TOPIC 4 PART III FILL AREA PRIMITIVES POLYGON FILL AREAS CGMB214: Introduction to Computer Graphics

What we are going to learn To be able to understand the concept of fill attributes To be able to understand the concept of polygon

Fill Area Primitives

Filled-Area Primitives Two ways of area filling on raster system  By determining the overlaps intervals for scan lines that cross the area.  By starting from interior position outward until specified boundary condition is encountered.

Filling 2D Shapes Types of filling Pattern FillSolid FillTexture Fill

Filling 2D Shapes Some requirements  A digital representation of the shape  The shape must be closed  It must have a well defines inside and outside  A test for determining if a point is inside or outside of the shape  A rule or procedure for determining the colors of points inside the shape

Representing Filled Shapes Digital images  Inside determined by a color or range of colors Original Image Pink pixels have been filled with yellow

Representing Filled Shapes A digital outline and a seed point indicating the interior Digital outline and seed points Filled outlines

Representing Filled Shapes An implicit function representing a shape’s interior The inside of a circle of radius R The inside of a unit square

Representing Filled Shapes An equation or list of edges representing a shape’s boundary and a rule for determining its interior  E.g.  Edge list Line from (0,0) to (1,0) Line from (1,0) to (1,1) Line from (1,1) to (0,1) Line from (0,1) to (0,0)  Rule for interior points All points to the right of all of the (ordered) edges

Representing Filled Shapes Edge list  Line from (0,0) to (1,0)  Line from (1,0) to (1,1)  Line from (1,1) to (0,1)  Line from (0,1) to (0,0) Rule for interior points  All points to the right of all of the (ordered) edges

Representing Filled Shapes Edge list  Line from (0,0) to (1,0)  Line from (1,0) to (1,1)  Line from (1,1) to (0,1)  Line from (0,1) to (0,0) Rule for interior points  All points to the right of all of the (ordered) edges Filled shape

Fill Options How to set pixel colors for points inside the shape? Solid Fill Pattern Fill Texture Fill

Seed Fill Approach  Select a seed point inside a region  Move outwards from the seed point, setting neighboring pixels until the region is filled Seed pointMove outwards to neighbors Stop when the region is filled

Selecting the Seed Point Difficult to place the seed point automatically  Seed fill works best in an interactive application where the user sets the seed point What is the inside of this shape? It depends on the user’s intent

Seed Fill Basic algorithm select seed pixel initialize a fill list to contain seed pixel while (fill list not empty) { pixel  get next pixel from fill list setPixel(pixel) for (each of the pixel’s neighbors) { if (neighbor is inside region AND neighbor not set) add neighbor to fill list }

Which neighbors should be tested? There are two types of 2D regions  4-connected region (test 4 neighbors)  Two pixels are 4-connected if they are vertical or horizontal neighbors  8-connected region (test 8 neighbors)  Two pixels are 8-connected if they are vertical, horizontal, or diagonal neighbors

Which neighbors should be tested? Using 4-connected and 8-connected neighbors gives different results Magnified area Original boundary Fill using 4-connected neighbors Fill using 8-connected neighbors

When is a Neighbor Inside the Region? There are two types of tests, resulting in two filling approaches  Boundary fill  Flood fill

Fill condition  The region is defined by a set of boundary pixels  A neighbor of an inside pixel is also inside if it is not a boundary pixel Boundary Fill Boundary pixel Seed pixel Original image and seed point Image after 4-connected boundary fill

Fill condition  The region is defined by a patch of like-colored pixels  A neighbor of an inside pixel is also inside if its color is within a range of the seed pixel’s original color  The range of inside colors can be specified in the application Flood Fill Seed pixel Original image and seed point Image after 4-connected flood fill

Improving Performance Problems with the basic algorithm  We don’t know how big the fill list should be  Worst case, all the image pixels  Slow  Pixels may be checked many times to see if they have already been set (especially for 8- connected regions)

Improving Performance Use coherence (logical connection) to improve performance and reduce memory requirements  Neighbor coherence  Neighboring pixels tend to be in the same region  Span coherence  Neighboring pixels along a given scan line tend to be in the same region  Scan-line coherence  The filling patterns of adjacent scan lines tends to be similar

Span-based seed fill algorithm Improving Performance Seed point

Improving Performance Span-based seed fill algorithm  Start from the seed point  Fill the entire horizontal span of pixels inside the region Seed point

Improving Performance Span-based seed fill algorithm  Determine spans of pixels in the rows above and below the current row that are connected to the current span  Add the left-most pixel of these spans to the fill list

