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How to minimize a non-linear function with constraints in c#?

I would like to minimize the following function
with constraints
in C#. I tried to do it with Math.Net's Newton Method, but I can't figure out how to do it. How can I minimize the function programmatically in C# for given $F_1, F_2$?
Update:
After the comment of #MinosIllyrien I tried the following, but I don't get the syntax:
_f1 = 0.3; // Global fields.
_f2 = 0.7;
var minimizer = new NewtonMinimizer(1E-4, 100, false);
var objectiveFunction = ObjectiveFunction.ScalarDerivative(FunctionToMinimize, GradientOfFunctionToMinimize);
var firstGuess = CreateVector.DenseOfArray(new[] {0.5});
var minimalWeight1 = minimizer.FindMinimum(objectiveFunction, firstGuess).MinimizingPoint;
private double GradientOfFunctionToMinimize(double w1){
return _f1 - (w1 * _f2) / Math.Sqrt(1 - Math.Pow(w1, 2));
}
private double FunctionToMinimize(double w1){
return w1 * _f1 + Math.Sqrt(1 - Math.Pow(w1, 2)) * _f2;
}
This does not work, because FindMinimum method requires IObjectiveFunction as function and not IScalarObjectiveFunction...
Update 2:
I tried a solution from Google:
var solver = Solver.CreateSolver("GLOP");
Variable w1 = solver.MakeNumVar(0.0, double.PositiveInfinity, "w1");
Variable w2 = solver.MakeNumVar(0.0, double.PositiveInfinity, "w2");
solver.Add(Math.Sqrt(w1*w1 + w2*w2) == 1);
This throws the error that *-operator cannot be used for "Variable" and "Variable". Someone any ideas?

w₁² + w₂² = 1 is basically the unit circle.
The unit circle can also be described by the following parametric equation:
(cos t, sin t)
In other words, for every pair (w₁, w₂),
there is an angle t for which w₁ = cos t and w₂ = sin t.
With that substitution, the function becomes:
y = F₁ cos t + F₂ sin t
w₁ ≥ 0, w₂ ≥ 0 restricts t to a single quadrant. This leaves you with a very simple constraint, that consists of a single variable:
0 ≤ t ≤ ½π
By the way, the function can be simplified to:
y = R cos(t - α)
where R = √(F₁² + F₂²) and α = atan2(F₂, F₁)
This is a simple sine wave. Without the constraint on t, its range would be [-R, R], making the minimum -R. But the constraint limits the domain and thereby the range:
If F₁ < 0 and F₂ < 0, then the minimum is at w₁ = -F₁ / R, w₂ = -F₂ / R, with y = -R
For 0 < F₁ ≤ F₂, a minimum is at w₁ = 1, w₂ = 0, with y = F₁
For 0 < F₂ ≤ F₁, a minimum is at w₁ = 0, w₂ = 1, with y = F₂
Notes:
if F₁ = F₂ > 0, then you have two minima.
if F₁ = F₂ = 0, then y is just flat zero everywhere.
In code:
_f1 = 0.3;
_f2 = 0.7;
if (_f1 == 0.0 && _f2 == 0.0) {
Console.WriteLine("Constant y = 0 across the entire domain");
}
else if (_f1 < 0.0 && _f2 < 0.0) {
var R = Math.sqrt(_f1 * _f1 + _f2 * _f2);
Console.WriteLine($"Minimum y = {-R} at w1 = {-_f1 / R}, w2 = {-_f2 / R}");
}
else {
if (_f1 <= _f2) {
Console.WriteLine($"Minimum y = {_f1} at w1 = 1, w2 = 0");
}
if (_f1 >= _f2) {
Console.WriteLine($"Minimum y = {_f2} at w1 = 0, w2 = 1");
}
}

