Computer Graphics One of the central components of three-dimensional graphics has been a basic system that renders objects represented by a set of polygons One approach to rendering three-dimensional objects is to build a basic renderer then add on enhancements The basic renderer may be one which incorporates a local reflection model, such as the Phong model, into a Phong incremental shader

Computer Graphics Advantages gained by this approach include: Modeling objects using polygons is straighforward Piecewise linearities are rendered invisible by the shading technique Geometric information is only stored at the polygonal vertices - information required for the reflection model that evaluates a shade at each pixel is interpolated from this information This allows fast, hardware-based shading

Computer Graphics More accurate representations made up of a set of bicubic patches can be converted to a polygon representation and fed to such a renderer One drawback of using polygons to model objects is that a large number of polygons are required to achieve much detail for complex objects

Computer Graphics The main steps in rendering a polygonal object are: 1. Polygons representing an object are extracted from the database and transformed into the world coordinate system using linear transformations such as translation and scaling 2. A scene constructed in this way is transformed into a coordinate system based on a view point or view direction. 3. The polygons are then subjected to a visibility test. This is called ‘backface elimination’ or ‘culling’ and removes those polygons that face away from the viewer.

Computer Graphics The main steps in rendering a polygonal object are: 4. Unculled polygons are clipped against a three-dimensional view volume. 5. Clipped polygons are then projected onto a view plane or image plane 6. Projected polygons are then shaded by an incremental shading algorithm. First, the polygon is rasterized, or those pixels that the edges of the polygon contain are determined. Second, a depth for each pixel is evaluated and a hidden surface calculation is performed. Third, the polygon is shaded.

Polygonal representation of three-dimensional objects Objects that possess curved surfaces have these surfaces approximated by polygonal facets The error introduced by this representation may be visually diminished by using interpolative shading algorithms

Polygonal Representation of Curved Objects

Polygonal representation of three-dimensional objects For complex objects, a number of polygons in excess of 100,000 is not uncommon Another problem occurs when objects are scaled up An object adequately represented at one size may degrade when the object is enlarged This has been called ‘geometric aliasing’

Polygonal representation of three-dimensional objects Polygonal representations can be generated manually from a designer’s abstraction, or automatically from real objects by using devices such as laser rangers in conjunction with the appropriate software Complete information necessary to shade a polygon is usually stored in a hierarchical data structure - objects; surfaces; vertices and normals

Coordinate systems and rendering One view of the geometric part of the rendering process is that it consists of the application of a series of coordinate transformations that takes an object database through a series of coordinate systems For ease of modeling and application of local transformations, it makes sense to store the vertices of an object with respect to some point conveniently located in or near the the object This is called the local coordinate system

Coordinate systems and rendering Once an object has been modeled, the next stage is to place it in the scene that we wish to render The global coordinate system of the scene is known as the world coordinate system All the objects have to be placed into this common space in order to have their relative spatial relationships defined The act of placing an object in a scene defines the transformation required to take the object from local space to global space If this object is being animated, then the animation provides a time-varying transformation that takes the object into world space on a frame-by-frame basis

Coordinate systems and rendering The scene is lit in world space Light sources are specified The eye or camera coordinate system is a a space used to establish viewing parameters and view volume A virtual camera can be positioned anywhere in the world space and can point in any direction The scene is projected on a view plane

Backface elimination and culling This operation removes entire polygons that face away from the viewer When dealing with a single convex object, culling completely solves the hidden surface problem If an object contains a concavity, or if we have multiple objects in a scene, a general hidden surface removal algorithm is needed as well as culling We can determine whether a polygon is visible from a view point by a simple geometric test The geometric normal to the polygon is calculated and the angle between this and the line-of-sight vector is determined. (The line-of-sight vector is the vector from the polygon to the view point) If this angle is greater than 90 degrees, then the polygon is invisible.

