Fractals/Computer graphic techniques/2D/algorithms
AlgorithmsEdit
postprocessing = modification of the image = graphic algorithms = image processing
Algorithms by graphic type
- raster algorithms
- vector algorithms
Five classes of image processing algorithms:
- image restoration
- image analysis
- image synthesis
- image enhancement
- image compression.
- Morphological operations on binary images[6][7]
- morphological closing = dilation followed by erosion
- morphological opening = erosion followed by dilation
Postprocessing
- Two types of edge detection
- Pseudo-3D projections
- Star field generator
- Random dot stereograms (aka Magic Eye)
- Motion blur for animations
- Interlacing
- Embossing
- Antialiasing
- Palette emulation to allow color cycling on true-color displays
- True color emulation that provides dithering on 256-color display
Algorithms:
- coloring after and independently of the fractal creation using gradient maps
- gamma correction of dense images[8]
- Histogram equalization in wikipedia
- (matrix) convolution ( = discrete convolution) is an element-wise multiplication of two matrices followed by a sum
- image convolution
- LIC
- image convolution
typesEdit
rangeEdit
The image processing operators ( algorithms) can be classified into the 4 categories
- pixel algorithms = pixel processing: the point operator acts on individual pixels : g = H(f)
- add two images: g(x)=f1(x)+f2(x)
- the histogram
- look-up table (LUT)
- windowing
- local algorithms = kernaling = local processing: "A local operator calculates the pixel value g(x) based not only on the value in the same position in the input image, i.e. f(x) , but it used several values in nearby points f(x+y). Local operators are at the heart of almost all image processing tasks."[9]
- spatial location filtering (convolution),
- spatial frequency filtering using high- and low-pass digital filters,
- the unsharp masking technique[10]
- geometric processing = geometric transformations: "Whereas a point operator changes the value of all pixels a geometrical operator doesn’t change the value but instead it ‘moves’ a pixel to a new position."
- Global Operators: "A global operator (potentially) needs al pixel values in the input image to calculate the value for just one pixel in the output image."
LibrariesEdit
Digital topologyEdit
lightEdit
- image-based-lighting by Mikael Hvidtfeldt Christensen
- Global Illumination Compendium September 29, 2003 by Philip Dutré
-
Light sources
gamma correctionEdit
dense imageEdit
Dense image[11][12][13][14][15]
- downscaling with gamma correction[16]
- path finding[17]
- aliasing [18]
- changing algorithm ( representation function) from discrete to continous, like from level set method of escape time to continous ( DEM )
"the denser the area, the more heavy the anti-aliasing have to be in order to make it look good." knighty[19]
"the details are smaller than pixel spacing, so all that remains is the bands of colour shift from period-doubling of features making it even denser" claude[20]
path findingEdit
path finding
- old fractal forums : pathfinding-in-the-mandelbrot-set/
- fractalforums.org : pathfinding-in-the-mandelbrot-set-revisited
- 5-ways-to-find-the-shortest-path-in-a-graph by Johannes Baum
- mathreference - graph
Algorithms[21]
- Depth-First Search (DFS) = the simplest algorithm
- Breadth-First Search (BFS)
- Bidirectional Search
- Dijkstra's algorithm (or Dijkstra's Shortest Path First algorithm, SPF algorithm)
- Bellman-Ford Algorithm
resizingEdit
Methods:
polygonsEdit
How to tell whether a point is to the right or left side of a line ?Edit
/*
How to tell whether a point is to the right or left side of a line ?
http://stackoverflow.com/questions/1560492/how-to-tell-whether-a-point-is-to-the-right-or-left-side-of-a-line
a, b = points
line = ab
pont to check = z
position = sign((Bx - Ax) * (Y - Ay) - (By - Ay) * (X - Ax))
It is 0 on the line, and +1 on one side, -1 on the other side.
