it means is that if: f = f(a,b) then Δf ≠ f(Δa,Δb) Variational calculus:

7 Error propagation Young's modulus for a beam of length L 0, area A 0 which lengthens ΔL under force F What is the error in E. Clue: divide the error propagation equation by E.

8 Poisson Distribution (counting experiments) - a quantity that does not vary continuously. Eg: measuring radiation 'events' or the same quantity several times. Displayed as No of counts (frequency of counts) per count interval. The standard deviation of an experiment measuring a quantity N times, with standard deviation σ is

9 Propagate errors

10 Error propagation An amorphous square solar cell of dimensions (1.0±0.1) cm is illuminated with (0.155 ± 0.001) W. The maximum current density measured from the device is J m = (18.9 ± 0.1) mA/cm 2 and maximum voltage is V m = (0.950 ± 0.001) V. What is the efficiency of the device and the error?

11 Peak position A spectrum = counts per energy/ frequency/ wavelength interval – Poisson statistics. The standard deviation in the position of a peak scales with 1/√N (no. of data points used in fitting). We can compare peak-shift positions say to meV even when the energy interval (sampling resolution) is 200meV. How many data points? What energy interval?

12 Mean, Median, Mode, Skewness, etc Mean = sum of observations / no. of observations (the expected value). Median = the number separating the higher half of observations from the lower half of observations. Mode = the value that occurs most frequently. Skewness = asymmetry of the distribution. Negative – long left tail, positive – longer right tail. Kurtosis (bulging -greek) : high: sharp peak with fat tails vs low: rounded peak with thin tails.

13 Image processing and analysis Intensity histograms; Histogram transformations. Geometric transformations: scaling, rotation, interpolation, binning. Color images. Filtering Particle analysis Bi-variate tri-variate histograms.

14 Intensity Histogram. Transformations

15 Intensity Histogram. Transformations

16 Interpolation and Resampling Interpolation: Nearest-neighbour: assign the value of the nearest pixel Bilinear: weighted sum of the 4 nearest pixels. Resampling (binning) combine two or more pixels into a weighted average → decrease of resolution but faster read-out. Can be done on CCD capture device.

17 Colour RGB = (red, green, blue) : vector; each component CMY, CMYK (blacK) Magenta Cyan Yellow

18 Filtering Filtering = used for smoothing, noise reduction or for emphasizing a pattern/feature. Convolution

19 Filtering Smooth Sharpen Band pass Edge detection

20 Particle analysis 1)Flat-fielding (uniform illumination, uniform response) 2)Filtering + binarization (convert grey scale to black- white). 3)Thresholding (select information) 4)Analyze particles. ImageJ Plugins

21 Graphs Usually all data should look good represented in an image a journal column-wide. Font min 11, size 8.5 x (up to a page length) cm. Minimise empty space (scaling, etc) Captions caption captions! Descriptive and informative. The reader should not need to refer to the text to understand what is shown in the graph. Definitely NOT 'plot of this vs that'

22 Examples of captions

23 Examples of captions