Abstract:
A system renders a primitive of an image to be displayed, for instance in a mobile 3D graphic pipeline, the primitive including a set of pixels. The system locates the pixels in the area of the primitive, generates, for each pixel located in the area, a set of associated sub-pixels, borrows a set of sub-pixels from neighboring pixels, subjects the set of associated sub-pixels and the borrowed set of pixels to adaptive filtering to create an adaptively filtered set of sub-pixels, and further filters the adaptively filtered set of sub-pixels to compute a final pixel for display. Preferably, the set of associated sub-pixels fulfils at least one of the following: the set includes two associated sub-pixels and the set includes associated sub-pixels placed on triangle edges.

Description:
BACKGROUND OF THE INVENTION  
       [0001]     1. Field of the Invention  
         [0002]     The present invention relates to 3D graphic technology.  
         [0003]     2. Description of the Related Art  
         [0004]     While significant progress over the recent years has greatly contributed to increasing the realism and overall quality of rendering of images, a type of artifact that still affects computer generated pictures is the aliasing effect referred to as “staircasing” or “jaggies”.  
         [0005]     That effect, which is mostly manifest at polygon edges, is due to lack of sufficient pixel availability to render smoothly lines, points and polygon edges as defined mathematically. When a scene is processed by the rendering pipeline and transposed in a pixel matrix, a color is assigned to each pixel depending on whether it is covered by another object belonging to the scene. This approach does not take in consideration adequately the possibility for a pixel to be covered partially by an object as it may typically happen at object edges. The consequence is an abrupt color variation between the object and the background.  
         [0006]     Aliasing is in fact very annoying. A typical 3D scene has many objects composed of a thousand (and even more) triangles and aliasing artifacts are very visible at object edges. In particular when the scene has moving portions, the artifacts become evident as flickering details and instability of the contours. This decreases the realism perceived.  
         [0007]     Since the size of those artifacts cannot exceed the pixel dimension, the net effect is decreased by using increased picture resolution. For example, an image without any anti-aliasing will look better with 1600×1200 resolution than with 1024×768 resolution. The human eye can detect artifacts up to about 600 points per inch but unfortunately no commercial monitor has such resolution.  
         [0008]     Methods to decrease aliasing artifacts, named anti-aliasing techniques, have the goal of making those transitions smoother by processing edge pixels. Anti-aliasing techniques are needed to decrease the visual impairment and provide the illusion of a higher-than-real screen resolution.  
         [0009]     Such techniques have been developed for a long time and many of them have been used successfully in off-line rendering. Over the last ten years, computational power available run-time reached a level that permits implementation of these techniques with hardware support in real time.  
         [0010]     A generic OpenGL pipeline for 3D interactive graphic will be now described in order to provide a general introduction to the specific technology considered. This description fulfils the double aim of i) facilitating a discussion of the related art and the intrinsic disadvantages of prior art solutions, and ii) highlighting the specific contribution of the invention to the prior art in question.  
         [0011]     Briefly, an OpenGL pipeline for 3D interactive graphics is able to implement a set of algorithms that process a tri-dimensional synthetic scene involving motion, and is able to show it on a bi-dimensional display. Ideally, the process of generating and displaying the scene is decomposed in consecutive steps or pipeline stages. Each stage has a specific task and provides the input for the next one. The stages have a varying complexity, which varies from one stage to another: typically stages that process and affect pixels (e.g., the late stages of the pipeline) are more complex than those that process the 3D scene details.  
         [0012]     Roughly speaking, 70% of the computational resources are needed by rasterizer and fragment operations while the remaining is needed by the geometry stages. This is because the final shape of each pixel is obtained by applying repeatedly the same conceptual steps on the same pixel, while other operations are executed only once per 3D object.  
         [0013]     For a 3D pipeline it is thus vital to use as few operations as possible while achieving a good level of detail and a minimum guaranteed frame rate.  
         [0014]     A plurality of 3D objects (or models) composes a 3D scene that is placed and moves inside a tri-dimensional space of coordinates. Each object is comprised of a number of graphic primitives. The appearance of the object depends on the number of primitives that compose it: the higher the number of the primitives, the higher the level of detail perceived from the visual point of view.  