Rotate 2D points using System.Numerics.Vectors

I'm looking to optimize a program that is basing a lot of its calculations on the rotation of a lot of 2D Points. I've search around to see if it's possible to do these calculations using SIMD in C#.
I found a c++ answer here that seems to do what I want, but I can't seem to translate this into C# using the System.Numerics.Vectors package.
Optimising 2D rotation
Can anyone point me in the right direction for how this can be done?
The below code shows the regular method without SIMD. Where Point is a struct with doubles X and Y.
public static Point[] RotatePoints(Point[] points, double cosAngle, double sinAngle)
{
var pointsLength = points.Length;
var results = new Point[pointsLength];
for (var i = 0; i < pointsLength; i++)
{
results[i].X = (points[i].X * cosAngle) - (points[i].Y * sinAngle);
results[i].Y = (points[i].X * sinAngle) + (points[i].Y * cosAngle);
}
return results;
}
Edit:
I've managed to get an implementation working using two Vector< float> but from benchmarking this, this seems to be a lot slower than the previous implementation.
private static void RotatePoints(float[] x, float[] y, float cosAngle, float sinAngle)
{
var chunkSize = Vector<float>.Count;
var resultX = new float[x.Length];
var resultY = new float[x.Length];
Vector<float> vectorChunk1;
Vector<float> vectorChunk2;
for (var i = 0; i < x.Length; i += chunkSize)
{
vectorChunk1 = new Vector<float>(x, i);
vectorChunk2 = new Vector<float>(y, i);
Vector.Subtract(Vector.Multiply(vectorChunk1, cosAngle), Vector.Multiply(vectorChunk2, sinAngle)).CopyTo(resultX, i);
Vector.Add(Vector.Multiply(vectorChunk1, sinAngle), Vector.Multiply(vectorChunk2, cosAngle)).CopyTo(resultY, i);
}
}

The code added in the edit is a good start, however the codegen for Vector.Multiply(Vector<float>, float) is extremely bad so this function should be avoided. It's an easy change to avoid it though, just broadcast outside the loop and multiply by a vector. I also added a more proper loop bound and "scalar epilog" in case the vector size does not neatly divide the size of the input arrays.
private static void RotatePoints(float[] x, float[] y, float cosAngle, float sinAngle)
{
var chunkSize = Vector<float>.Count;
var resultX = new float[x.Length];
var resultY = new float[x.Length];
Vector<float> vectorChunk1;
Vector<float> vectorChunk2;
Vector<float> vcosAngle = new Vector<float>(cosAngle);
Vector<float> vsinAngle = new Vector<float>(sinAngle);
int i;
for (i = 0; i + chunkSize - 1 < x.Length; i += chunkSize)
{
vectorChunk1 = new Vector<float>(x, i);
vectorChunk2 = new Vector<float>(y, i);
Vector.Subtract(Vector.Multiply(vectorChunk1, vcosAngle), Vector.Multiply(vectorChunk2, vsinAngle)).CopyTo(resultX, i);
Vector.Add(Vector.Multiply(vectorChunk1, vsinAngle), Vector.Multiply(vectorChunk2, vcosAngle)).CopyTo(resultY, i);
}
for (; i < x.Length; i++)
{
resultX[i] = x[i] * cosAngle - y[i] * sinAngle;
resultY[i] = x[i] * sinAngle + y[i] * cosAngle;
}
}

How would i work out magnitude quickly for 3 values?

