1. Creating and Manipulating Images using Maths and a Spreadsheet LTA Conference 19 June 2013 Jeff Waldock Department of Engineering and Mathematics, Faculty of ACES 1 http://tinyurl.com/n7u5x6p

2. Creating and Manipulating Images using Maths and a SpreadsheetJeff Waldock 2 Experiential/active learning -Engagement -Motivation -Understanding -Practical applicability -Contextualises learning -Employability BSc Mathematics at SHU is distinctive -it emphasises real practical applications -it emphasises the importance of mathematical modelling

3. Creating and Manipulating Images using Maths and a SpreadsheetJeff Waldock 3 Almost all digital photo post-processing uses Mathematics: Reflection Rotation Changing brightness Changing contrast Tinting, altering colour balance Blurring Sharpening Re-sizing Edge-detection Distortions Pattern detection Students are familiar with digital images, and in post-processing them Face recognition SHUMaths students are also very adept with Excel. … led to a final year case study combining these elements, developing mathematical skills through experiential/action learning.

4. Creating and Manipulating Images using Maths and a SpreadsheetJeff Waldock 4 Digital images are composed of a pixel array, with each pixel itself comprising separate red, green and blue components Each of these components can take values between 0 and 255. Normally expressed as hexadecimal values – in the form #RRGGBB, so red, for example, is #FF0000. Colours.exe

5. Creating and Manipulating Images using Maths and a SpreadsheetJeff Waldock 5 The most common format for digital photographs is JPG, or JPEG. An Excel add-in has been created which can: Import JPG images, placing the red, green and blue components as (decimal) integers in separate worksheets. Resize images to fit the number of columns Recompile JPG images from the values in the R,G and B worksheets Students can therefore devise and apply a variety of effects by - developing a mathematical technique to create the effect - importing the image to be processed - implementing the effect using existing spreadsheet skills - exporting the new image

6. Creating and Manipulating Images using Maths and a SpreadsheetJeff Waldock 6 Contrast / Brightness Picturevalues.exe

7. Creating and Manipulating Images using Maths and a SpreadsheetJeff Waldock 7 Brightness: Straight lines. Contrast: Straight lines and curves. e.g. Image Gamma: Monochrome, colour casts, tints: Arithmetic Blur, Sharpen (unsharp mask): Matrices Translation, shear, stretch: Linear transformations Distortion (e.g. ripple effect): Geometry, trigonometry Rotation: Matrices Noise: Random numbers Colour control: Histograms, Statistics Morphing: Linear Interpolation Edge-detection: Finite differences Filtering: 2-D FFT The range of mathematics that can be explored in this way is broad: Images.xlsx

8. Creating and Manipulating Images using Maths and a SpreadsheetJeff Waldock 8 The Excel Add-in ImageAnalysis The student case study

9. Creating and Manipulating Images using Maths and a SpreadsheetJeff Waldock 9 “We're increasingly surrounded by digitally altered images in everyday life and it was fascinating to see the mathematics behind these" Student comments: “I thought that it was really fun, different and something you wouldn’t initially expect to be related to mathematics!” "The Excel skills that I have gained will be very useful in future employment" "The way the case study is set up encourages interest and research into the topic"

10. Creating and Manipulating Images using Maths and a SpreadsheetJeff Waldock 10

11. Creating and Manipulating Images using Maths and a SpreadsheetJeff Waldock 11 Can generate images based on mathematical functions:

12. Creating and Manipulating Images using Maths and a SpreadsheetJeff Waldock 12 http://maths.shu.ac.uk/units/55-5767/images/One_colour_grad6.php?w=500&h=3