Exercises

1.

Generate a square grid such as that in the background of Fig.2.6, if necessary using pen and paper, and mark in some pixels. Repeat with another grid and mark the origin in both grids. Now compute carefully the dilation and erosion of one by the other. Write a program to dilate and or erode one pixel array by another and display all three arrays.

2.

With the program of the last exercise, explore alternate erosions and dilations on some .tif files obtained from a scanner or from disk.

3.

Use erosion techniques to count the objects in Fig.2.10 or Fig.2.12.

4.

Write a program to find the lines of text and the spaces between them in some scanned image of text such as Fig.2.4. You may use any method you choose.

5.

Extend the previous program so as to separate out each line of text into words.

6.

Extend the previous program so as to isolate single characters. What does it do to an image such as Fig.2.2?

7.

Write a boundary tracing program and try it out on a .tif file of your choice. Alternatively, you might like to start on a bitmap file if you have access to a unix workstation and some familiarity with the beast. See the .c files on disk for a helping hand.

8.

By hook or by crook put a box around each of some characters in a printed word, either from Fig.2.4 or somewhere else. Normalise the boxes to a standard size, having first worked out what a good standard size is. Print your normalised images and ask whether you could recognise them by eye! Recall that a 2 by 2 matrix applied to each pixel will give the effect of a linear transformation. It is recommended that you expand or shrink each axis separately to a standard size as your first move.

9.

How many scan lines would you think necessary to distinguish the characters in your sample?

Turn each character into a vector by scanline methods. Can you tell by looking at the vectors what the characters are? (It might be useful to start of with a set of digits only, and then to investigate to see how things change a the alphabet size increases.)

10.

Are there any alternative mask shapes you feel tempted to use in order to convert each character into a vector?

11.

Segment by hand an image such as Fig.2.12 and normalise to occupy the unit disk with the centroid of the function representing the region at the origin. Divide the disk up into sectors and some the quantity of grey in each sector. Can the resulting vectors distinguish shapes by eye? Can you devise any modifications of this simple scheme to improve the recognition?

12.

Obtain images of shapes which have some kind of structure which is complicated; orient them in some canonical way and look at them hard. Can you devise a suitable collection of masks to discriminate the shapes? Characters under grey levels present an interesting problem set. Try getting camera images of stamped digits (easily made with modelling clay and a set of house numbers) under different illuminations.

13.

Compute the central moments of one of your characters up to order two by hand. Write a program for doing it automatically and confirm it gives the right answer. Extend your program to compute central moments up to any order. Compute them up to order n for all your characters, and check by eye to see if you can tell them apart by looking at the vectors, for different values of n.

14.

Write a program to normalise any array of characters into the unit disk with the centroid as origin.

15.

Write a program to compute the first 12 Zernike moments for a pixel array which has been normalised into the unit disk.

16.

Use a border following algorithm to extract external borders of your characters and then apply the last program to see if you can still recognise the characters by looking at the vectors obtained from taking moments of the boundaries.

17.

Find an algorithm for finding edges in greyscale images and use it to differentiate an image, bringing the borders of objects into relief. The disk programs might be places to start looking, but you may find other books of value.

18.

Write a program which smooths a time series of real numbers by applying a moving average filter. The program should ask for the filter coefficients from -n to n after finding out what n is. To be more explicit, let x(n) be a time series of real numbers; to get a good one, take the daily exchange rate for the Dollar in terms of the Yen for the last few weeks from the past issues of any good newspaper. Let g(n) be defined to be zero for |n| > 5 and assign positive values summing to 1 for $-5 \leq n \leq 5$. You could make them all equal, or have a hump in the middle, or try some negative values if you feel brave.

Your program should generate a new time series y(n) defined by

$$y(n) = \sum_{j=-\infty}^{j=\infty} x(n-t)g(t)$$

and plot both x and y.

19.

Modify your program to deal with a two dimensional filter and experiment with it; get it to remove `salt and pepper' noise introduced into a grey scale image by hand.

20.

Beef up the above program even further, and use it for differentiating images. Try to segment images like Fig.2.12 by these means.

21.

Generate a binary waveform x(n) as follows: if the preceding value is black, make the current value white, and if the preceding value is white, make the current one black. Initialise at random. This gives a trivial alternating sequence, so scrap that one and go back the preceding two values.If the two preceding values are both black, make the current value white; if the last two values were black then white, make the current value white; if the last two values were white make the current value black, and if the last two values were white then black, make the current value black. Well, this is still pretty trivial, so to jazz it up, make it probabilistic. Given the two previous values, make a random choice of integer between 0 and 9. If both the preceding values were black, and the integer is less than 8 make the current value white, if it is 8 or 9 make it black. Make up similar rules for the other three cases. Now run the sequence from some initial value and see what you get.

22.

Could you, given a sequence and knowing that it was generated according to probabilities

based on the preceding k values as in the last problem, work out what the probabilities are?

23.

Instead of having k = 2, does it get more or less interesting if k is increased?

24.

Instead of generating a sequence of binary values, generate a sequence of grey scale values between 0 and 1 by the same idea.

25.

Can you generalise this to the case of two dimensional functions? On a rectangular pixel array, make up some rules which fix the value of a pixel at 0 or 1 depending on the values of pixels in the three cells to the North, West and North-West. Put a border around the array and see if you can find rules for generating a chess-board pattern. Make it probabilistic and see if you can generate an approximate texture. Do it with grey levels instead of just black and white.

26.

Make up a probabilistic rule which decides what a cell is going to have as its value given all the neighbouring pixel values. Generate a pixel array at random. Now mutate this initial array by putting a three by three mask on, and using the surrounding cells to recompute the middle one. Put your mask down at random repeatedly. Do the following: always make the centre pixel the average of the surrounding pixels. Now make the top and left edge black, the bottom and right edge white, and randomise the original array. Can you see what must happen if you repeat the rewrite operation indefinitely? Run this case on your program if you can't see it.

27.

Can you, in the last set of problems, go back and infer the probabilistic rules from looking at the cell values? Suppose one part of an array was obtained by using one set of values and another by using a different set, can you tell? Can you find a way of segmenting regions by texture?

28.

Take an image of some gravel on white paper, then sieve the mixture and take another image of what gets through the sieve and another of what gets left in the sieve. Repeat six times. Can you sort the resulting images by any kind of texture analysis? Try both high and low angle illumination in collecting your data.