*/
double CheckSide(double Zx, double Zy, double Ax, double Ay, double Bx, double By)
{
return ((Bx - Ax) * (Zy - Ay) - (By - Ay) * (Zx - Ax));
}
Testing if point is inside triangleEdit
/*
c console program
gcc t.c -Wall
./a.out
*/
# include <stdio.h>
// 3 points define triangle
double Zax = -0.250000000000000;
double Zay = 0.433012701892219;
// left when y
double Zlx = -0.112538773749444;
double Zly = 0.436719687479814 ;
double Zrx = -0.335875821657728;
double Zry = 0.316782798339332;
// points to test
// = inside triangle
double Zx = -0.209881783739630;
double Zy = +0.4;
// outside triangle
double Zxo = -0.193503885412548 ;
double Zyo = 0.521747636163664;
double Zxo2 = -0.338750000000000;
double Zyo2 = +0.440690927838329;
// ============ http://stackoverflow.com/questions/2049582/how-to-determine-a-point-in-a-2d-triangle
// In general, the simplest (and quite optimal) algorithm is checking on which side of the half-plane created by the edges the point is.
double sign (double x1, double y1, double x2, double y2, double x3, double y3)
{
return (x1 - x3) * (y2 - y3) - (x2 - x3) * (y1 - y3);
}
int PointInTriangle (double x, double y, double x1, double y1, double x2, double y2, double x3, double y3)
{
double b1, b2, b3;
b1 = sign(x, y, x1, y1, x2, y2) < 0.0;
b2 = sign(x, y, x2, y2, x3, y3) < 0.0;
b3 = sign(x, y, x3, y3, x1, y1) < 0.0;
return ((b1 == b2) && (b2 == b3));
}
int Describe_Position(double Zx, double Zy){
if (PointInTriangle( Zx, Zy, Zax, Zay, Zlx, Zly, Zrx, Zry))
printf(" Z is inside \n");
else printf(" Z is outside \n");
return 0;
}
// ======================================
int main(void){
Describe_Position(Zx, Zy);
Describe_Position(Zxo, Zyo);
Describe_Position(Zxo2, Zyo2);
return 0;
}
Orientation and area of the triangleEdit
Orientation and area of the triangle : how to do it ?
// gcc t.c -Wall
// ./a.out
# include <stdio.h>
// http://ncalculators.com/geometry/triangle-area-by-3-points.htm
double GiveTriangleArea(double xa, double ya, double xb, double yb, double xc, double yc)
{
return ((xb*ya-xa*yb)+(xc*yb-xb*yc)+(xa*yc-xc*ya))/2.0;
}
/*
wiki Curve_orientation
[http://mathoverflow.net/questions/44096/detecting-whether-directed-cycle-is-clockwise-or-counterclockwise]
The orientation of a triangle (clockwise/counterclockwise) is the sign of the determinant
matrix = { {1 , x1, y1}, {1 ,x2, y2} , {1, x3, y3}}
where
(x_1,y_1), (x_2,y_2), (x_3,y_3)$
are the Cartesian coordinates of the three vertices of the triangle.
:<math>\mathbf{O} = \begin{bmatrix}
1 & x_{A} & y_{A} \\
1 & x_{B} & y_{B} \\
1 & x_{C} & y_{C}\end{bmatrix}.</math>
A formula for its determinant may be obtained, e.g., using the method of [[cofactor expansion]]:
:<math>\begin{align}
\det(O) &= 1\begin{vmatrix}x_{B}&y_{B}\\x_{C}&y_{C}\end{vmatrix}
-x_{A}\begin{vmatrix}1&y_{B}\\1&y_{C}\end{vmatrix}
+y_{A}\begin{vmatrix}1&x_{B}\\1&x_{C}\end{vmatrix} \\
&= x_{B}y_{C}-y_{B}x_{C}-x_{A}y_{C}+x_{A}y_{B}+y_{A}x_{C}-y_{A}x_{B} \\
&= (x_{B}y_{C}+x_{A}y_{B}+y_{A}x_{C})-(y_{A}x_{B}+y_{B}x_{C}+x_{A}y_{C}).
\end{align}
</math>
If the determinant is negative, then the polygon is oriented clockwise. If the determinant is positive, the polygon is oriented counterclockwise. The determinant is non-zero if points A, B, and C are non-[[collinear]]. In the above example, with points ordered A, B, C, etc., the determinant is negative, and therefore the polygon is clockwise.
