====== Introduction to MATLAB ======

MATLAB will be used for the MPV labs. 
In case you are not familiar with matlab, study the following parts of the  [[http://www.mathworks.com/access/helpdesk/help/techdoc/learn_matlab/bqr_2pl.html|"Getting started with
MATLAB (MathWorks)"]]:

  * Matrices and Arrays - (Expressions, Working with Matrices, More About Matrices and Arrays)
  * Graphics - (Editing Plots, Mesh and Surface Plots, Images)
  * Programming - (Flow Control, Other Data Structures, Scripts and Functions)

(home study)

Other useful tutorials are:

[[http://rt.uits.iu.edu/visualization/analytics/math/matlab-getting-started.php|"Getting started with MATLAB (Indiana University)"]]

[[http://www.cyclismo.org/tutorial/matlab/|"Matlab Tutorial (Clarkson University)"]]


 
====== Basics of Image Processing in Matlab ======

===== Convolution, Image Smoothing and Gradient =====

  * The Gaussian function is often used in image processing as a low pass filter for noise reduction, or as a windowing function weighting points in a neighbourhood. Implement the function ''G=gauss(x,sigma)'' that computes values of a (1D) Gaussian with zero mean and variance $\sigma^2$:\\ $$
G = \frac{1}{\sqrt{2\pi}\sigma}\cdot e^{-\frac{x^2}{2\sigma^2}}
$$\\ in points specified by vector ''x''.
  * Implement function ''D=dgauss(x,sigma)'' that returns the first derivative of a Gaussian\\ $$\frac{d}{dx}G(x) = \frac{d}{dx}\frac{1}{\sqrt{2\pi}\sigma}\cdot e^{-\frac{x^2}{2\sigma^2}} = -\frac{1}{\sqrt{2\pi}\sigma^3}\cdot x\cdot e^{-\frac{x^2}{2\sigma^2}} = -\frac{x}{\sigma^2}G(x)$$\\ in points specified by vector ''x''.
  * Get acquainted with the function ''conv2''.
  * The effect of filtering with Gaussian and its derivative can be best visualized using an impulse (1-nonzero-pixel) image:<code>
sigma = 6.0;
x = [-ceil(3.0*sigma):ceil(3.0*sigma)];
G = gauss(x, sigma);
D = dgauss(x, sigma);
imp = zeros(49); imp(25,25) = 1.0;
out = conv2(D, G, imp, 'same');
...

imagesc(out); % or surf(out);</code> try to find out impulse responses of other combinations of the Gaussian and its derivatives.
  * Examples of impulse responses: \\ {{courses:a4m33mpv:cviceni:1_uvod:derivatives.png}} \\ 
  * Write a function ''out=gaussfilter(in,sigma)'' that implements smoothing of an input image //in// with a Gaussian filter of width ''2*ceil(sigma*3.0)+1'' and variance $\sigma^2$ and returns the smoothed image //out// (e.g. [[https://cw.felk.cvut.cz/lib/exe/fetch.php/courses/a4m33mpv/cviceni/1_uvod/lenna.png|Lenna]]). Exploit the separability property of Gaussian filter and implement the smoothing as two convolutions with one dimensional Gaussian filter (see function ''conv2'').
  * Modify function gaussfilter to a new function ''[dx,dy]=gaussderiv(in,sigma)'' that returns the estimate of the gradient (gx, gy) in each point of the input image //in// (MxN matrix) after smoothing with Gaussian with variance $\sigma^2$. Use either first derivative of Gaussian or the convolution and symmetric difference to estimate the gradient.
  * Implement function ''[dxx,dxy,dyy]=gaussderiv2(in,sigma)'' that returns all second derivatives of input image //in// (MxN matrix) after smoothing with Gaussian of variance $\sigma^2$.
 

==== References ==== 
[[http://www.bmia.bmt.tue.nl/education/courses/FEV/book/index.html|Additional study material: chapters [3,4,5] on the Gaussian function, its derivatives and their role in image processing]]

=====  Geometric Transformations and Interpolation of the Image =====
  * Implement function ''A=affine(x1,y1,x2,y2,x3,y3)'' that returns 3x3 transformation matrix A which transforms point in homogeneous coordinates from canonical coordinate system into image: (0,0,1)->(x1,y1,1), (1,0,1)->(x2,y2,1), (0,1,1)->(x3,y3,1). \\ {{ courses:a4m33mpv:cviceni:1_uvod:affine.png?500 }} \\ 
  * Write function ''out=affinetr(in,A,ps,ext)'' that warps a patch from image //in// (MxN matrix) into canonical coordinate system. Affine transformation matrix //A// (3x3 elements) is a transformation matrix from the canonical coordinate system into image from previous task. The parameter //ps// defines the dimensions of the output image (the length of each side) and //ext// is a real number that defines the extent of the patch in coordinates of the canonical coordinate system. E.g. ''out=affinetr(in,A,41,3.0)'', returns the patch of size 41x41 pixels that corresponds to the rectangle (-3.0,-3.0)x(3.0,3.0) in the canonical coordinate system. Top left corner of the image has coordinates (0,0). Use bilinear interpolation for image warping. Check the functionality on [[https://cw.felk.cvut.cz/lib/exe/fetch.php/courses/a4m33mpv/cviceni/1_uvod/img1.png|image]].  \\ {{ courses:a4m33mpv:cviceni:1_uvod:affinetr.png?600 }} \\ 

===== What should you upload? =====
Upload all the Matlab functions implemented in this lab: ''gauss.m'', ''dgauss.m'', ''gaussfilter.m'', ''gaussderiv.m'', ''gaussderiv2.m'', ''affine.m'' and ''affinetr.m'' into the upload system in a .zip archive. Keep all functions in the root of the .zip archive together with all helper functions required. Follow closely the specification and order of the input/output arguments.
==== References ====
[[http://www.cs.princeton.edu/courses/archive/fall00/cs426/lectures/transform/transform.pdf|Geometric transformations - hierarchy of transformations, homogeneous coordinates]]\\
[[http://cmp.felk.cvut.cz/~hlavac/TeachPresCz/11DigZprObr/18BrightGeomTxCz.pdf|Geometric transformations - review of course Digital image processing]]

===== Checking Your Results =====
You can check results of the functions required in this lab using the Matlab function publish. Copy the test script {{courses:a4m33mpv:cviceni:1_uvod:test.m|test.m}} into MATLAB path (directory with implemented functions) a run. Compare your {{courses:a4m33mpv:cviceni:1_uvod:results.zip|results}} to the [[http://cmp.felk.cvut.cz/~perdom1/mpv/01_uvod/test.html|reference solution]].