How can I use a Fast Magnitude calculation for 3 values (instead of using square root)? (+/- 3% is good enough)
public void RGBToComparison(Color32[] color)
{
DateTime start = DateTime.Now;
foreach (Color32 i in color)
{
var r = PivotRgb(i.r / 255.0);
var g = PivotRgb(i.g / 255.0);
var b = PivotRgb(i.b / 255.0);
var X = r * 0.4124 + g * 0.3576 + b * 0.1805;
var Y = r * 0.2126 + g * 0.7152 + b * 0.0722;
var Z = r * 0.0193 + g * 0.1192 + b * 0.9505;
var LB = PivotXyz(X / 95.047);
var AB = PivotXyz(Y / 100);
var BB = PivotXyz(Z / 108.883);
var L = Math.Max(0, 116 * AB - 16);
var A = 500 * (LB - AB);
var B = 200 * (AB - BB);
totalDifference += Math.Sqrt((L-LT)*(L-LT) + (A-AT)*(A-AT) + (B-BT)*(B-BT));
}
totalDifference = totalDifference / color.Length;
text.text = "Amount of Pixels: " + color.Length + " Time(MilliSeconds):" + DateTime.Now.Subtract(start).TotalMilliseconds + " Score (0 to 100)" + (totalDifference).ToString();
RandomOrNot();
}
private static double PivotRgb(double n)
{
return (n > 0.04045 ? Math.Pow((n + 0.055) / 1.055, 2.4) : n / 12.92) * 100.0;
}
private static double PivotXyz(double n)
{
return n > 0.008856 ? CubicRoot(n) : (903.3 * n + 16) / 116;
}
private static double CubicRoot(double n)
{
return Math.Pow(n, 1.0 / 3.0);
}
This is the important part: totalDifference += Math.Sqrt((L-LT)*(L-LT) + (A-AT)*(A-AT) + (B-BT)*(B-BT));
I know there are FastMagnitude calculations online, but all the ones online are for two values, not three. For example, could i use the difference between the values to get a precise answer? (By implementing the difference value into the equation, and if the difference percentage-wise is big, falling back onto square root?)
Adding up the values and iterating the square root every 4 pixels is a last resort that I could do. But firstly, I want to find out if it is possible to have a good FastMagnitude calculation for 3 values.
I know I can multi-thread and parllelize it, but I want to optimize my code before I do that.

If you just want to compare the values, why not leave the square root out and work with the length squared?
Or use the taylor series of the square root of 1+x and cut off early :)

Restart program from a certain line with an if statement?

could anyone help me restart my program from line 46 if the user enters 1 (just after the comment where it states that the next code is going to ask the user for 2 inputs) and if the user enters -1 end it. I cannot think how to do it. I'm new to C# any help you could give would be great!
class Program
{
static void Main(string[] args)
{
//Displays data in correct Format
List<float> inputList = new List<float>();
TextReader tr = new StreamReader("c:/users/tom/documents/visual studio 2010/Projects/DistanceCalculator3/DistanceCalculator3/TextFile1.txt");
String input = Convert.ToString(tr.ReadToEnd());
String[] items = input.Split(',');
Console.WriteLine("Point Latitude Longtitude Elevation");
for (int i = 0; i < items.Length; i++)
{
if (i % 3 == 0)
{
Console.Write((i / 3) + "\t\t");
}
Console.Write(items[i]);
Console.Write("\t\t");
if (((i - 2) % 3) == 0)
{
Console.WriteLine();
}
}
Console.WriteLine();
Console.WriteLine();
// Ask for two inputs from the user which is then converted into 6 floats and transfered in class Coordinates
Console.WriteLine("Please enter the two points that you wish to know the distance between:");
string point = Console.ReadLine();
string[] pointInput = point.Split(' ');
int pointNumber = Convert.ToInt16(pointInput[0]);
int pointNumber2 = Convert.ToInt16(pointInput[1]);
Coordinates distance = new Coordinates();
distance.latitude = (Convert.ToDouble(items[pointNumber * 3]));
distance.longtitude = (Convert.ToDouble(items[(pointNumber * 3) + 1]));
distance.elevation = (Convert.ToDouble(items[(pointNumber * 3) + 2]));
distance.latitude2 = (Convert.ToDouble(items[pointNumber2 * 3]));
distance.longtitude2 = (Convert.ToDouble(items[(pointNumber2 * 3) + 1]));
distance.elevation2 = (Convert.ToDouble(items[(pointNumber2 * 3) + 2]));
//Calculate the distance between two points
const double PIx = 3.141592653589793;
const double RADIO = 6371;
double dlat = ((distance.latitude2) * (PIx / 180)) - ((distance.latitude) * (PIx / 180));
double dlon = ((distance.longtitude2) * (PIx / 180)) - ((distance.longtitude) * (PIx / 180));
double a = (Math.Sin(dlat / 2) * Math.Sin(dlat / 2)) + Math.Cos((distance.latitude) * (PIx / 180)) * Math.Cos((distance.latitude2) * (PIx / 180)) * (Math.Sin(dlon / 2) * Math.Sin(dlon / 2));
double angle = 2 * Math.Atan2(Math.Sqrt(a), Math.Sqrt(1 - a));
double ultimateDistance = (angle * RADIO);
Console.WriteLine("The distance between your two points is...");
Console.WriteLine(ultimateDistance);
//Repeat the program if the user enters 1, end the program if the user enters -1
Console.WriteLine("If you wish to calculate another distance type 1 and return, if you wish to end the program, type -1.");
Console.ReadLine();
if (Convert.ToInt16(Console.ReadLine()) == 1);
{
//here is where I need it to repeat
}