         [0015]     A geometric primitive should be a simple entity that can be easily processed by a real pipeline. The algorithms used during this processing depend on the geometric nature of the primitives. Examples of primitives are: 
        points     lines     triangles     quads (squares)     polygons     higher-order surfaces        
 
         [0022]     Excluding lines and points, an adequate object representation is provided via triangles and such a representation will be assumed by way of simplicity throughout the description that follows. In fact, a triangle is a simple, easy-to-process planar convex surface. Quads, instead, are not planar and more difficult to manage. In brief, the higher the number of triangles composing an object, the higher the graphic detail achieved.  
         [0023]     Each triangle is defined by its vertices. Thus, each 3D scene can be completely defined by a list of vertices, which are tri-dimensional points placed in a reference coordinate system. Those vertices are associated to each primitive to which they belong. Each vertex can be defined by associating to it the following information: 
        X, Y, Z real coordinates     R, G, B color components        
 
         [0026]     Additional information can be the following: 
        alpha factors (one or more),     three real coordinates to specify the normal vector associated to the vertex. This vector is used by the lighting stage to apply a light to the primitive,     two or more coordinates used as addresses of suitable 2 or 3 maps, from which it is possible to extract pixel information associated to the vertices in order to contribute to their final colors.        
 
         [0030]     For example, if a vertex is defined by: XYZ, RGB, alpha, UV texture coordinates and Nx, Ny, Nz normal coordinates each expressed as a 32 bit single precision floating point number, then the memory requirement for storing each vertices will be 48 bytes.  
         [0031]     To locate a 3D object within a scene, certain geometric transformations are needed. These can be decomposed in a set of simple affine transformations like rotation, translation, scaling etc. Each object can be moved independently from the others by transforming its vertices at a certain instant. Natural motion is achieved by carefully applying transformations in sequence in order to obtain a discrete-linear approximation of real, continuous motion. Such transformations are not incremental with respect to the previous ones. The position of the object at a time “t” is obtained by applying transformations to the object geometry considered at an initial time.  
         [0032]     Usually, another requirement is permitting the scene to be watched from a given position where the observer can stay. This can be thought of as a window through which the 3D scene is watched. Global motion is a result of the object motion and the motion (if any) of the viewpoint. Alternatively, the viewpoint can be like a camera that samples the 3D scene.  
         [0033]     The 3D scene is displayed on a monitor through a bi-dimensional grid of pixels each one defined by (R,G,B) colors. Then some operations are applied to each pixel before display. These operations are those demanding most of the computational power. The pixels compose a frame that is a snapshot from the camera viewpoint, taken a number of times per second as defined by the frame rate. Typically a frame rate of 20-30 frames per second (fps) is needed to achieve a fluid motion perception by the viewer.  
         [0034]     Before describing the pipeline stages, it is advantageous to describe how 3D objects are defined, placed and moved inside the virtual tri-dimensional world.  
         [0035]     Different Cartesian reference systems are used in 3D pipelines:  
         [0036]     model space: each models corresponds to a “local” reference system. The origin can be at a point of it, such as, e.g., the lowest one,  
         [0037]     world space: all models are placed within this unique system (OpenGL does not use this and goes directly from the “model space” to the “view space”),  
         [0038]     view space (or Camera Space): the viewpoint is placed in this space (alternatively, it may be placed in a “world space” point). The world space is transformed in order to have the “camera space” with the camera placed at the origin and directed to the negative z-axis. In the case of OpenGL, the “GL_MODELVIEW” matrix set the transformation from “model space” to “view space”. In this phase, it is possible to detect what objects are potentially visible (i.e., have a negative x coordinate in the “view space”) while culling the others that are not visible,  
         [0039]     clip space: after “camera space” there is “clip space” ; there it is possible to detect what primitives are visible using the “clipping” process and send them through the pipeline. “Clip space” is a space obtained through a perspective transform, in order to have a “view volume”, that is a parallelepiped if the transform is orthogonal and a “frustum” if the transform is of a perspective type. It projects visible 3D objects onto a bi-dimensional plane that has a projection center placed at the camera itself (projective transform) or at infinity (orthogonal transform). In OpenGL, the “view volume” is normalized as a unity cube defined by two vertices with (−1, −1, −1) and (1, 1, 1) coordinates in order to have more efficient clipping operations. Besides, a primitive with “clip space” coordinates within the cube is potentially visible. The “GL_PROJECTION” matrix is able to transform a 3D object from “camera space” coordinate to projected normalized coordinates,  
         [0040]     screen space: 3D primitive coordinates in the “clip space” are re-scaled into 2D screen coordinates depending on the real screen resolution. In OpenGL, this happens thanks to the “glViewPort( )” function by fixing screen coordinates.  
