Game Rendering » Optimization

Grid Traversal

Math, Optimizations | Monday 20 July 2009 10:03 pm

Probably the best and most famous grid traversal algorithm uses a simple loop that for each iteration takes one more step in the grid. It works in both 2D and 3D.

Java code:

   public boolean isFreePath(float startX, float startZ, float endX, float endZ)
   {
      // calculate the direction of the ray (linear algebra)
      dirX = (endX - startX);
      dirY = (endZ - startZ);
      float length = (float) Math.sqrt(dirX * dirX + dirY * dirY);
      dirX /= length; // normalize the direction vector
      dirY /= length;
 
      tDeltaX = MAP_RESOLUTION / Math.abs(dirX); // how far we must move in the ray direction before we encounter a new voxel in x-direction
      tDeltaY = MAP_RESOLUTION / Math.abs(dirY); // same but y-direction
 
      // start voxel coordinates
      x = (int) (worldCoordToMapCoord(startX));  // use your transformer function here
      y = (int) (worldCoordToMapCoord(startZ));
 
      // end voxel coordinates
      endX1 = (int) (worldCoordToMapCoord(endX));
      endY1 = (int) (worldCoordToMapCoord(endZ));
 
      // decide which direction to start walking in
      stepX = (int) Math.signum(dirX);
      stepY = (int) Math.signum(dirY);
 
      float tMaxX, tMaxY;
      // calculate distance to first intersection in the voxel we start from
      if(dirX &lt; 0)
      {
         tMaxX = (mapToWorld(x)-startX) / dirX;
      }else
      {
         tMaxX = (mapToWorld(x+1)-startX) / dirX;
      }
 
      if(dirY &lt; 0)
      {
         tMaxY = (mapToWorld(y)-startY) / dirY;
      }else
      {
         tMaxY = (mapToWorld(y+1)-startY) / dirY;
      }
 
      // check if first is occupied
      if (isMapPositionOccupied(x, y)) // use your function here
         return false;
 
      boolean reachedX = false, reachedY = false;
 
      while (true)
      {
         if (tMaxX &lt; tMaxY)
         {
            tMaxX += tDeltaX;
            x += stepX;
         } else
         {
            tMaxY += tDeltaY;
            y += stepY;
         }
 
         if (stepX &gt; 0.0f)
         {
            if (x &gt;= endX1)
            {
               reachedX = true;
            }
         } else if (x &lt;= endX1)
         {
            reachedX = true;
         }
 
         if (stepY &gt; 0.0f)
         {
            if (y &gt;= endY1)
            {
               reachedY = true;
            }
         } else if (y &lt;= endY1)
         {
            reachedY = true;
         }
 
         if (isMapPositionOccupied(x, y))
         {
            return false;
         }
 
         if (reachedX &amp;&amp; reachedY)
         {
            break;
         }
      }
 
      return true;
   }

Here’s an applet showing it in action.
The white tiles are free, the gray tiles are blocking. The ray starts at the green dot and ends at the red dot. The gray spheres are the visited grid sections. Use leftclick to position the green dot and the right button to position the red dot.

Hi There

Link to paper:
http://www.flipcode.com/archives/A%20faster%20voxel%20traversal%20algorithm%20for%20ray%20tracing.pdf

Some discussion on GameDev:
http://www.gamedev.net/community/forums/topic.asp?topic_id=385303

Implementations:
http://www.clockworkcoders.com/oglsl/rt/gpurt3.htm
http://www.devmaster.net/articles/raytracing_series/part4.php

Full source of the applet:

import java.awt.*;
import java.awt.event.MouseEvent;
import java.awt.event.MouseListener;
import java.awt.event.MouseMotionListener;
import java.applet.*;
 
public class GTApplet extends Applet implements MouseListener,
		MouseMotionListener
{
	enum mapNotations
	{
		FREE, OCCUPIED, BOOKED, RESERVED
	};
 
	private static final int GRID_PAINT_SIZE = 50;
 
	private static final int MAP_RESOLUTION = GRID_PAINT_SIZE;
 
	private static final int MAP_SIZE = 4 * 2;
 
	private final mapNotations[][] map = new mapNotations[MAP_SIZE][MAP_SIZE];
 
	private final int[][] markedMap = new int[MAP_SIZE][MAP_SIZE];
 
	// Specify variables that will be needed everywhere, anytime here
	// The font variable
	Font bigFont, smallFont, smallestFont;
 
