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A depth buffer, also known as a z-buffer, is a type of
data buffer In computer science, a data buffer (or just buffer) is a region of a memory used to temporarily store 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 (such a ...
used in
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
to represent depth information of objects in 3D space from a particular perspective. Depth buffers are an aid to rendering a scene to ensure that the correct
polygons In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
properly occlude other polygons. Z-buffering was first described in 1974 by Wolfgang Straßer in his PhD thesis on fast algorithms for rendering occluded objects. A similar solution to determining overlapping polygons is the
painter's algorithm The painter’s algorithm (also depth-sort algorithm and priority fill) is an algorithm for visible surface determination in 3D computer graphics that works on a polygon-by-polygon basis rather than a pixel-by-pixel, row by row, or area by are ...
, which is capable of handling non-opaque scene elements, though at the cost of efficiency and incorrect results. In a 3D-rendering pipeline, when an object is projected on the screen, the depth (z-value) of a generated fragment in the projected screen image is compared to the value already stored in the buffer (depth test), and replaces it if the new value is closer. It works in tandem with the rasterizer, which computes the colored values. The fragment output by the rasterizer is saved if it is not overlapped by another fragment. When viewing an image containing partially or fully overlapping opaque objects or surfaces, it is not possible to fully see those objects that are farthest away from the viewer and behind other objects (i.e., some surfaces are hidden behind others). If there were no mechanism for managing overlapping surfaces, surfaces would render on top of each other, not caring if they are meant to be behind other objects. The identification and removal of these surfaces are called the hidden-surface problem. To check for overlap, the computer calculates the z-value of a pixel corresponding to the first object and compares it with the z-value at the same pixel location in the z-buffer. If the calculated z-value is smaller than the z-value already in the z-buffer (i.e., the new pixel is closer), then the current z-value in the z-buffer is replaced with the calculated value. This is repeated for all objects and surfaces in the scene (often in
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of ...
). In the end, the z-buffer will allow correct reproduction of the usual depth perception: a close object hides one further away. This is called z-culling. The z-buffer has the same internal data structure as an image, namely a 2D-array, with the only difference being that it stores a single value for each screen pixel instead of color images that use 3 values to create color. This makes the z-buffer appear black-and-white because it is not storing color information. The buffer has the same dimensions as the screen buffer for consistency. Primary visibility tests (such as
back-face culling In computer graphics, back-face culling determines whether a polygon of a graphical object is drawn. It is a step in the graphical pipeline that tests whether the points in the polygon appear in clockwise or counter-clockwise order when projected ...
) and secondary visibility tests (such as overlap checks and screen clipping) are usually performed on objects' polygons in order to skip specific polygons that are unnecessary to render. Z-buffer, by comparison, is comparatively expensive, so performing primary and secondary visibility tests relieve the z-buffer of some duty. The granularity of a z-buffer has a great influence on the scene quality: the traditional
16-bit 16-bit microcomputers are microcomputers that use 16-bit microprocessors. A 16-bit register can store 216 different values. The range of integer values that can be stored in 16 bits depends on the integer representation used. With the two mos ...
z-buffer can result in artifacts (called "
z-fighting Demonstration of z-fighting with multiple colors and textures over a grey background Z-fighting, also called stitching, or planefighting, is a phenomenon in 3D rendering that occurs when two or more primitives have very similar distances to th ...
" or stitching) when two objects are very close to each other. A more modern 24-bit or
32-bit In computer architecture, 32-bit computing refers to computer systems with a processor, memory, and other major system components that operate on data in 32-bit units. Compared to smaller bit widths, 32-bit computers can perform large calculation ...
z-buffer behaves much better, although the problem cannot be eliminated without additional algorithms. An
8-bit In computer architecture, 8-bit Integer (computer science), integers or other Data (computing), data units are those that are 8 bits wide (1 octet (computing), octet). Also, 8-bit central processing unit (CPU) and arithmetic logic unit (ALU) arc ...
z-buffer is almost never used since it has too little precision.