         [0041]     There are no univocal conventions for the number of stage or types of operations implemented. A typical configuration will be described below by way of general reference.  
         [0042]      FIG. 1  is a block diagram illustrating a typical 3D graphic system  100  and its various stages.  
         [0043]     A useful classification of the operations implemented is based on the input type processed by each stage. Accordingly, the pipeline operations can be classified in the following modes:  
         [0044]     operations per-vertex: these are applied to the vertexes associated to each primitive that composes the scene. Geometric transforms are examples of these operations;  
         [0045]     operations per-triangle: these are applied to the primitives that compose the scene. “Clipping” and “culling” are good examples: the former defines what parts of the primitives are inside of “view volume” and, depending on this intersection, defines how these are cut; the latter determines if those points are outside the “view volume” or back facing the “view point”;  
         [0046]     operations per-pixel: they operate directly to the pixels. Examples are rendering, per-pixel lighting, per pixel texturing, fragment operations where one or more fragments are associated to each pixel. Another difference with the other types of operations considered previously is that “per-pixel” or “per fragment” operations can be applied repeatedly to the same displayed pixel. This is the main reason why high complexity operations reside in this stage: in fact they will influence also the frame rate. Up-conversion methods may reduce the number of “per pixel” operations in order to save computational power associated with them.  
         [0047]     On the basis of the above definitions, the 3D graphic system  100  shown in  FIG. 1  can be subdivided in four main stages:  
         [0048]     an application/scene stage  101   
         [0049]     a geometry stage  111 ;  
         [0050]     a triangle setup stage  112 ;  
         [0051]     a rendering/rasterization stage  113 .  
         [0052]     The geometry stage  111 , the triangle setup stage  112  and the rendering/rasterization stage  113  are part of a pipelined graphic engine  110 .  
         [0053]     Lights  107  and textures  108  represent the information regarding 3D objects  106  input to the application/scene stage  101 .  
         [0054]     Operations in the application/scene stage  101  are defined and set by the programmer using a API (Application Protocol Interface): OpenGL is an example of a set of such APIs. Typically, the operations implemented are database management, motion set-up of 3D objects and camera description of the content of a 3D animation, control of primitive visibility (“occlusion culling”), level of scene details (LOD, the number of primitives used to render the 3D scene).  
         [0055]     The geometry stage  111  is the first pipeline stage. The associated operations are mainly “per-vertex” operations and the geometry stage  111  includes operations suitable for moving 3D objects. Motion is obtained by applying geometric transforms to vertices of the primitives in each single object of the scene. Transforms are of the affine type and defined in an affine space where two entities are defined: points and vectors. Results of transformation are vectors or points.  
         [0056]     The transforms used in a 3D pipeline can be grouped in two main families, i.e.:  
         [0057]     rigid body transforms, i.e., roto-translation plus scaling where the results is still within the 3D space; and  
         [0058]     project transforms (perspective, orthogonal etc.) that move a primitive from the 3D space to the 2D space where those objects are located to be displayed on a screen.  
         [0059]     These transforms use suitable square 4×4 arrays. Each affine transform can be obtained by properly defining each coefficient in such arrays.  
         [0060]     In the geometry stage  111  other operations such as “culling”, “clipping”, “screen space transform”, “lighting”, “back-face culling”, are also implemented, fully or partly, depending on the kind of implementation.  