	Color bgColor;
	Color gridLineColor;
	Color gridOccupiedColor;
	Color startColor;
	Color endColor;
	Color helperLineColor;
	Color markerColor;
	Color textColor;
 
	int startX = 50, startY = 50, endX = 50, endY = 50;
 
	int markerDiameter = 6;
	int mapMarkerDiameter = 26;
 
	boolean isFreePath = false;
 
	public void init()
	{
		// init the map
		for (int x = 0; x &lt; MAP_SIZE; x++)
		{
			for (int y = 0; y &lt; MAP_SIZE; y++) 			{ 				if (Math.random() &gt; 0.2f)
				{
					map[x][y] = mapNotations.FREE;
				} else
				{
					map[x][y] = mapNotations.OCCUPIED;
				}
			}
		}
 
		bigFont = new Font("Arial", Font.BOLD, 16);
		bigFont = new Font("Arial", Font.BOLD, 12);
		smallestFont = new Font("Calibri", Font.PLAIN, 10);
 
		gridLineColor = Color.DARK_GRAY;
		gridOccupiedColor = Color.LIGHT_GRAY;
		startColor = Color.green;
		endColor = Color.red;
		helperLineColor = Color.black;
		markerColor = Color.gray;
		textColor = Color.DARK_GRAY;
 
		bgColor = new Color(230, 231, 232);
 
		setBackground(bgColor);
 
		addMouseListener(this);
		addMouseMotionListener(this);
 
	}
 
	public void mouseEntered(MouseEvent e)
	{
		// called when the pointer enters the applet's rectangular area
	}
 
	public void mouseExited(MouseEvent e)
	{
		// called when the pointer leaves the applet's rectangular area
	}
 
	public void mouseClicked(MouseEvent e)
	{
		// called after a press and release of a mouse button
		// with no motion in between
		// (If the user presses, drags, and then releases, there will be
		// no click event generated.)
	}
 
	public void mousePressed(MouseEvent e)
	{ // called after a button is pressed down
		if (e.getButton() == MouseEvent.BUTTON1)
		{
			startX = e.getX();
			startY = e.getY();
		} else
		{
			endX = e.getX();
			endY = e.getY();
		}
 
		resetMarkedMap();
		isFreePath = isFreePath(startX, startY, endX, endY);
 
		repaint();
		// "Consume" the event so it won't be processed in the
		// default manner by the source which generated it.
		e.consume();
	}
 
	public void mouseReleased(MouseEvent e)
	{ // called after a button is released
	//        repaint();
		e.consume();
	}
 
	public void mouseMoved(MouseEvent e)
	{ // called during motion when no buttons are down
	//        repaint();
		e.consume();
	}
 
	public void mouseDragged(MouseEvent e)
	{ // called during motion with buttons down
	//        repaint();
		e.consume();
	}
 
	public void stop()
	{
	}
 
	// just some simple and rude drawing
	public void paint(Graphics g)
	{
		g.setColor(gridLineColor);
		for (int x = 0; x &lt; MAP_SIZE + 1; x++)
			g.drawLine(x * GRID_PAINT_SIZE, 0, x * GRID_PAINT_SIZE, MAP_SIZE
					* GRID_PAINT_SIZE);
 
		for (int y = 0; y &lt; MAP_SIZE + 1; y++)
			g.drawLine(0, y * GRID_PAINT_SIZE, MAP_SIZE * GRID_PAINT_SIZE, y
					* GRID_PAINT_SIZE);
 
		g.setColor(gridOccupiedColor);
		for (int x = 0; x &lt; MAP_SIZE; x++)
		{
			for (int y = 0; y &lt; MAP_SIZE; y++)
			{
				if (map[x][y] == mapNotations.OCCUPIED)
				{
					g.fillRect(x * GRID_PAINT_SIZE + 1,
							y * GRID_PAINT_SIZE + 1, GRID_PAINT_SIZE - 1,
							GRID_PAINT_SIZE - 1);
				}
			}
		}
 
		g.setColor(textColor);
 
		g.setFont(smallestFont);
 
		for (int x = 0; x &lt; MAP_SIZE; x++)
		{
			for (int y = 0; y &lt; MAP_SIZE; y++)
			{
				g.drawString("(" + x + "," + y + ")", x * GRID_PAINT_SIZE + 2,
						y * GRID_PAINT_SIZE + 10);
			}
		}
 
