# Fractals/Computer graphic techniques/2D

All tasks ( image processing [1]) can be done using :

• own programs ( GUI or console )
• own programs and graphic libraries
• graphic programs like GIMP
• fractal programs like :

One can use free graphic libraries :

# Creating graphicEdit

Here are 3 targets / tasks :

• memory array ( processing image )
• screen pixels ( displaying image )

Graphic files

## Memory arrayEdit

Image in memory is a matrix :

• A 24-bit color image is an (Width x Height x 3) matrix.
• Gray-level and black-and-white images are of size (Width x Height) .

The color depth of the image :

• 8-bit for gray
• 24 or 32-bit for color,
• 1-bit for black and white.

## Screen pixelsEdit

glxinfo | grep OpenGL

glxinfo | grep "direct rendering"


### DRIEdit

Direct Rendering Infrastructure (DRI2)[10]

# ColorEdit

Palette graphics, palette replacement mechanism

# CurveEdit

Field line [11]

## TracingEdit

Tracing curve [12]

## Curve rasterisationEdit

### RayEdit

Ray can be parametrised with radius ( r)

### Closed curveEdit

Simple closed curve ("a connected curve that does not cross itself and ends at the same point where it begins" [13] = having no endpoints) can be parametrized with angle ( t).

## Edge detectionEdit

### Sobel filterEdit

#### Short introductionEdit

Sobel filter G consist of 2 filters (masks):

• Gh for horizontal changes.
• Gv for vertical changes.
##### Sobel kernelsEdit

8-point neighborhood on a 2D grid

The Sobel kernel contains weights for each pixel from the 8-point neighbourhood of a tested pixel. These are 3x3 kernels.

There are 2 Sobel kernels, one for computing horizontal changes and other for computing vertical changes. Notice that a large horizontal change may indicate a vertical border, and a large vertical change may indicate a horizontal border. The x-coordinate is here defined as increasing in the "right"-direction, and the y-coordinate is defined as increasing in the "down"-direction.

The Sobel kernel for computing horizontal changes is:

${\displaystyle \mathbf {H} ={\begin{bmatrix}H_{1}&H_{2}&H_{3}\\H_{4}&H_{5}&H_{6}\\H_{7}&H_{8}&H_{9}\end{bmatrix}}={\begin{bmatrix}-1&0&+1\\-2&0&+2\\-1&0&+1\end{bmatrix}}}$

The Sobel kernel for computing vertical changes is:

${\displaystyle \mathbf {V} ={\begin{bmatrix}-1&-2&-1\\\ \ 0&\ \ 0&\ \ 0\\+1&+2&+1\end{bmatrix}}}$

Note that :

• sum of weights of kernels are zero

${\displaystyle \sum _{i=1}^{9}H_{i}=0}$

${\displaystyle \sum _{i=1}^{9}V_{i}=0}$

• One kernel is simply the other rotated by 90 degrees [15]
• 3 weights in each kernal are zero
##### Pixel kernelEdit

Pixel kernel A containing central pixel ${\displaystyle A_{5}}$  with its 3x3 neghbourhood  :

${\displaystyle \mathbf {A} ={\begin{bmatrix}A_{1}&A_{2}&A_{3}\\A_{4}&A_{5}&A_{6}\\A_{7}&A_{8}&A_{9}\end{bmatrix}}}$

Other notations for pixel kernel :

${\displaystyle \mathbf {A} ={\begin{bmatrix}A_{1}&A_{2}&A_{3}\\A_{4}&A_{5}&A_{6}\\A_{7}&A_{8}&A_{9}\end{bmatrix}}={\begin{bmatrix}ul&um&ur\\ml&mm&mr\\ll&lm&lr\end{bmatrix}}}$

where :[16]

unsigned char ul, // upper left
unsigned char um, // upper middle
unsigned char ur, // upper right
unsigned char ml, // middle left
unsigned char mm, // middle = central pixel
unsigned char mr, // middle right
unsigned char ll, // lower left
unsigned char lm, // lower middle
unsigned char lr, // lower right


Pixel 3x3 neighbourhood (with Y axis directed down)

In array notation it is :[17]

${\displaystyle \mathbf {A} ={\begin{bmatrix}A[x-1][y+1]&A[x][y+1]&A[x+1][y+1]\\A[x-1][y]&A[x][y]&A[x+1][y]\\A[x-1][y-1]&A[x][y-1]&A[x+1][y-1]\end{bmatrix}}}$

In geographic notation usede in cellular aotomats it is central pixel of Moore neighbourhood.

So central ( tested ) pixel is :

${\displaystyle A_{5}=mm=A[x][y]\,}$

##### Sobel filtersEdit

Compute sobel filters ( where ${\displaystyle *}$  here denotes the 2-dimensional convolution operation not matrix multiplication ). It is a sum of products of pixel and its weghts :

${\displaystyle \mathbf {G} _{h}=\mathbf {H} *A=A_{1}H_{1}+A_{2}H_{2}+\cdots +A_{9}H_{9}=\sum _{r=1}^{9}A_{r}H_{r},}$
${\displaystyle \mathbf {G} _{v}=\mathbf {V} *A=A_{1}V_{1}+A_{2}V_{2}+\cdots +A_{9}V_{9}=\sum _{r=1}^{9}A_{r}V_{r},}$

Because 3 weights in each kernal are zero so there are only 6 products.[18]

short Gh = ur + 2*mr + lr - ul - 2*ml - ll;
short Gv = ul + 2*um + ur - ll - 2*lm - lr;

##### ResultEdit

Result is computed (magnitude of gradient):

${\displaystyle \mathbf {G} (A_{5})={\sqrt {{\mathbf {G} _{h}}^{2}+{\mathbf {G} _{v}}^{2}}}}$

It is a color of tested pixel .