Screen space The fundamental transformation that takes us into screen space is the perspective transformation which takes a point in the scene and projects it onto a view plane positioned at disance D away from the view point and oriented normal to the viewing direction

Perspective Transformation

Screen Space Screen space is defined to act within a closed volume called the viewing frustum that delineates the volume of space which is to be rendered Objects that lie outside the viewing frustum are not rendered

View Frustum

Homogeneous Coordinates A point in three dimensional space requires 3 coordinates - x, y and z - to describe its location In homoegeneous coordinates, an extra coordinate (which can be thought of as a scaling term) is added By using homogeneous coordinates, we are able to represent transformations as matrix multiplications Then in order to compute the transformation of some object, we need only perform matrix multiplications for the vertices of the polygons making up the object’s representation

Homogeneous Coordinates The transformations we are most interested in are scaling, translation and rotation These can be defined as follows

Scaling Transformation

Scaling Transformation

Translation Transformation

Translation Transformation

Rotation About the X-Axis

Rotation About the X-Axis

Homogeneous Coordinates Rotations about the y and z axis are similar A series of operations can be combined by performing a series of multiplications If the scaling factor is greater than one, the resulting coordinates must be divided by that number at the end of the operation

Perspective Transformation The near plane is the view plane at distance D. The far plane is at distance F If the height of the view plane is 2h, then the perspective transformation can be given by the following matrix: 1 0 0 0 0 1 0 1 0 0 hF/(D(F-D)) h/D 0 0 -hF/(F-D) 0

Perspective Transformation This transformation can simply be concatenated with all of the other transformations in the rendering pipeline Clipping is performed against the viewing frustum Objects lying completely outside the frustum are discarded Objects inside the frustum are transformed into screen space and then rendered Objects intersecting the frustum are clipped and then transformed into screen space

Pixel-Level Processes Once the objects in a scene have been translated into screen space, the processing is oriented towards pixels rather than polygons Rasterization, hidden-surface removal and shading are all pixel-level processes

Rasterization Rasterization is the process of finding out which pixels a polygon projects onto in screen space Calculate, for each scan line within a polygon, by linear interpolation, the x values of the edge pixels, xstart and xend This results in a span of pixels between two edges that have to be rendered Care must be taken in these calculations, otherwise we will have aliasing problems (e.g. a straight line may appear to be a series of steps) Most of these problems are caused by rounding errors in the interpolation of polygon edges

Hidden Surface Removal We now know which pixels contain which objects, however since some pixels may contain two or more objects we must calculate which of these objects is visible and which are hidden Hidden surface removal is generally accomplished using the Z-buffer algorithm

Hidden Surface Removal In this algorithm, we set aside a two-dimensional array of memory (the Z-buffer) of the same size as the screen (#rows x #columns) This is in addition to the buffer we will use to store the values of pixels which will be displayed (color values) The Z-buffer will hold values which are depths (or z-values) The buffer is initialized so that each element has the value of the far clipping plane (the largest possible z-value after clipping has been performed) The other buffer is initialized so that each element contains a value which is the background color

Hidden Surface Removal Now for each polygon we have a set of pixel values which that polygon covers For each one of these pixels, we compare its interpolated depth (z-value) with the value of the corresponding element already stored in the Z-buffer If this value is less than the previously stored value, the pixel is nearer the viewer than previously encountered pixels Replace the old value of the Z- buffer with the new, interpolated value and replace the old value of the other buffer with the value (color) of the pixel Repeat for all polygons in the image

Interpolative or Incremental Shading

Interpolative or Incremental Shading A major reason for the growth of popularity of polygon-based renderers is the existence of two interpolative or incremental shading algorithms usually known as Gouraud shading and Phong shading Gouraud shading is faster than the other method, but cannot produce accurate highlights. It is usually used in applications where speed is important

Interpolative or Incremental Shading Phong shading gives higher quality images, but is more expensive Gouraud shading calculates intensities at polygon vertices only, using a local reflection model, and interpolates these for pixels within the polygon Phong shading interpolates vertex normals and applies a local reflection model at each pixel

Interpolative or Incremental Shading The motivation of both schemes is efficiency and the ‘convincing’ rendering of polygonal objects This means that as well as giving an impression of solidity or three-dimensionality to the model, the shared edges in adjacent polygons that approximate a curved surface, are made invisible

Gouraud Shading Generally, the intensity of light at a polygon vertex is described as a function of the orientation of vertex normals with respect to both the light source and the eye This function is called a reflection model Gouraud shading applies a local reflection model at each vertex of a polygon to calculate a set of vertex intensities These intensities are linearly interpolated across the polygon interior on a scan line basis

Phong Shading In Phong shading, the vertex normals (calculated exactly as in Gouraud shading) are linearly interpolated The local reflection model is then applied at each pixel using the interpolated normal This solves the highlight problem and Mach banding, but is more computationally expensive