         [0061]     These geometry operations can be briefly described as follows:  
         [0062]     culling: 3D objects are now in the “clip space”. In openGL “clip space” is a unit cube defined by (−1, −1, −1) and (1, 1, 1) coordinate vertices. All those triangles whose vertices are inside the cube are potentially visible (but some of them might occlude partially or totally the vision of others). All the triangles that have vertices falling outside the unit cube, from the same side, are culled and will not be processed by later stages of the pipeline; thus only a decreased set of triangles needs to be processed;  
         [0063]     clipping: this is similar to culling but is more complex. Clipping processes triangles that are partially inside and partially outside the unit cube. For example, if two vertices of a triangle are inside the cube and only one outside, then the triangle is partially visible. In particular, visible parts will have a quadrilateral shape where two more vertices are defined as intersections between the triangle and the faces of the unit cube. The clipping operation computes the two intersections and, since the pipeline processes only triangles, the quad is re-tasseled and then decomposed in triangles. This is performed on potentially visible triangles and the new triangle list is concatenated with the visible tingles obtained after culling;  
         [0064]     back-face culling: it is quite convenient to cull the triangles in earlier stages of the pipeline since this will appreciably decrease the “potentially visible” triangles submitted to later stages and processing needed. Assuming that a 3D model is composed of triangles inside the unit cube (which means that they are potentially visible), the triangles can be of two types: those that are visible to the observer in that they are “front facing” and those that are not visible in that they are “back facing”. “Back-face culling” is able to remove from the list of potentially visible triangles those of the “back-face” type. For symmetric objects, 50% of the triangles are front facing while the others are back facing. Removing 50% of the triangles means avoiding all the subsequent pipeline computations on their vertices;  
         [0065]     lighting: this takes place after the scene has been transformed and positioned in the “view space”. After this it is possible to apply correctly the lights on the 3D objects. The model used is very simplified with respect to the real world situation. For example, an object can be lit by some sources without reflections, which happens in reality. A very detailed and realistic lighting scheme easily becomes very complex from the computational viewpoint to be implemented in a 3D pipeline.  
         [0066]     A simplified lighting model briefly described herein includes two main lights, global and local, giving rise to three main types of lighting:  
         [0067]     directional: it is a global type and the source is placed at a distance notionally infinite from the objects in the scene. The light of the sun is a good example of this type of lighting: all the rays are parallel;  
         [0068]     point light: it is local and positioned at a point inside the “view space”. Its rays emanate to all directions: a light bulb is a good example of this type of lighting;  
         [0069]     spot: rays are emitted in only one direction within a limited cone of light; car headlights are typical examples of this kind of lighting.  
         [0070]     All effects are mixed in order to have a global i tot  effect.  
         i   tot     =       i   amb     +     i   diff     +     i   spec           
 
         [0071]     When a single light affects an object, then the final effect also depends on the material properties of the lit object. For that reason, openGL and direct 3D allow to specify the “material color properties” that contribute to the definition of the final lighting effect for a surface.  
         [0072]     These properties are defined as:  
         [0073]     m amb : Ambient Material Color  
         [0074]     m diff : Diffuse Material Color  
         [0075]     m spec : Specular Material Color  
         [0076]     m shi : Shininess Material Color  
         [0077]     m emi : Emissive Material Color  
         [0078]     These are placed in the equations for the lighting model which, even though simplified, turns out to be a highly complex one especially if more lights influence the scene.  
         [0079]     To reach a compromise between sophisticated lighting schemes and acceptable complexity, certain pipelines use “light maps”. These are pre-computed maps of lights adapted to be super imposed on the primitives during the texturing stage. Hybrid techniques exist that mix lights computed in the “geometry stage” (the so-called “lighting per-vertex”), and light-maps (the so-called “lighting per-pixel”). For example, a directional light can be computed with “lighting per-vertex”, while spot lights are simulated with ad-hoc light-map.  
         [0080]     Screen space transform (o ViewPort Transform) is the last operation performed before the “triangle setup” stage. The updated list of triangles is full of potentially visible primitives without back facing triangles. Vertices have (x, y, z) coordinates expressed in the “clip space” system of coordinates. They range between [−1, 1] and have been already projected.  
         [0081]     The next stage, i.e., “triangle setup”, implements some preliminary operations on the list of visible triangles to prepare and optimize the next rendering stage. In particular, coordinates need to be transformed from the “clip space” into the “screen space” related to the real screen resolution in order to properly run the rasterizer stage.  
         [0082]     The triangle setup stage  112  performs a step that prepares the following rendering stage  113 . In the triangle setup stage  112  operations on data from the geometry stage  111  are organized to compute a new set of data necessary for the next stage. Specifically, a new set of data is generated and associated with each vertex.  
