Z-buffering - AbsoluteAstronomy.com

Computer graphics

Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....

, z-buffering is the management of image depth coordinates in three-dimensional (3-D) graphics, usually done in hardware

Hardware

Hardware is a general term for equipment such as keys, locks, hinges, latches, handles, wire, chains, plumbing supplies, tools, utensils, cutlery and machine parts. Household hardware is typically sold in hardware stores....

, sometimes in software. It is one solution to the visibility problem, which is the problem of deciding which elements of a rendered scene are visible, and which are hidden. The painter's algorithm

Painter's algorithm

The painter's algorithm, also known as a priority fill, is one of the simplest solutions to the visibility problem in 3D computer graphics...

is another common solution which, though less efficient, can also handle non-opaque scene elements. Z-buffering is also known as depth buffering.

When an object is rendered by a 3D graphics card, the depth of a generated pixel

Pixel

In digital imaging, a pixel, or pel, is a single point in a raster image, or the smallest addressable screen element in a display device; it is the smallest unit of picture that can be represented or controlled....

(z coordinate) is stored in a buffer

Buffer (computer science)

In computer science, a buffer is a region of a physical memory storage used to temporarily hold data while it is being moved from one place to another. Typically, the data is stored in a buffer as it is retrieved from an input device or just before it is sent to an output device...

(the z-buffer or depth buffer). This buffer is usually arranged as a two-dimensional array (x-y) with one element for each screen pixel. If another object of the scene must be rendered in the same pixel, the graphics card compares the two depths and chooses the one closer to the observer. The chosen depth is then saved to the z-buffer, replacing the old one. In the end, the z-buffer will allow the graphics card to correctly reproduce the usual depth perception: a close object hides a farther one. This is called z-culling.

The granularity of a z-buffer has a great influence on the scene quality: a 16-bit

16-bit

-16-bit architecture:The HP BPC, introduced in 1975, was the world's first 16-bit microprocessor. Prominent 16-bit processors include the PDP-11, Intel 8086, Intel 80286 and the WDC 65C816. The Intel 8088 was program-compatible with the Intel 8086, and was 16-bit in that its registers were 16...

z-buffer can result in artifacts (called "z-fighting

Z-fighting

Z-fighting is a phenomenon in 3D rendering that occurs when two or more primitives have similar values in the z-buffer. It is particularly prevalent with coplanar polygons, where two faces occupy essentially the same space, with neither in front...

") when two objects are very close to each other. A 24-bit

24-bit

Notable 24-bit machines include the ICT 1900 series and the Harris H series.The IBM System/360, announced in 1964, was a popular computer system with 24-bit addressing and 32-bit general registers and arithmetic...

or 32-bit

32-bit

The range of integer values that can be stored in 32 bits is 0 through 4,294,967,295. Hence, a processor with 32-bit memory addresses can directly access 4 GB of byte-addressable memory....

z-buffer behaves much better, although the problem cannot be entirely eliminated without additional algorithms. An 8-bit

8-bit

The first widely adopted 8-bit microprocessor was the Intel 8080, being used in many hobbyist computers of the late 1970s and early 1980s, often running the CP/M operating system. The Zilog Z80 and the Motorola 6800 were also used in similar computers...

z-buffer is almost never used since it has too little precision.

Uses

Z-buffer data in the area of video editing permits one to combine 2D video elements in 3D space, permitting virtual sets, "ghostly passing through wall" effects, and complex effects like mapping of video on surfaces. An application for Maya

Maya (software)

Autodesk Maya , commonly shortened to Maya, is 3D computer graphics software that runs on Microsoft Windows, Mac OS and Linux, originally developed by Alias Systems Corporation and currently owned and developed by Autodesk, Inc. It is used to create interactive 3D applications, including video...

, called IPR, permits one to perform post-rendering texturing on objects, utilizing multiple buffers like z-buffers, alpha, object id, UV coordinates and any data deemed as useful to the post-production process, saving time otherwise wasted in re-rendering of the video.

Z-buffer data obtained from rendering a surface from a light's POV permits the creation of shadows in a scanline renderer, by projecting the z-buffer data onto the ground and affected surfaces below the object. This is the same process used in non-raytracing

Ray tracing

In computer graphics, ray tracing is a technique for generating an image by tracing the path of light through pixels in an image plane and simulating the effects of its encounters with virtual objects. The technique is capable of producing a very high degree of visual realism, usually higher than...

modes by the free and open sourced 3D application Blender

Blender (software)

Blender is a free and open-source 3D computer graphics software product used for creating animated films, visual effects, interactive 3D applications or video games. The current release version is 2.60, and was released on October 19, 2011...