Uses

The Z-buffer is a technology used in almost all contemporary computers, laptops, and mobile phones for performing
3D computer graphics 3D computer graphics, or “3D graphics,” sometimes called CGI, 3D-CGI or three-dimensional computer graphics are graphics that use a three-dimensional representation of geometric data (often Cartesian) that is stored in the computer for th ...
. The primary use now is for
video games Video games, also known as computer games, are electronic games that involves interaction with a user interface or input device such as a joystick, game controller, controller, computer keyboard, keyboard, or motion sensing device to gener ...
, which require fast and accurate processing of 3D scenes. The Z-buffer is implemented in hardware within consumer
graphics cards A graphics card (also called a video card, display card, graphics adapter, VGA card/VGA, video adapter, display adapter, or mistakenly GPU) is an expansion card which generates a feed of output images to a display device, such as a computer moni ...
. The Z-buffer is also used (implemented as software as opposed to hardware) for producing computer-generated special effects for films. Furthermore, Z-buffer data obtained from rendering a surface from a light's point-of-view permits the creation of shadows by the
shadow mapping Shadow mapping or shadowing projection is a process by which shadows are added to 3D computer graphics. This concept was introduced by Lance Williams in 1978, in a paper entitled "Casting curved shadows on curved surfaces." Since then, it has b ...
technique.


Developments

Even with small enough granularity, quality problems may arise when
precision Precision, precise or precisely may refer to: Science, and technology, and mathematics Mathematics and computing (general) * Accuracy and precision, measurement deviation from true value and its scatter * Significant figures, the number of digit ...
in the z-buffer's distance values are not spread evenly over distance. Nearer values are much more precise (and hence can display closer objects better) than values that 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. The invention of the z-buffer concept is most often attributed to
Edwin Catmull Edwin Earl "Ed" Catmull (born March 31, 1945) is an American computer scientist who is the co-founder of Pixar and was the President of Walt Disney Animation Studios. He has been honored for his contributions to 3D computer graphics, including th ...
, although Wolfgang Straßer described this idea in his 1974 Ph.D. thesis months before Catmull's invention. On more recent PC graphics cards (1999–2005), z-buffer management uses a significant chunk of the available
memory Memory is the faculty of the mind by which data or information is encoded, stored, and retrieved when needed. It is the retention of information over time for the purpose of influencing future action. If past events could not be remembered, ...
bandwidth Bandwidth commonly refers to: * Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range * Bandwidth (computing), the rate of data transfer, bit rate or thr ...
. 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, 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 the 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 a pixel that would not be
visible Visibility, in meteorology, is a measure of the distance at which an object or light can be seen. Visibility may also refer to: * A measure of turbidity in water quality control * Interferometric visibility, which quantifies interference contrast ...
anyway. Also, time-consuming
pixel shader In computer graphics, a shader is a computer program that calculates the appropriate levels of light, darkness, and color during the Rendering (computer graphics), rendering of a 3D scene - a process known as ''shading''. Shaders have evolved ...
s will generally not be executed for the culled pixels. This makes z-culling a good optimization candidate in situations where
fillrate In computer graphics, a video card's pixel fillrate refers to the number of pixels that can be rendered on the screen and written to video memory in one second. Pixel fillrates are given in megapixels per second or in gigapixels per second (in ...
, lighting, texturing, or pixel shaders are the main bottlenecks. While z-buffering allows the geometry to be unsorted, sorting
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
s by increasing depth (thus using a reverse
painter's algorithm The painter’s algorithm (also depth-sort algorithm and priority fill) is an algorithm for visible surface determination in 3D computer graphics that works on a polygon-by-polygon basis rather than a pixel-by-pixel, row by row, or area by are ...
) 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 In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
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.