bool exit = false;
do
{
Console.WriteLine("Please enter the two points that you wish to know the distance between:");
...
Console.WriteLine("If you wish to calculate another distance type 1 and return, if you wish to end the program, type -1.");
string input;
do
{
input = Console.ReadLine().Trim();
}
while (input != "1" && input != "-1");
if (input == -1) exit = true;
}
while (!exit);
But you would do much better to think about pushing logic into methods and functions such that you program is built up of much smaller building blocks.
You should be aiming towards something like this:
bool exit = false;
do
{
Point[] points = ReadCoordinates();
Whatever result = CalculateWhatever();
DisplayResults(results);
exit = ShouldExit();
}
while (!exit);
This makes the outer loop of your program self documenting and the methods self explanatory.

if (Convert.ToInt16(Console.ReadLine()) == 1);
{
Main(args)
}

That's quite an odd thing to do.
Have you looked into the while loop?
You could make a method of the above structure and loop it, while userinput equals 1
if it equals -1, call the following statement
Application.Exit()

Can use Goto to make the program loop on user input.
public static void Main(string[] args)
{
RestartApplication:
//// Displays data in correct Format
TextReader textReader = new StreamReader("c:/users/tom/documents/visual studio 2010/Projects/DistanceCalculator3/DistanceCalculator3/TextFile1.txt");
var input = Convert.ToString(textReader.ReadToEnd());
var items = input.Split(',');
Console.WriteLine("Point Latitude Longtitude Elevation");
for (var i = 0; i < items.Length; i++)
{
if (i % 3 == 0)
{
Console.Write((i / 3) + "\t\t");
}
Console.Write(items[i]);
Console.Write("\t\t");
if (((i - 2) % 3) == 0)
{
Console.WriteLine();
}
}
Console.WriteLine();
Console.WriteLine();
//// Ask for two inputs from the user which is then converted into 6 floats and transferred in class Coordinates
Console.WriteLine("Please enter the two points that you wish to know the distance between:");
var point = Console.ReadLine();
string[] pointInput;
if (point != null)
{
pointInput = point.Split(' ');
}
else
{
goto RestartApplication;
}
var pointNumber = Convert.ToInt16(pointInput[0]);
var pointNumber2 = Convert.ToInt16(pointInput[1]);
var distance = new Coordinates
{
Latitude = Convert.ToDouble(items[pointNumber * 3]),
Longtitude = Convert.ToDouble(items[(pointNumber * 3) + 1]),
Elevation = Convert.ToDouble(items[(pointNumber * 3) + 2]),
Latitude2 = Convert.ToDouble(items[pointNumber2 * 3]),
Longtitude2 = Convert.ToDouble(items[(pointNumber2 * 3) + 1]),
Elevation2 = Convert.ToDouble(items[(pointNumber2 * 3) + 2])
};
//// Calculate the distance between two points
const double PIx = 3.141592653589793;
const double Radio = 6371;
var dlat = (distance.Latitude2 * (PIx / 180)) - (distance.Latitude * (PIx / 180));
var dlon = (distance.Longtitude2 * (PIx / 180)) - (distance.Longtitude * (PIx / 180));
var a = (Math.Sin(dlat / 2) * Math.Sin(dlat / 2)) + Math.Cos(distance.Latitude * (PIx / 180)) * Math.Cos(distance.Latitude2 * (PIx / 180)) * (Math.Sin(dlon / 2) * Math.Sin(dlon / 2));
var angle = 2 * Math.Atan2(Math.Sqrt(a), Math.Sqrt(1 - a));
var ultimateDistance = angle * Radio;
Console.WriteLine("The distance between your two points is...");
Console.WriteLine(ultimateDistance);
//// Repeat the program if the user enters 1, end the program if the user enters -1
Console.WriteLine("If you wish to calculate another distance type 1 and return, if you wish to end the program, type -1.");
var userInput = Console.ReadLine();
if (Convert.ToInt16(userInput) == 1)
{
//// Here is where the application will repeat.
goto RestartApplication;
}
}
also did a bit of code formatting. Hope it helps.