Improving Performance Span-based seed fill algorithm  Repeat until the fill list is empty

Improving Performance Span-based seed fill algorithm  Repeat until the fill list is empty

Improving Performance Span-based seed fill algorithm  Repeat until the fill list is empty

Improving Performance Span-based seed fill algorithm  Repeat until the fill list is empty

Filling Axis-Aligned Rectangles An axis-aligned rectangle is defined by its corner points (X min, Y min ) and (X max, Y max ) (X min, Y min ) (X max, Y max )

Filling Axis-Aligned Rectangles Filling can be done in a nested loop for (j = Ymin, j < Ymax, j++) { for (i = Xmin, i < Xmax, i++) { setPixel(i, j, fillColor) } (X min, Y min ) (X max, Y max )

Polygon Fill Areas

What is polygon? A plane figure specified by a set of three or more coordinate positions (called vertices), that are connected in sequence by straight line segments (called edges or sides of the polygon). Polygon must have  All vertices within a single plane  No edge crossing

Polygon Classification An interior angle of a polygon is an angle inside the polygon boundary formed by two adjacent edges If all interior angles of a polygon are less than or equal to 180 degree, the polygon is said to be convex If there is at least one interior angle greater than 180 degree, the polygon is said to be concave The order of vertices for a polygon can be either clockwise or anti-clockwise

Polygon  Concave  Convex Polygon Classification

Setup vectors for all edges Perform cross product to adjacent vectors to test for concavity Perform dot product if we want to determine the angle between two edges All vector products will be the same value (positive or negative) for convex polygon If there are some cross products yield a positive and some yield a negative value, we have a concave polygon Identifying Concave Polygon

V1 V2 V3 V4 V5 V6 E1 E2 E3 E4 E5 E6 (E1 X E2) > 0 (E2 X E3) > 0 (E3 X E4) < 0 (E4 X E5) > 0 (E5 X E6) > 0 (E6 X E1) > 0 E J X E K = E JX E KY - E JY E KX Ej = (V nx – V mx, V ny – V my ) VmVm VnVn EjEj Identifying Concave Polygon

Example:  Given 6 vertices:  V1 = (1,1)  V2 = (5,1)  V3 = (7,3)  V4 = (4,5)  V5 = (4,10)  V6 = (1,10)  Prove that these vertices is for concave polygon. What can you say about the cross product values if we change the order of these vertices (v6 becomes v1, v5 becomes v2, etc…).

Exact angle between two adjacent edges Use dot product operation  a.b = |a||b|cos θ  |a| means the magnitude of vector a  |a| = Angle between Edges θ

Angle Between Edges Example:  Given 2 vectors a = (2,3) and b = (6,3). Determine the angle between these two vectors  Determine the angle between E3 and E4

Filling General Polygons Representing general polygons  Defined by a list of connected line segments  The line segments must form a closed shape (i.e. the boundary must connected)  General polygons  Can be self intersecting  Can have interior holes

Filling General Polygons Specifying the interior  Must be able to determine which points are inside the polygon  Need a fill rule

Filling General Polygons Inside-Outside Tests  Filling means coloring a region  How to identify interior or exterior region?  Once determined only then interior to be filled accordingly

Filling General Polygons Specifying the interior  There are two commonly used fill rules  Even-odd parity rule  Non-zero winding rule Filled using even-odd parity rule Filled using none-zero winding rule

Inside-Outside Tests: Even-Odd Rule Even-Odd Rule  Also known as odd-parity and odd-even rule.  How its work?  Pick a point of P in the region of interest  Draw a line from P to a distant point which lower than the smallest x  Move from P along the line to the distant point  Count the number of region edges the line crosses  If the number of crossed is odd then P is inside the interior region  If the number of crossed is even then P is inside the exterior region

Inside-Outside Tests: Even-Odd Rule To determine if a point P is inside or outside  Draw a line from P to infinity  Count the number of times the line crosses an edge  If the number of crossing is odd, the point is inside  If the number of crossing is even, the point is outside

Inside-Outside Tests: Non-Zero Winding Rule Non-Zero Winding Number Rule  Each boundary is given a direction number and then sum the numbers.  Rules  The line chosen must not pass through any vertices.  If first y values less than second y value Then give direction number –1  If first y values greater than second y value Then give direction number 1.  Move from P along the line to the distant point.  Add or minus based on the direction number when crossing the edges.  Interior regions have non-zero winding numbers.  Exterior regions have a winding number of 0.