		g.setFont(bigFont);
		g.drawString("Start: (" + startX + "," + startY + ")", 16, MAP_SIZE
				* GRID_PAINT_SIZE + 30 + 10);
		g.drawString("End: (" + endX + "," + endY + ")", 16, MAP_SIZE
				* GRID_PAINT_SIZE + 48 + 10);
		g.drawString("Is free path: " + isFreePath, 16, MAP_SIZE
				* GRID_PAINT_SIZE + 48 + 18 + 10);
		g.setFont(smallFont);
		g.drawString("Start (map coords): (" + sx + "," + sy + ")", 16,
				MAP_SIZE * GRID_PAINT_SIZE + 10 + 48 + 18 + 18 + 0 * 14);
		g.drawString("End (map coords): (" + endX1 + "," + endY1 + ")", 16,
				MAP_SIZE * GRID_PAINT_SIZE + 10 + 48 + 18 + 18 + 1 * 14);
		g.drawString("Step (map coords): (" + stepX + "," + stepY + ")", 16,
				MAP_SIZE * GRID_PAINT_SIZE + 10 + 48 + 18 + 18 + 2 * 14);
		g.drawString("Dir: (" + dirX + "," + dirY + ")", 16, MAP_SIZE
				* GRID_PAINT_SIZE + 10 + 48 + 18 + 18 + 3 * 14);
		g.drawString("tDelta: (" + tDeltaX + "," + tDeltaY + ")", 16, MAP_SIZE
				* GRID_PAINT_SIZE + 10 + 48 + 18 + 18 + 4 * 14);
		g.drawString("sTMax: (" + sTMaxX + "," + sTMaxY + ")", 16, MAP_SIZE
				* GRID_PAINT_SIZE + 10 + 48 + 18 + 18 + 5 * 14);
 
		g.setColor(markerColor);
 
		for (int x = 0; x &lt; MAP_SIZE; x++)
		{
			for (int y = 0; y &lt; MAP_SIZE; y++)
			{
				if (markedMap[x][y] == 1)
				{
					g.fillOval(x * GRID_PAINT_SIZE - mapMarkerDiameter / 2
							+ GRID_PAINT_SIZE / 2, y * GRID_PAINT_SIZE
							- mapMarkerDiameter / 2 + GRID_PAINT_SIZE / 2,
							mapMarkerDiameter, mapMarkerDiameter);
				}
			}
		}
 
		g.setColor(startColor);
		// draw start and end markers
		g.drawOval(startX - markerDiameter / 2, startY - markerDiameter / 2,
				markerDiameter, markerDiameter);
 
		g.setColor(endColor);
		// draw start and end markers
		g.drawOval(endX - markerDiameter / 2, endY - markerDiameter / 2,
				markerDiameter, markerDiameter);
 
		g.setColor(helperLineColor);
		g.drawLine(startX, startY, endX, endY);
	}
 
	public void resetMarkedMap()
	{
		for (int x = 0; x &lt; MAP_SIZE; x++)
		{
			for (int y = 0; y &lt; MAP_SIZE; y++)
			{
				markedMap[x][y] = 0;
			}
		}
	}
 
	int sx, sy, x, y, endX1, endY1, stepX, stepY;
	float tDeltaX, tDeltaY;
	float dirX, dirY;
	float sTMaxX, sTMaxY;
 
	public boolean isFreePath(float startX, float startZ, float endX, float endZ)
	{
		// calculate the direction of the ray (linear algebra)
		dirX = (endX - startX);
		dirY = (endZ - startZ);
		float length = (float) Math.sqrt(dirX * dirX + dirY * dirY);
		dirX /= length; // normalize the direction vector
		dirY /= length;
 
		tDeltaX = MAP_RESOLUTION / Math.abs(dirX); // how far we must move in the ray direction before we encounter a new voxel in x-direction
		tDeltaY = MAP_RESOLUTION / Math.abs(dirY); // same but y-direction
 
		// start voxel coordinates
		sx = (int) (worldCoordToMapCoord(startX));  // use your transformer function here
		sy = (int) (worldCoordToMapCoord(startZ));
 
		// end voxel coordinates
		endX1 = (int) (worldCoordToMapCoord(endX));
		endY1 = (int) (worldCoordToMapCoord(endZ));
 
		x = sx;
		y = sy;
 
		// decide which direction to start walking in
		stepX = (int) Math.signum(dirX);
		stepY = (int) Math.signum(dirY);
 
		float tMaxX, tMaxY;
		// calculate distance to first intersection in the voxel we start from
		if(dirX &lt; 0)
		{
			tMaxX = (mapToWorld(x)-startX) / dirX;
		}else
		{
			tMaxX = (mapToWorld(x+1)-startX) / dirX;
		}
 
		if(dirY &lt; 0)
		{
			tMaxY = (mapToWorld(y)-startY) / dirY;
		}else
		{
			tMaxY = (mapToWorld(y+1)-startY) / dirY;
		}
 
		sTMaxX = tMaxX;
		sTMaxY = tMaxY;
 