Developments

Even with small enough granularity, quality problems may arise when precision

Accuracy and precision

In the fields of science, engineering, industry and statistics, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's actual value. The precision of a measurement system, also called reproducibility or repeatability, is the degree to which...

in the z-buffer's distance values is not spread evenly over distance. Nearer values are much more precise (and hence can display closer objects better) than values which are farther away. Generally, this is desirable, but sometimes it will cause artifacts to appear as objects become more distant. A variation on z-buffering which results in more evenly distributed precision is called w-buffering (see below).

At the start of a new scene, the z-buffer must be cleared to a defined value, usually 1.0, because this value is the upper limit (on a scale of 0 to 1) of depth, meaning that no object is present at this point through the viewing frustum

Viewing frustum

In 3D computer graphics, the viewing frustum or view frustum is the region of space in the modeled world that may appear on the screen; it is the field of view of the notional camera. The exact shape of this region varies depending on what kind of camera lens is being simulated, but typically it is...

.

The invention of the z-buffer concept is most often attributed to Edwin Catmull

Edwin Catmull

Dr. Edwin Earl Catmull, Ph.D. is a computer scientist and current president of Walt Disney Animation Studios and Pixar Animation Studios. As a computer scientist, Catmull has contributed to many important developments in computer graphics....

, although Wolfgang Straßer also described this idea in his 1974 Ph.D. thesis¹.

On recent PC graphics cards (1999–2005), z-buffer management uses a significant chunk of the available memory

Computer storage

Computer data storage, often called storage or memory, refers to computer components and recording media that retain digital data. Data storage is one of the core functions and fundamental components of computers....

bandwidth

Bandwidth (computing)

In computer networking and computer science, bandwidth, network bandwidth, data bandwidth, or digital bandwidth is a measure of available or consumed data communication resources expressed in bits/second or multiples of it .Note that in textbooks on wireless communications, modem data transmission,...

. Various methods have been employed to reduce the performance cost of z-buffering, such as lossless compression (computer resources to compress/decompress are cheaper than bandwidth) and ultra fast hardware z-clear that makes obsolete the "one frame positive, one frame negative" trick (skipping inter-frame clear altogether using signed numbers to cleverly check depths).

Z-culling

In rendering

Rendering (computer graphics)

Rendering is the process of generating an image from a model , by means of computer programs. A scene file contains objects in a strictly defined language or data structure; it would contain geometry, viewpoint, texture, lighting, and shading information as a description of the virtual scene...

, z-culling is early pixel elimination based on depth, a method that provides an increase in performance when rendering of hidden surfaces is costly. It is a direct consequence of z-buffering, where the depth of each pixel candidate is compared to the depth of existing geometry behind which it might be hidden.

When using a z-buffer, a pixel can be culled (discarded) as soon as its depth is known, which makes it possible to skip the entire process of lighting and texturing

Texture mapping

Texture mapping is a method for adding detail, surface texture , or color to a computer-generated graphic or 3D model. Its application to 3D graphics was pioneered by Dr Edwin Catmull in his Ph.D. thesis of 1974.-Texture mapping:...

a pixel that would not be visible

Visibility (geometry)

Visibility is a mathematical abstraction of the real-life notion of visibility.Given a set of obstacles in the Euclidean space, two points in the space are said to be visible to each other, if the line segment that joins them does not intersect any obstacles.Computation of visibility is among the...

anyway. Also, time-consuming pixel shaders will generally not be executed for the culled pixels. This makes z-culling a good optimization candidate in situations where fillrate

Fillrate

The term fillrate usually refers to the number of pixels a video card can render and write to video memory in a second. In this case, fillrates are given in megapixels per second or in gigapixels per second , and they are obtained by multiplying the number of raster operations by the clock...

, lighting, texturing or pixel shaders are the main bottlenecks.

While z-buffering allows the geometry to be unsorted, sorting polygon

Polygon

In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

s by increasing depth (thus using a reverse painter's algorithm

Painter's algorithm

The painter's algorithm, also known as a priority fill, is one of the simplest solutions to the visibility problem in 3D computer graphics...

) allows each screen pixel to be rendered fewer times. This can increase performance in fillrate-limited scenes with large amounts of overdraw, but if not combined with z-buffering it suffers from severe problems such as:

polygons might occlude one another in a cycle (e.g. : triangle A occludes B, B occludes C, C occludes A), and
there is no canonical "closest" point on a triangle (e.g.: no matter whether one sorts triangles by their centroid
Centroid
In geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...

or closest point or furthest point, one can always find two triangles A and B such that A is "closer" but in reality B should be drawn first).