         [0083]     By way of example, an incremental scan conversion process can be implemented in the triangle setup stage  112 : for each item in the list of triangles that are potentially visible and available from the geometry stage  111 , the process locates each triangle over the pixel grid depending also on the z depth value associated to each vertex.  
         [0084]     The rendering/rasterization stage  113  is one of the most complex stages from the computational viewpoint within the 3D graphic engine  110 : in fact, the stage in question performs “per pixel” computations, that are typically iterated many times in order to provide the final appearance of each pixel actually displayed.  
         [0085]     The order of the operations involved depends on the specific implementation. However, the outcome of the operations performed in the stages from  101  to  113  is finally supplied to a display frame device  130 .  
         [0086]     The description above highlights that the rendering stage  113  operates “per-pixel” and it is the stage in which the most part of the complexity of the pipeline resides. The workload on the rendering stage  113  affects the frame rate that can be achieved at the display  130 : when the maximum workload that the rendering stage  113  can sustain is fixed, then the complexity of an application in term of number and type of primitives may temporarily exceed the fixed computational power, thus originating a variable time for rendering a frame and variable frame rate. This also occurs if algorithms with different complexities (such as various shading techniques) are used at different times.  
         [0087]     Operations in the rendering/rasterization stage  113  comprise a shading operation that computes the color of each pixel composing the projected triangle. Different methods exist for implementing 3D shading. The most commonly used include: “flat shading”, “Gouraud shading” and “phong shading”. These methods span from low (flat) to high (phong) complexity and realism achieved.  
         [0088]     More specifically, shading computes the color of each pixel that composes the projected triangle. A local reflection model is assumed to be available. Different methods are possibly used that work on a normal surface vector n of the current point, a vector v oriented toward the viewpoint and a vector I oriented toward the light source. Typically, a normal vector n is associated with each vertex of a triangle. Each triangle has a vertex list and a normal-per-vertex list. A set of triangles approximates a real surface and includes for each vertex a normal vector to the surface in this point. In other words, the surface is sampled through the vertices that approximate it.  
         [0089]     There are different methods to implement 3D shading.  
         [0090]     In flat shading each triangle has a unique color. If the normal is defined per triangle, then the color depends on it and on the local reflection model available. Since the normal is typically associated on a per vertex basis, the method computes on a single vertex (the first found in the list) a color that is then applied to all the pixels onto which the triangle is projected. This approach is very simple and the lighting equation is computed once per triangle. On the other side the quality is usually very poor since there is a color discontinuity between adjacent triangles.  
         [0091]     Gouraud shading is a good compromise in order to produce a quasi-continuous color change within a triangle. Essentially, this approach:  
         [0092]     takes the vertex normals;  
         [0093]     computes the color for each vertex using the local reflection model;  
         [0094]     computes the color per each pixel within the triangle as a bilinear interpolation of the vertex colors.  
         [0095]     If the normal is not available, it is possible to compute the vertex normal by using the average of the normals associated to adjacent polygons that share the same vertex. If the normals are the same, the method performs like “flat shading”. “Gouraud shading” makes the color transitions between adjacent triangles smoother. It is simple and fast, being just a bit more complex than flat shading. However, specular reflections (highlights) are not rendered realistically.  
         [0096]     Phong shading offers high realism if compared with previous methods. The vertex normals are interpolated inside the triangle. The interpolated normal is used in the local reflection model to compute the lighting equation for the color of each pixel inside the triangle. Phong shading captures specular reflections as it fully exploits the local reflection model per pixel. OpenGL does not support this approach since the normals are discarded after prospective projection.  
         [0097]     Other possible operations include depth cueing, texture mapping, anti-aliasing and rasterization.  
         [0098]     Depth cueing is useful in order to give a depth look to the rendered picture.  
         [0099]     Texture mapping exploits the fact that reflection models are quite versatile. With a suitable selection of related parameters, different materials can be rendered in a quasi-realistic way. A limitation of shading lies in the impossibility of simulating very high details unless the model geometry is significantly modified. A way to solve this problem is to find a way to add details as a part of the rendering process itself. The methods considered here influence the shading process and are implemented as part of it. A method uses a bidimensional map, named texture map, in order to add details to an object surface.  