*/
double IsTriangleCounterclockwise(double xa, double ya, double xb, double yb, double xc, double yc)
{return ((xb*yc + xa*yb +ya*xc) - (ya*xb +yb*xc + xa*yc)); }
int DescribeTriangle(double xa, double ya, double xb, double yb, double xc, double yc)
{
double t = IsTriangleCounterclockwise( xa, ya, xb, yb, xc, yc);
double a = GiveTriangleArea( xa, ya, xb, yb, xc, yc);
if (t>0) printf("this triangle is oriented counterclockwise, determinent = %f ; area = %f\n", t,a);
if (t<0) printf("this triangle is oriented clockwise, determinent = %f; area = %f\n", t,a);
if (t==0) printf("this triangle is degenerate: colinear or identical points, determinent = %f; area = %f\n", t,a);
return 0;
}
int main()
{
// clockwise oriented triangles
DescribeTriangle(-94, 0, 92, 68, 400, 180); // https://www-sop.inria.fr/prisme/fiches/Arithmetique/index.html.en
DescribeTriangle(4.0, 1.0, 0.0, 9.0, 8.0, 3.0); // clockwise orientation https://people.sc.fsu.edu/~jburkardt/datasets/triangles/tex5.txt
// counterclockwise oriented triangles
DescribeTriangle(-50.00, 0.00, 50.00, 0.00, 0.00, 0.02); // a "cap" triangle. This example has an area of 1.
DescribeTriangle(0.0, 0.0, 3.0, 0.0, 0.0, 4.0); // a right triangle. This example has an area of (?? 3 ??) = 6
DescribeTriangle(4.0, 1.0, 8.0, 3.0, 0.0, 9.0); // https://people.sc.fsu.edu/~jburkardt/datasets/triangles/tex1.txt
DescribeTriangle(-0.5, 0.0, 0.5, 0.0, 0.0, 0.866025403784439); // an equilateral triangle. This triangle has an area of sqrt(3)/4.
// degenerate triangles
DescribeTriangle(1.0, 0.0, 2.0, 2.0, 3.0, 4.0); // This triangle is degenerate: 3 colinear points. https://people.sc.fsu.edu/~jburkardt/datasets/triangles/tex6.txt
DescribeTriangle(4.0, 1.0, 0.0, 9.0, 4.0, 1.0); //2 identical points
DescribeTriangle(2.0, 3.0, 2.0, 3.0, 2.0, 3.0); // 3 identical points
return 0;
}
Testing if point is inside polygonEdit
- stackoverflow[27]
- description by W Muła[28]
- point_in_polygon by Patrick Glauner
- PNPOLY - Point Inclusion in Polygon Test - W. Randolph Franklin (WRF)
/*
gcc p.c -Wall
./a.out
----------- git --------------------
cd existing_folder
git init
git remote add origin git@gitlab.com:adammajewski/PointInPolygonTest_c.git
git add .
git commit
git push -u origin master
*/
#include <stdio.h>
#define LENGTH 6
/*
Argument Meaning
nvert Number of vertices in the polygon. Whether to repeat the first vertex at the end is discussed below.
vertx, verty Arrays containing the x- and y-coordinates of the polygon's vertices.
testx, testy X- and y-coordinate of the test point.
https://www.ecse.rpi.edu/Homepages/wrf/Research/Short_Notes/pnpoly.html
PNPOLY - Point Inclusion in Polygon Test
W. Randolph Franklin (WRF)
I run a semi-infinite ray horizontally (increasing x, fixed y) out from the test point,
and count how many edges it crosses.
At each crossing, the ray switches between inside and outside.
This is called the Jordan curve theorem.
The case of the ray going thru a vertex is handled correctly via a careful selection of inequalities.
Don't mess with this code unless you're familiar with the idea of Simulation of Simplicity.
This pretends to shift the ray infinitesimally down so that it either clearly intersects, or clearly doesn't touch.
Since this is merely a conceptual, infinitesimal, shift, it never creates an intersection that didn't exist before,
and never destroys an intersection that clearly existed before.
The ray is tested against each edge thus:
Is the point in the half-plane to the left of the extended edge? and
Is the point's Y coordinate within the edge's Y-range?
Handling endpoints here is tricky.
I run a semi-infinite ray horizontally (increasing x, fixed y) out from the test point,
and count how many edges it crosses. At each crossing,
the ray switches between inside and outside. This is called the Jordan curve theorem.