As such, a reverse painter's algorithm cannot be used as an alternative to Z-culling (without strenuous re-engineering), except as an optimization to Z-culling. For example, an optimization might be to keep polygons sorted according to x/y-location and z-depth to provide bounds, in an effort to quickly determine if two polygons might possibly have an occlusion interaction.

Algorithm

Given: A list of polygons {P1,P2,.....Pn}

Output: A COLOR array, which displays the intensity of the visible polygon surfaces.

Initialize:
note : z-depth and z-buffer(x,y) is positive........
z-buffer(x,y)=max depth; and
COLOR(x,y)=background color.
Begin:

for(each polygon P in the polygon list) do{
for(each pixel(x,y) that intersects P) do{
Calculate z-depth of P at (x,y)
If (z-depth < z-buffer[x,y]) then{
z-buffer[x,y]=z-depth;
COLOR(x,y)=Intensity of P at(x,y);
}
}
}
display COLOR array.

Mathematics

The range of depth values in camera space

Space

Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

(see 3D projection

3D projection

3D projection is any method of mapping three-dimensional points to a two-dimensional plane. As most current methods for displaying graphical data are based on planar two-dimensional media, the use of this type of projection is widespread, especially in computer graphics, engineering and drafting.-...

) to be rendered is often defined between a

and

value of

. After a perspective transformation, the new value of

, or

, is defined by:

After an orthographic projection

Orthographic projection

Orthographic projection is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface...

, the new value of

, or

, is defined by:

where

is the old value of

in camera space, and is sometimes called

.

The resulting values of

are normalized between the values of -1 and 1, where the

plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

is at -1 and the

plane is at 1. Values outside of this range correspond to points which are not in the viewing frustum

Frustum

In geometry, a frustum is the portion of a solid that lies between two parallel planes cutting it....

, and shouldn't be rendered.

Fixed-point representation

Typically, these values are stored in the z-buffer of the hardware graphics accelerator in fixed point

Fixed-point arithmetic

In computing, a fixed-point number representation is a real data type for a number that has a fixed number of digits after the radix point...

format. First they are normalized to a more common range which is [0,1] by substituting the appropriate conversion

into the previous formula:

Second, the above formula is multiplied by

where d is the depth of the z-buffer (usually 16, 24 or 32 bits) and rounding the result to an integer:

This formula can be inversed and derivated in order to calculate the z-buffer resolution (the 'granularity' mentioned earlier). The inverse of the above

where

The z-buffer resolution in terms of camera space

Space

would be the incremental value resulted from the smallest change in the integer stored in the z-buffer, which is +1 or -1. Therefore this resolution can be calculated from the derivative of

as a function of

Expressing it back in camera space

Space

terms, by substituting

by the above

This shows that the values of

are grouped much more densely near the

plane, and much more sparsely farther away, resulting in better precision closer to the camera. The smaller the

ratio is, the less precision there is far away—having the

plane set too closely is a common cause of undesirable rendering artifacts in more distant objects.

To implement a z-buffer, the values of

are linearly interpolated

Linear interpolation

Linear interpolation is a method of curve fitting using linear polynomials. Lerp is an abbreviation for linear interpolation, which can also be used as a verb .-Linear interpolation between two known points:...

across screen space between the vertices

Vertex (geometry)

In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

of the current polygon

Polygon

, and these intermediate values are generally stored in the z-buffer in fixed point

Fixed-point arithmetic

In computing, a fixed-point number representation is a real data type for a number that has a fixed number of digits after the radix point...

format.

W-buffer

To implement a w-buffer, the old values of

in camera space, or

, are stored in the buffer, generally in floating point

Floating point

In computing, floating point describes a method of representing real numbers in a way that can support a wide range of values. Numbers are, in general, represented approximately to a fixed number of significant digits and scaled using an exponent. The base for the scaling is normally 2, 10 or 16...

format. However, these values cannot be linearly interpolated across screen space from the vertices—they usually have to be inverted, interpolated, and then inverted again. The resulting values of

, as opposed to

, are spaced evenly between

and

. There are implementations of the w-buffer that avoid the inversions altogether.

Whether a z-buffer or w-buffer results in a better image depends on the application.

External links

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

Uses

Developments

Z-culling

Algorithm

Mathematics

Fixed-point representation

W-buffer

See also

External links