Inside-Outside Tests: Non-Zero Winding Rule The outline of the shape must be directed  The line segments must have a consistent direction so that they formed a continuous, closed path

Inside-Outside Tests: Non-Zero Winding Rule To determine if a points is inside or outside  Determine the winding number (i.e. the number of times the edge winds around the point in either a clockwise or counterclockwise direction)  Points are outside if the winding number is zero  Point are inside if the winding number is not zero

Inside-Outside Tests: Non-Zero Winding Rule To determine the winding number at a point P  Initialize the winding number to zero and draw a line (e.g. horizontal) from P to infinity  If the line crosses an edge directed bottom to up  Add 1 to the winding number  If the line crosses an edge directed top to bottom  Subtract 1 from the winding number

Inside-Outside Tests: Non-Zero Winding Rule The non-zero winding number rule and the even-odd parity rule can give different results for general polygons  When polygons self intersect  When polygons have interior holes Even-odd parity Non-zero winding

Inside-Outside Tests Standard polygons  Standard polygons (e.g. triangles, rectangles, octagons) do not self intersect and do not contain holes  The non-zero winding number rule and the even-odd parity rule give the same results for standard polygons

Shared Vertices Edges share vertices  If the line drawn for the fill rule intersects a vertex, the edge crossing would be counted twice  This yields incorrect and inconsistent even-odd parity checks and winding numbers Line pierces the outline - Should count as one crossing Line grazes the outline - Should count as no crossings

Dealing with Shared Vertices 1. Check the vertex type (piercing or grazing)  If the vertex is between two upwards or two downwards edges, the line pierces the edge  Process a single edge crossing  If the vertex is between an upwards and a downwards edge, the line grazes the vertex  Don’t process any edge crossings Vertex between two upwards edges - Process a single crossing Vertex between upwards and downwards edges - Process no crossings

Dealing with Shared Vertices 2. Ensure that the line does not intersect a vertex  Use a different line if the first line intersects a vertex  Could be costly if you have to try several lines  If using horizontal scan line for the inside-outside test  Preprocess edge vertices to make sure that none of them fall on a scan line Add a small floating point value to each vertex y-position

Filling Polygons via Boundary Fill Polygons are defined by their edges

Filling Polygons via Boundary Fill Polygons are defined by their edges  Use a line drawing algorithm to draw edges of the polygon with a boundary color

Filling Polygons via Boundary Fill Polygons are defined by their edges  Fill the inside of the polygon using a boundary fill

Filling Polygons via Boundary Fill Problems 1.Pixels are drawn on both sides of the line  The polygon contains pixels outside of the outline  Polygons with shared edges will have overlapping pixels 2.Efficiency  Drawing outlines and then filling can be less efficient that combining the edge drawing and filling in one step

Raster-Based Filling Fill polygons in raster-scan order  Fill spans of pixels inside the polygon along each horizontal scan line  More efficient addressing by accessing spans of pixels  Only test pixels at the span endpoints

Raster-Based Filling For each scan line  Determine points where the scan line intersects the polygon

Raster-Based Filling For each scan line  Set pixels between intersection points (using a fill rule)  Even-odd parity rule: set pixels between pairs of intersections  Non-zero winding rule: set pixels according to the winding number

Raster-Based Filling Basic algorithm (with even-odd parity rule) for (each scan line, j) { find the x-intersections between the scan line and each edge sort the x-intersections by increasing x-value for (each pair of intersection points, x1 and x2) { while (x1 < i < x2) setPixel(i, j, fillColor) }

Conventions for Setting Edge Pixels Adjacent polygons share edges  When rendered, some pixels along the edges are shared  Need to know what color to use for shared edge pixels

Conventions for Setting Edge Pixels If we draw all edge pixels for each polygon  Shared pixels will be rendered more than once  If setPixel() overwrites the current pixel, the last polygon drawn will look larger Green triangle written last

Conventions for Setting Edge Pixels If we draw all edge pixels for each polygon  Shared pixels will be rendered more than once  If setPixel() overwrites the current pixel, the last polygon drawn will look larger Blue triangle written last

Conventions for Setting Edge Pixels If we draw all edge pixels for each polygon  Shared pixels will be rendered more than once  If setPixel() blends the background color with the foreground color, shared edge pixels will have a blended color Edge color different than either triangle

If we draw none of the edge pixels  Only interior pixels are drawn  Gaps appear between polygons and the background shows through Conventions for Setting Edge Pixels Gaps between adjacent triangles

Conclusion You have learn on:  The basic tools  to draw points, lines, curves  To fill color areas  Three methods on developing straight line  DDA  Bresenham Algo  Midpoint Algo  How to draw circle or ellipse efficiency by taking the symmetrical value into account  Using Inside-Outside test to test the interior and the exterior of an area