One can also approximate result by sum of 2 magnitudes :

${\displaystyle \mathbf {G} (A_{5})=\left|\mathbf {G} _{h}\right|+\left|\mathbf {G} _{v}\right|}$

which is much faster to compute.[19]

#### AlgorithmEdit

• choose pixel and its 3x3 neighberhood A
• compute sobel filter for horizontal Gh and vertical lines Gv
• compute sobel filter G
• compute color of pixel

#### ProgrammingEdit

Sobel filters ( 2 filters 3x3 ) : image and full c code

Skipped pixel - some points from its neighbourhood are out of the image

Lets take array of 8-bit colors ( image) called data. To find borders in this image simply do :

for(iY=1;iY<iYmax-1;++iY){
for(iX=1;iX<iXmax-1;++iX){
Gv= - data[iY-1][iX-1] - 2*data[iY-1][iX] - data[iY-1][iX+1] + data[iY+1][iX-1] + 2*data[iY+1][iX] + data[iY+1][iX+1];
Gh= - data[iY+1][iX-1] + data[iY-1][iX+1] - 2*data[iY][iX-1] + 2*data[iY][iX+1] - data[iY-1][iX-1] + data[iY+1][iX+1];
G = sqrt(Gh*Gh + Gv*Gv);
if (G==0) {edge[iY][iX]=255;} /* background */
else {edge[iY][iX]=0;}  /* boundary */
}
}


Note that here points on borders of array ( iY= 0 , iY = iYmax , iX=0, iX=iXmax) are skipped

Result is saved to another array called edge ( with the same size).

One can save edge array to file showing only borders, or merge 2 arrays  :

for(iY=1;iY<iYmax-1;++iY){
for(iX=1;iX<iXmax-1;++iX){ if (edge[iY][iX]==0) data[iY][iX]=0;}}


to have new image with marked borders.

Above example is for 8-bit or indexed color. For higher bit colors "the formula is applied to all three color channels separately" ( from RoboRealm doc).

Other implementations :

#### ProblemsEdit

Bad edge position seen in the middle of image. Lines are not meeting in good points, like z = 0

Edge position :

In ImageMagic as "you can see, the edge is added only to areas with a color gradient that is more than 50% white! I don't know if this is a bug or intentional, but it means that the edge in the above is located almost completely in the white parts of the original mask image. This fact can be extremely important when making use of the results of the "-edge" operator." [20]

The result is :

• doubling edges ; "if you are edge detecting an image containing an black outline, the "-edge" operator will 'twin' the black lines, producing a weird result."[21]
• lines are not meeting in good points

See also new operators from 6 version of Image Magic : EdgeIn and EdgeOut from Morphology [22]

## Edge thickeningEdit

dilation [23][24][25]

convert $tmp0 -convolve "1,1,1,1,1,1,1,1,1" -threshold 0$outfile


# Filling contourEdit

Filling contour - simple procedure in c

# Quality of imageEdit

## AntialiasingEdit

Aliased chessboard - image and c src code

### SupersamplingEdit

example of supersampled image

Cpp code of supersampling

Other names :

• antigrain geometry
• Supersampling ( downsampling) [34][35]
• downsizing
• downscaling[36]
• subpixel accuracy

Examples :

 // subpixels finished -> make arithmetic mean
char pixel[3];
for (int c = 0; c < 3; c++)
pixel[c] = (int)(255.0 * sum[c] / (subpix * subpix)  + 0.5);
fwrite(pixel, 1, 3, image_file);
//pixel finished

• command line version of Aptus ( python and c code ) by Ned Batchelder [37] ( see aptuscmd.py ) is using a high-quality downsampling filter thru PIL function resize [38]
• Java code by Josef Jelinek:[39] supersampling with grid algorithm, computes 4 new points (corners), resulting color is an avarage of each color component :
 //Created by Josef Jelinek
// http://java.rubikscube.info/
Color c0 = color(dx, dy); // color of central point
// computation of 4 new points for antialiasing
if (antialias) { // computes 4 new points (corners)
Color c1 = color(dx - 0.25 * r, dy - 0.25 * r);
Color c2 = color(dx + 0.25 * r, dy - 0.25 * r);
Color c3 = color(dx + 0.25 * r, dy + 0.25 * r);
Color c4 = color(dx - 0.25 * r, dy + 0.25 * r);
// resulting color; each component of color is an avarage of 5 values ( central point and 4 corners )
int red = (c0.getRed() + c1.getRed() + c2.getRed() + c3.getRed() + c4.getRed()) / 5;
int green = (c0.getGreen() + c1.getGreen() + c2.getGreen() + c3.getGreen() + c4.getGreen()) / 5;
int blue = (c0.getBlue() + c1.getBlue() + c2.getBlue() + c3.getBlue() + c4.getBlue()) / 5;
color = new Color(red, green, blue);
}

• one can make big image ( like 10 000 x 10 000 ) and convert/resize it ( downsize ). For example using ImageMagic :
convert big.ppm -resize 2000x2000 m.png