		// check if first is occupied
		if (isMapPositionOccupied(x, y)) // use your function here
			return false;
 
		boolean reachedX = false, reachedY = false;
 
		while (true)
		{
			if (tMaxX &lt; tMaxY) 			{ 				tMaxX += tDeltaX; 				x += stepX; 			} else 			{ 				tMaxY += tDeltaY; 				y += stepY; 			} 			if (stepX &gt; 0.0f)
			{
				if (x &gt;= endX1)
				{
					reachedX = true;
				}
			} else if (x &lt;= endX1) 			{ 				reachedX = true; 			} 			if (stepY &gt; 0.0f)
			{
				if (y &gt;= endY1)
				{
					reachedY = true;
				}
			} else if (y &lt;= endY1)
			{
				reachedY = true;
			}
 
			if (isMapPositionOccupied(x, y))
			{
				return false;
			}
 
			if (reachedX &amp;&amp; reachedY)
			{
				break;
			}
		}
 
		return true;
	}
 
	private float mapToWorld(int coord)
	{
		return coord*MAP_RESOLUTION;
	}
 
	private float worldCoordToMapCoord(float coord)
	{
		return coord / MAP_RESOLUTION;
	}
 
	private boolean isMapPositionOccupied(int x, int y)
	{
		if(x &lt; 0 || y &lt; 0)
			return true; 
 
		// save this for debug use
		markedMap[x][y] = 1;
 
		return map[x][y] == mapNotations.OCCUPIED;
	}
}

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Comments (4)

Percentage Closer Filtering for Shadow Mapping

Shadow Mapping | Saturday 15 November 2008 5:08 pm

This is a technique for making softer shadows when doing shadow mapping. It works by filtering the result of the depth comparison. So when comparing a depth, some depths around should also be compared and the result should be averaged. This will give a softer look on the shadow edges.

An example of the the soft shadows when using PCF Shadow Mapping

It can be implemented as simple as this in a the pixel shader:

float result;
result = shadow2DProj(shadowMap,texCoord+offset[0]);
result += shadow2DProj(shadowMap,texCoord+offset[1]);
result += shadow2DProj(shadowMap,texCoord+offset[2]);
result += shadow2DProj(shadowMap,texCoord+offset[3]);
result /= 4.0; // now result will hold the average shading

The samples are often either taken in a grid-based square around the original sample location or randomly scattered around it.

Optimization:

NVIDIA has built in hardware support for doing bilinear interpolation between four samples.

ATI has fetch4 which will fetch four texture samples at the same time.

The original paper for PCF:
http://graphics.pixar.com/library/ShadowMaps/paper.pdf

A technique that is similar (percentage-closer soft shadows)
http://developer.download.nvidia.com/shaderlibrary/docs/shadow_PCSS.pdf

A paper including lot’s of shadow mapping information, including different ways to do PCF (the image in this article is borrowed from this presentation):
http://developer.amd.com/media/gpu_assets/Isidoro-ShadowMapping.pdf

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Basic Culling Techniques

Culling | Sunday 2 November 2008 2:46 am

The best optimization when rendering is to not render anything unnecessary. And that is what culling is about, to find out what can be skipped when rendering because it cannot be visible anyway. Below are the basic culling techniques which most renderers’ implements. The image is from a course slide.

Back Face Culling

Faces that faces away from the camera can not be visible on the screen so they don’t need to be drawn. This is so often used that it’s implemented by the hardware. It roughly cuts the amount of faces drawn in half. Just remember to turn it on!

View frustum Culling

The faces that is outside the view frustum can not be visible on the screen (we don’t bother about reflections now) so they can be culled. This check is done by checking if the geometries bounding volume is outside the view frustum volume or not. So the check will not be done on every face, as that would cost to much. Sometimes, the view frustum culling can cost more than what is gain ( for example when doing instancing). One way to speed up view frustum culling is to use a suitable spatial structure for the scene (octree, BSP or so).

All kind of bounding volume test that you can imagine can be found on this page:
http://www.realtimerendering.com/intersections.html

Info about frustum culling:
http://www.flipcode.com/archives/Frustum_Culling.shtml

Portal Culling

A technique that divides the scene into cells with portals between. When rendering, the camera will be in one of the rooms and that room will be rendered normally. But for each portal that is visible in the room a view frustum is set up for the size of the portal and then the room behind it is rendered. This will work recursively. The result will be that a lot of geometry can be culled by view frustum culling when rendering the other rooms. A very useful technique for indoor scenes.

Detail Culling

When a geometry is so far away that it’s not visible then there is no need to draw it at all so it can safely be culled. A more advanced scheme of detail culling that decreases the amount of details with the distance is LOD (level of detail).

Occlusion Culling

The hardest culling technique to implement. Geometry that is occluded by other geometry does not need to be rendered. One solution is to use the Z-buffer and sort the geometry in a front to back order. But this does not always work and all pixels needs to be checked against the Z-buffer so it will be costly for big scenes. Better occlusion culling techniques culls the geometry before it’s even sent to the GPU.

A good link to a couple of occlusion culling techniques
http://www.gamasutra.com/features/19991109/moller_haines_01.htm

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Quadtree

Spatial Data Structure | Friday 17 October 2008 1:05 am

Nearly the same as a Octree but instead of dividing the space in cuboids in 3D it’s rectangles in 2D space. This is useful for rendering terrain and you want to cull a lot of the terrain fast.