====== Image Processing and Normalizing Transforms ======

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

  * The Gaussian function is often used in image processing as a low pass filter for noise reduction or so-called windowing function for weighting contribution of points in the neighbourhood. Write an implementation of the function ''G=gauss(x,sigma)'' that computes discrete samples of Gaussian\\ <latex>G = \frac{1}{\sqrt{2\pi}\sigma}\cdot e^\frac{-x^2}{2\sigma^2}</latex>\\ in points given by vector <latex>x</latex> with variance <latex>\sigma^2</latex> and zero mean.
  * Implement function ''D=dgauss(x,sigma)'' that returns a discrete samples of first derivative of Gaussian\\ <latex>\frac{d}{dx}G = \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}} </latex>\\ v in points given by vector <latex>x</latex> with variance <latex>\sigma^2</latex> and zero mean.
  * Get acquainted with the function ''conv2''.
  * The effect of filtering with Gaussian can be studied on an impulse response:<code>
sigma = 6.0;
x = [-ceil(3.0*sigma):ceil(3.0*sigma)];
G = gauss(x, sigma);
D = dgauss(x, sigma);
imp = zeros(51); imp(25,25) = 255;
out = conv2(G,G,imp);
...

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*(sigma*3.0)+1'' and variance <latex>\sigma^2</latex> 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 <latex>\sigma^2</latex>. 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 <latex>\sigma^2</latex>.
 

==== 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 submit? =====
Submit 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]]