You could do that with goto, be aware however that this is considered bad practice.
static void Main(string[] args)
{
...
MyLabel:
...
if (Convert.ToInt16(Console.ReadLine()) == 1);
{
//here is where I need it to repeat
goto MyLabel;
}

Connected-component labeling algorithm optimization

I need some help with optimisation of my CCL algorithm implementation. I use it to detect black areas on the image. On a 2000x2000 it takes 11 seconds, which is pretty much. I need to reduce the running time to the lowest value possible to achieve. Also, I would be glad to know if there is any other algorithm out there which allows you to do the same thing, but faster than this one. So here is my code:
//The method returns a dictionary, where the key is the label
//and the list contains all the pixels with that label
public Dictionary<short, LinkedList<Point>> ProcessCCL()
{
Color backgroundColor = this.image.Palette.Entries[1];
//Matrix to store pixels' labels
short[,] labels = new short[this.image.Width, this.image.Height];
//I particulary don't like how I store the label equality table
//But I don't know how else can I store it
//I use LinkedList to add and remove items faster
Dictionary<short, LinkedList<short>> equalityTable = new Dictionary<short, LinkedList<short>>();
//Current label
short currentKey = 1;
for (int x = 1; x < this.bitmap.Width; x++)
{
for (int y = 1; y < this.bitmap.Height; y++)
{
if (!GetPixelColor(x, y).Equals(backgroundColor))
{
//Minumum label of the neighbours' labels
short label = Math.Min(labels[x - 1, y], labels[x, y - 1]);
//If there are no neighbours
if (label == 0)
{
//Create a new unique label
labels[x, y] = currentKey;
equalityTable.Add(currentKey, new LinkedList<short>());
equalityTable[currentKey].AddFirst(currentKey);
currentKey++;
}
else
{
labels[x, y] = label;
short west = labels[x - 1, y], north = labels[x, y - 1];
//A little trick:
//Because of those "ifs" the lowest label value
//will always be the first in the list
//but I'm afraid that because of them
//the running time also increases
if (!equalityTable[label].Contains(west))
if (west < equalityTable[label].First.Value)
equalityTable[label].AddFirst(west);
if (!equalityTable[label].Contains(north))
if (north < equalityTable[label].First.Value)
equalityTable[label].AddFirst(north);
}
}
}
}
//This dictionary will be returned as the result
//I'm not proud of using dictionary here too, I guess there
//is a better way to store the result
Dictionary<short, LinkedList<Point>> result = new Dictionary<short, LinkedList<Point>>();
//I define the variable outside the loops in order
//to reuse the memory address
short cellValue;
for (int x = 0; x < this.bitmap.Width; x++)
{
for (int y = 0; y < this.bitmap.Height; y++)
{
cellValue = labels[x, y];
//If the pixel is not a background
if (cellValue != 0)
{
//Take the minimum value from the label equality table
short value = equalityTable[cellValue].First.Value;
//I'd like to get rid of these lines
if (!result.ContainsKey(value))
result.Add(value, new LinkedList<Point>());
result[value].AddLast(new Point(x, y));
}
}
}
return result;
}
Thanks in advance!

You could split your picture in multiple sub-pictures and process them in parallel and then merge the results.
1 pass: 4 tasks, each processing a 1000x1000 sub-picture
2 pass: 2 tasks, each processing 2 of the sub-pictures from pass 1
3 pass: 1 task, processing the result of pass 2
For C# I recommend the Task Parallel Library (TPL), which allows to easily define tasks depending and waiting for each other. Following code project articel gives you a basic introduction into the TPL: The Basics of Task Parallelism via C#.