The variable c is switching from 0 to 1 and 1 to 0 each time the horizontal ray crosses any edge.
So basically it's keeping track of whether the number of edges crossed are even or odd.
0 means even and 1 means odd.
*/
int pnpoly(int nvert, double *vertx, double *verty, double testx, double testy)
{
int i, j, c = 0;
for (i = 0, j = nvert-1; i < nvert; j = i++) {
if ( ((verty[i]>testy) != (verty[j]>testy)) &&
(testx < (vertx[j]-vertx[i]) * (testy-verty[i]) / (verty[j]-verty[i]) + vertx[i]) )
c = !c;
}
return c;
}
void CheckPoint(int nvert, double *vertx, double *verty, double testx, double testy){
int flag;
flag = pnpoly(nvert, vertx, verty, testx, testy);
switch(flag){
case 0 : printf("outside\n"); break;
case 1 : printf("inside\n"); break;
default : printf(" ??? \n");
}
}
int main (){
// values from http://stackoverflow.com/questions/217578/how-can-i-determine-whether-a-2d-point-is-within-a-polygon
// number from 0 to (LENGTH-1)
double zzx[LENGTH] = { 13.5, 6.0, 13.5, 42.5, 39.5, 42.5};
double zzy[LENGTH] = {100.0, 70.5, 41.5, 56.5, 69.5, 84.5};
CheckPoint(LENGTH, zzx, zzy, zzx[4]-0.001, zzy[4]);
CheckPoint(LENGTH, zzx, zzy, zzx[4]+0.001, zzy[4]);
return 0;
}
curveEdit
types:
- closed/open
- with/without multiple points
methods
- tracing /drawing : Generating Discrete Curves
- sketching
- sampling
- Pathfinding or pathing is the plotting, by a computer application, of the shortest route between two points
- clipping
- Approximation of digitized curves (with cubic Bézier splines )
- curve fitting[29]
- Mending broken lines [30]
- edge detection
Examples
- circle [36]
curve simplifyingEdit
reduce the number of points, but still keep the overall shape of the curve = PolyLine Reduction
thick line drawingEdit
thick line
- The Beauty of Bresenham's Algorithm by Alois Zingl
- Murphy's Modified Bresenham Line Drawing Algorithm [37]
- stackoverflow question: how-do-i-create-a-line-of-arbitrary-thickness-using-bresenham
curve drawingEdit
- Curve-Drawing Algorithms for Raster Displays by JERRY VAN AKEN and MARK NOVAK
- A Rasterizing Algorithm for Drawing Curves by Alois Zingl Wien, 2012
- Curve-Drawing Algorithms for Raster Displays by JERRY VAN AKEN and MARK NOVAK
- A* Pathfinding for Beginners By Patrick Lester (Updated July 18, 2005)
- Max K. Agoston Computer Graphics and Geometric Modeling Implementation and Algorithms
neighborhood variants of 2D algorithms:
- 8-way stepping (8WS) for 8-direction neighbors of a pixel p(x, y)
- 4-way stepping (4WS) 4-direction neighbors of a pixel p(x, y)
- 2D neighborhood
-
4
-
8
curve samplingEdit
- uniform = gives equidistant points
- adaptive. " an adaptive method for sampling the domain with respect to local curvature. Samples concentration is in proportion to this curvature, resulting in a more efficient approximation—in the limit, a flat curve is approximated by merely two endpoints." [38]
Field linesEdit
Field line[39]
- Vector fied : Field line computing
- external rays
- gravitational field lines by Chris Thomasson. Images and AS3 code
TracingEdit
Tracing curve[40]
Methods
- general, (analytic or systematic) = curve sketching[41]
- local method
trace an external ray for angle t in turns, which means ( description by Claude Heiland-Allen)[42]
- starting at a large circle (e.g. r = 65536, x = r cos(2 pi t), y = r sin(2 pi t))
- following the escape lines (like the edges of binary decomposition colouring with large escape radius) in towards the Mandelbrot set.
this algorithm is O(period^2), which means doubling the period takes 4x as long, so it's only really feasible for periods up to a few 1000.
Three Curve Tracing Models[43]
- Pixel-by-Pixel tracing
- The bipartite receptive field operator
- The zoom lens operator
Images
Problems:
- intersection and Breadth-first search[44]
Examples
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"[45] = having no endpoints) can be parametrized with angle (t).
