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

TOPIC 4 PART I I IFILL AREA PRIMITIVESPOLYGON 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 Fill

Solid Fill

Texture 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 ImagePink pixels have been filled with

yellow


A digital outline and a seed point indicating the interior

Digital outline and seed points

Filled outlines


An implicit function representing a shape’s interior

The inside of a circle of radius R

The inside of a unit square


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







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 point

Move 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 algorithmselect seed pixelinitialize a fill list to contain seed pixelwhile (fill list not empty) {

pixel get next pixel from fill listsetPixel(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 approachesBoundary fillFlood 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)


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


Seed point


Span-based seed fill algorithm Start from the seed point Fill the entire horizontal span of pixels inside the

region

Seed point


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


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(Xmin, Ymin) and (Xmax, Ymax)

(Xmin, Ymin)

(Xmax, Ymax)

Filling Axis-Aligned Rectangles

Filling can be done in a nested loopfor (j = Ymin, j < Ymax, j++) { for (i = Xmin, i < Xmax, i++) { setPixel(i, j, fillColor) }}

(Xmin, Ymin)

(Xmax, Ymax)

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 edgesPerform cross product to adjacent vectors to test

for concavityPerform dot product if we want to determine the

angle between two edgesAll vector products will be the same value (positive

or negative) for convex polygonIf there are some cross products yield a positive

and some yield a negative value, we have a concave polygon

Identifying Concave Polygon

V1V2

V3

V4

V5V6

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

EJ X EK = EJXEKY - EJYEKX

Ej = (Vnx – Vmx , Vny – Vmy)

Vm

VnEj



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

22

yx aa

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


Specifying the interior Must be able to determine which points are

inside the polygon Need a fill rule


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

filled accordingly


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.


The outline of the shape must be directed The line segments must have a consistent

direction so that they formed a continuous, closed path


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


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


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


Polygons are defined by their edges Use a line drawing algorithm to draw edges of

the polygon with a boundary color


Polygons are defined by their edges Fill the inside of the polygon using a boundary fill


Problems1. 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


For each scan line Determine points where the scan line intersects

the polygon


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


Basic algorithm (with even-odd parity rule)

for (each scan line, j) {find the x-intersections between the scan line and each edgesort 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


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



If setPixel() overwrites the current pixel, the last polygon drawn will look larger

Blue triangle written last



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


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

Documents

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