I would process one scan line at a time, keeping track of the beginning and end of each run of black pixels.
Then I would, on each scan line, compare it to the runs on the previous line. If there is a run on the current line that does not overlap a run on the previous line, it represents a new blob. If there is a run on the previous line that overlaps a run on the current line, it gets the same blob label as the previous. etc. etc. You get the idea.
I would try not to use dictionaries and such.
In my experience, randomly halting the program shows that those things may make programming incrementally easier, but they can exact a serious performance cost due to new-ing.

The problem is about GetPixelColor(x, y), it take very long time to access image data.
Set/GetPixel function are terribly slow in C#, so if you need to use them a lot, you should use Bitmap.lockBits instead.
private void ProcessUsingLockbits(Bitmap ProcessedBitmap)
{
BitmapData bitmapData = ProcessedBitmap.LockBits(new Rectangle(0, 0, ProcessedBitmap.Width, ProcessedBitmap.Height), ImageLockMode.ReadWrite, ProcessedBitmap.PixelFormat);
int BytesPerPixel = System.Drawing.Bitmap.GetPixelFormatSize(ProcessedBitmap.PixelFormat) / 8;
int ByteCount = bitmapData.Stride * ProcessedBitmap.Height;
byte[] Pixels = new byte[ByteCount];
IntPtr PtrFirstPixel = bitmapData.Scan0;
Marshal.Copy(PtrFirstPixel, Pixels, 0, Pixels.Length);
int HeightInPixels = bitmapData.Height;
int WidthInBytes = bitmapData.Width * BytesPerPixel;
for (int y = 0; y < HeightInPixels; y++)
{
int CurrentLine = y * bitmapData.Stride;
for (int x = 0; x < WidthInBytes; x = x + BytesPerPixel)
{
int OldBlue = Pixels[CurrentLine + x];
int OldGreen = Pixels[CurrentLine + x + 1];
int OldRed = Pixels[CurrentLine + x + 2];
// Transform blue and clip to 255:
Pixels[CurrentLine + x] = (byte)((OldBlue + BlueMagnitudeToAdd > 255) ? 255 : OldBlue + BlueMagnitudeToAdd);
// Transform green and clip to 255:
Pixels[CurrentLine + x + 1] = (byte)((OldGreen + GreenMagnitudeToAdd > 255) ? 255 : OldGreen + GreenMagnitudeToAdd);
// Transform red and clip to 255:
Pixels[CurrentLine + x + 2] = (byte)((OldRed + RedMagnitudeToAdd > 255) ? 255 : OldRed + RedMagnitudeToAdd);
}
}
// Copy modified bytes back:
Marshal.Copy(Pixels, 0, PtrFirstPixel, Pixels.Length);
ProcessedBitmap.UnlockBits(bitmapData);
}
Here is the basic code to access pixel data.
And I made a function to transform this into a 2D matrix, it's easier to manipulate (but little slower)
private void bitmap_to_matrix()
{
unsafe
{
bitmapData = ProcessedBitmap.LockBits(new Rectangle(0, 0, ProcessedBitmap.Width, ProcessedBitmap.Height), ImageLockMode.ReadWrite, ProcessedBitmap.PixelFormat);
int BytesPerPixel = System.Drawing.Bitmap.GetPixelFormatSize(ProcessedBitmap.PixelFormat) / 8;
int HeightInPixels = ProcessedBitmap.Height;
int WidthInPixels = ProcessedBitmap.Width;
int WidthInBytes = ProcessedBitmap.Width * BytesPerPixel;
byte* PtrFirstPixel = (byte*)bitmapData.Scan0;
Parallel.For(0, HeightInPixels, y =>
{
byte* CurrentLine = PtrFirstPixel + (y * bitmapData.Stride);
for (int x = 0; x < WidthInBytes; x = x + BytesPerPixel)
{
// Conversion in grey level
double rst = CurrentLine[x] * 0.0721 + CurrentLine[x + 1] * 0.7154 + CurrentLine[x + 2] * 0.2125;
// Fill the grey matix
TG[x / 3, y] = (int)rst;
}
});
}
}
And the website where the code comes
"High performance SystemDrawingBitmap"
Thanks to the author for his really good job !
Hope this will help !