Border, boundary, contour, edgeEdit
- Filling contour
- Finds contours in a binary image
contour modelsEdit
- snakes = active contour models[46]
border tracingEdit
- Tracing Boundaries in 2D Images by V. Kovalevsky[47]
- wikipedia : Boundary tracing
- Calculating contour curves for 2D scalar fields
- the marching squares algorithm for tracing contour curves on a scalar 2D field[48]
- Smooth Contours[49]
- contour polygons[50]
- boundary in C++ by Shawn Halayka - 2019 and new code
- opencv_contour in C++ by Shawn Halayka
- CONREC A Contouring Subroutine Written by Paul Bourke July 1987 ( using triangulation)
Filling contourEdit
-
Filling contour - simple procedure in c
Edge detectionEdit
- Boundary scanning method for Julia set - BSM/J
- Boundary scanning method for Mandelbrot set - BSM/M
- edge detection in Matlab[51]
- Marching squares algorithm to generate contour lines
-
Level curves - edge detection (2 filters)
-
Edge detection of boundaries of level sets of escape time
-
Sobel filterEdit
Short introductionEdit
Sobel filter G consist of 2 kernels (masks):
- Gh for horizontal changes.
- Gv for vertical changes.
Sobel kernelsEdit
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:
The Sobel kernel for computing vertical changes is:
Note that:
- sum of weights of kernels are zero
- One kernel is simply the other rotated by 90 degrees.[52]
- 3 weights in each kernal are zero.
Pixel kernelEdit
Pixel kernel A containing central pixel with its 3x3 neighbourhood:
Other notations for pixel kernel:
where:[53]
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
In array notation it is:[54]
In geographic notation usede in cellular aotomats[check spelling] it is central pixel of Moore neighbourhood.
So central (tested) pixel is:
Sobel filtersEdit
Compute Sobel filters (where here denotes the 2-dimensional convolution operation not matrix multiplication). It is a sum of products of pixel and its weights:
Because 3 weights in each kernal are zero so there are only 6 products.[55]
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):
It is a color of tested pixel.
One can also approximate result by sum of 2 magnitudes:
which is much faster to compute.[56]
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
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:
- ImagMagic discussion: using a Sobel operator - edge detection
- Sobel in c from GIMP code
- Cpp code by Aaron Brooks - Brent Bolton - Jason O'Kane
- rosettacode: Image convolution
- C and opencl by royger
- C++ by Glenn Fiedler
- Convolution and Deconvolution
- Sobel Edge Detector by R. Fisher, S. Perkins, A. Walker and E. Wolfart with examples in Java
- Qt and GDI+ versions of the Sobel edge detection algorithm by Ken Earle
- Qt and OpenCV
ProblemsEdit
Edge position:
In ImageMagick 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."[57]
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."[58]
- lines are not meeting in good points.
See also new operators from 6 version of ImageMagick: EdgeIn and EdgeOut from Morphology[59]
Edge thickeningEdit
convert $tmp0 -convolve "1,1,1,1,1,1,1,1,1" -threshold 0 $outfile
SDF Signed Distance FunctionEdit
test external tangency of 2 circlesEdit
/*
distance between 2 points
z1 = x1 + y1*I
z2 = x2 + y2*I
en.wikipedia.org/wiki/Distance#Geometry
*/
double GiveDistance(int x1, int y1, int x2, int y2){
return sqrt((x1-x2)*(x1-x2) + (y1-y2)*(y1-y2));
}
/*
mutually and externally tangent circles
mathworld.wolfram.com/TangentCircles.html
Two circles are mutually and externally tangent if distance between their centers is equal to the sum of their radii
*/
double TestTangency(int x1, int y1, int r1, int x2, int y2, int r2){
double distance;
distance = GiveDistance(x1, y1, x2, y2);
return ( distance - (r1+r2));
// return should be zero
}
ReferencesEdit
- ↑ Michael Abrash's Graphics Programming Black Book Special Edition
- ↑ geometrictools Documentation
- ↑ list-of-algorithms by Ali Abbas
- ↑ Mitch Richling: 3D Mandelbrot Sets
- ↑ fractalforums : antialiasing-fractals-how-best-to-do-it/
- ↑ Matlab : images/morphological-filtering
- ↑ Tim Warburton : morphology in matlab
- ↑ A Cheritat wiki : see image showing gamma-correct downscale of dense part of Mandelbropt set
- ↑ Image processing by Rein van den Boomgaard.