Mathematics

The range of depth values in camera space to be rendered is often defined between a \textit and \textit value of z. After a
perspective transform A 3D projection (or graphical projection) is a design technique used to display a three-dimensional (3D) object on a two-dimensional (2D) surface. These projections rely on visual perspective and aspect analysis to project a complex object fo ...
ation, the new value of z, or z', is defined by: : z'= \frac + \frac \left(\frac\right) After an
orthographic projection Orthographic projection (also orthogonal projection and analemma) is a means of representing Three-dimensional space, three-dimensional objects in Two-dimensional space, two dimensions. Orthographic projection is a form of parallel projection in ...
, the new value of z, or z', is defined by: :z'= 2 \cdot \frac - 1 where z is the old value of z in camera space, and is sometimes called w or w'. The resulting values of z' are normalized between the values of -1 and 1, where the \textit
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
is at -1 and the \mathit plane is at 1. Values outside of this range correspond to points which are not in the viewing
frustum In geometry, a (from the Latin for "morsel"; plural: ''frusta'' or ''frustums'') is the portion of a solid (normally a pyramid or a cone) that lies between two parallel planes cutting this solid. In the case of a pyramid, the base faces are ...
, 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 format. First they are normalized to a more common range which is by substituting the appropriate conversion z'_2 = \frac\left(z'_1 + 1\right) into the previous formula: :z'= \frac + \frac + \frac \left(\frac\right) Simplifying: :z'= \frac + \frac \left(\frac\right) Second, the above formula is multiplied by S = 2^d - 1 where d is the depth of the z-buffer (usually 16, 24 or 32 bits) and rounding the result to an integer: :z' = f(z) = \left\lfloor \left(2^d - 1\right) \cdot \left(\frac + \frac \left(\frac\right) \right)\right\rfloor This formula can be inverted and derived in order to calculate the z-buffer resolution (the 'granularity' mentioned earlier). The inverse of the above f(z)\,: :z = \frac = \frac where S = 2^d - 1 The z-buffer resolution in terms of camera 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 z as a function of z': :\frac = \frac \cdot \left(\textit - \textit\right) Expressing it back in camera space terms, by substituting z' by the above f(z)\,: :\begin \frac &= \frac \\ &= \frac \\ &= \frac - \frac \approx \frac \end This shows that the values of z' are grouped much more densely near the \textit plane, and much more sparsely farther away, resulting in better precision closer to the camera. The smaller near is, the less precision there is far away—having the near plane set too closely is a common cause of undesirable rendering artifacts in more distant objects. To implement a z-buffer, the values of z' are linearly interpolated across screen space between the vertices of the current
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
, and these intermediate values are generally stored in the z-buffer in fixed point format.


W-buffer

To implement a w-buffer, the old values of z in camera space, or w, are stored in the buffer, generally in
floating point In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be ...
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 w, as opposed to z', are spaced evenly between \textit and \textit. 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.


Algorithmics

The following pseudocode demonstrates the process of z-buffering:
// First of all, initialize the depth of each pixel.
d(i, j) = infinite // Max length

// Initialize the color value for each pixel to the background color
c(i, j) = background color

// For each polygon, do the following steps :
for (each pixel in polygon's projection)


See also

*
Z-fighting Demonstration of z-fighting with multiple colors and textures over a grey background Z-fighting, also called stitching, or planefighting, is a phenomenon in 3D rendering that occurs when two or more primitives have very similar distances to th ...
*
Irregular Z-buffer The irregular Z-buffer is an algorithm designed to solve the visibility problem in real-time 3-d computer graphics. It is related to the classical Z-buffer in that it maintains a depth value for each image sample and uses these to determine which g ...
*
Z-order Z-order is an ordering of overlapping two-dimensional objects, such as Window (computing), windows in a stacking window manager, shapes in a vector graphics editor, or objects in a 3D application.Foley, James, Andries van Dam, Steven Feiner, and ...
*
A-buffer In computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized app ...
*
Depth map In 3D computer graphics and computer vision, a depth map is an image or image channel that contains information relating to the distance of the surfaces of scene objects from a viewpoint. The term is related (and may be analogous) to ''depth ...
*
HyperZ HyperZ is the brand for a set of processing techniques developed by ATI Technologies and later Advanced Micro Devices and implemented in their Radeon-GPUs. HyperZ was announced in November 2000 and was still available in the TeraScale-based Ra ...
*
Stencil buffer A stencil buffer is an extra data buffer, in addition to the ''color buffer'' and ''Z-buffer'', found on modern graphics hardware. The buffer is per pixel and works on integer values, usually with a depth of one byte per pixel. The Z-buffer and ...


References


External links


Learning to Love your Z-buffer



Notes

{{DEFAULTSORT:Z-Buffering 3D rendering