- ↑ Seeram E, Seeram D. Image Postprocessing in Digital Radiology-A Primer for Technologists. J Med Imaging Radiat Sci. 2008 Mar;39(1):23-41. doi: 10.1016/j.jmir.2008.01.004. Epub 2008 Mar 22. PMID: 31051771.
- ↑ wikipedi : dense_set
- ↑ mathoverflow question : is-there-an-almost-dense-set-of-quadratic-polynomials-which-is-not-in-the-inte/254533#254533
- ↑ fractalforums : dense-image
- ↑ fractalforums.org : m andelbrot-set-various-structures
- ↑ fractalforums.org : technical-challenge-discussion-the-lichtenberg-figure
- ↑ A Cheritat wiki : see image showing gamma-correct downscale of dense part of Mandelbropt set
- ↑ fractal forums : pathfinding-in-the-mandelbrot-set/
- ↑ serious_statistics_aliasing by Guest_Jim
- ↑ fractalforums.org : newton-raphson-zooming
- ↑ fractalforums : gerrit-images
- ↑ 5-ways-to-find-the-shortest-path-in-a-graph by Johannes Baum
- ↑ d3-polygon - Geometric operations for two-dimensional polygons from d3js.org
- ↑ github repo for d3-polygon
- ↑ accurate-point-in-triangle-test by Cedric Jules
- ↑ stackoverflow question : how-to-determine-a-point-in-a-2d-triangle
- ↑ js code
- ↑ stackoverflow questions : How can I determine whether a 2D Point is within a Polygon?
- ↑ Punkt wewnątrz wielokąta - W Muła
- ↑ Matlab examples : curvefitting
- ↑ Mending broken lines by Alan Gibson.
- ↑ The Beauty of Bresenham's Algorithm by Alois Zingl
- ↑ bresenhams-drawing-algorithms
- ↑ Bresenham’s Line Drawing Algorithm by Peter Occil
- ↑ Peter Occil
- ↑ Algorytm Bresenhama by Wojciech Muła
- ↑ wolfram : NumberTheoreticConstructionOfDigitalCircles
- ↑ Drawing Thick Lines
- ↑ IV.4 - Adaptive Sampling of Parametric Curves by Luiz Henrique deFigueiredo
- ↑ wikipedia: Field line
- ↑ Curve sketching in wikipedia
- ↑ slides from MALLA REDDY ENGINEERING COLLEGE
- ↑ fractalforums.org: help-with-ideas-for-fractal-art
- ↑ Predicting the shape of distance functions in curve tracing: Evidence for a zoom lens operator by PETER A. McCORMICK and PIERRE JOLICOEUR
- ↑ stackoverflow question: line-tracking-with-matlab
- ↑ mathwords: simple_closed_curve
- ↑ INSA : active-contour-models
- ↑ Tracing Boundaries in 2D Images by V. Kovalevsky
- ↑ Calculating contour curves for 2D scalar fields in Julia
- ↑ Smooth Contours from d3js.org
- ↑ d3-contour from d3js.org
- ↑ matrixlab - line-detection
- ↑ Sobel Edge Detector by R. Fisher, S. Perkins, A. Walker and E. Wolfart.
- ↑ NVIDIA Forums, CUDA GPU Computing discussion by kr1_karin
- ↑ Sobel Edge by RoboRealm
- ↑ nvidia forum: Sobel Filter Don't understand a little thing in the SDK example
- ↑ Sobel Edge Detector by R. Fisher, S. Perkins, A. Walker and E. Wolfart.
- ↑ ImageMagick doc
- ↑ Edge operator from ImageMagick docs
- ↑ ImageMagick doc: morphology / EdgeIn
- ↑ dilation at HIPR2 by Robert Fisher, Simon Perkins, Ashley Walker, Erik Wolfart
- ↑ ImageMagick doc: morphology, dilate
- ↑ Fred's ImageMagick Scripts