         [0100]     A univocal correspondence between object and texture is needed. A texture is an image or a set of RGB colors. To compute the color of the surface at a certain point the correspondent color of a texture is used. This technique, also known as “color mapping”, achieves good results as realism with a limited load from a computational point of view.  
         [0101]     However, using “texture mapping” instead of introducing more details in the model definition has three basic advantages:  
         [0102]     this approach is local and not global;  
         [0103]     textures require memory space and a suitable selection algorithm; an alternative is to produce them as result of a procedural method; and  
         [0104]     use of texture introduces aliasing artifacts and requires means to dispense with them.  
         [0105]     Texture-mapping needs a W( ) function for parametrization that associates a point of the texture (u,v) to a P point of the 3D surface (in other words it spreads the texture on the object surface). The point is mapped by the projection in a (x s ,y s ) screen point.  
         [0106]     Rendering of the texture defines the texture value to be associated to each pixel.  
         [0107]     Rasterization (or Scan Conversion) draws the current frame with the colors computed by the shading by addressing pixel inside a frame segment and the one inside a triangle.  
         [0108]     Two basic rasterization algorithms are:  
         [0109]     the Bresenham algorithm, which generates a sequence of pixels by using fixed arithmetic and choosing the pixel to be colored after the current one out of the 8 adjacent pixels;  
         [0110]     the scan-line algorithm, where pixels are chosen and colored by considering a scanline from top to bottom. A left-to-right scan is performed for each row and the triangles are filled between edges intersections to scan lines.  
         [0111]     As indicated in the introductory portion of this application, when rendering a frame, due to the discrete nature of its rows and columns, aliasing will decrease the perceived quality.  
         [0112]     Aliasing causes various artifacts in computer graphics as described, e.g., in E. Catmull: “A Hidden-Surface Algorithm With Anti-Aliasing”, Proceedings of the 5 th  Annual Conference On Computer Graphics And Interactive Techniques, pp. 6-11, ACM Press, 1978 or F. C. Crow: “The Aliasing Problem In Computer-Generated Shaded Images”, Communications of the ACM, 20(11): 799-805, 1977.  
         [0113]     Such artifacts include:  
         [0114]     jagged edges  
         [0115]     small objects that appear/disappear in consecutive frames  
         [0116]     moiré pattern is periodic images  
         [0117]     small detail loss  
         [0118]     Aliasing artifacts can occur also over time (temporal aliasing, see J. Korein et al.: “Temporal Anti-Aliasing In Computer Generated Animation”, Proceedings of the 10 th  Annual Conference On Computer Graphics And Interactive Techniques, pp. 377-388, 1983). The effects are typically visible when objects are rotating fast, like car gears that sometimes seem to rotate in the opposite direction or running slower than expected. This happens since the motion is not sampled properly from time to time. To reduce this, a motion blur technique is often used.  
         [0119]     As already indicated, a solution for decreasing aliasing is to create the effect of an increased screen resolution. From a historical point of view this is the best solution: “Jaggies” have dimensions not greater than the real pixel dimensions. By reducing the pixel dimension also those artifacts are minimized.  
         [0120]     However, a resolution change is not always feasible. This may be due, for example, to the maximum resolution being already used or the application itself having a resolution limit. Then a viable solution for increasing the resolution as perceived by the user is to use sophisticated solutions that compute the final pixel color.  
         [0121]     Even if anti-aliasing techniques have no well defined and widely accepted taxonomy, a well known classification defines them as pre-filtering techniques, post-filtering techniques and methods that use the so-called A-Buffer.  
         [0122]     Pre-filtering methods remove image frequencies, greater than the Nyquist′ frequency, before sampling the signal. This involves a low-pass filtering of the image and subsequent sampling. In other words, this approach computes the contribution of the area for each polygon for each pixel: this is the only method that notionally allows the artifacts to be removed completely because limits are imposed on the ideal spectrum before sampling.  
         [0123]     Post-filtering methods consider the pixel coverage only at discrete points as an approximation of the exact area computation. This approach leads to methods that are very easy to implement in hardware form and are quite efficient in terms of real-time rendering. However, due their discrete nature, post-filtering methods cannot remove aliasing completely. They can appreciably increase the threshold value, at the frequency level, below which the aliasing artifacts are no longer visible.  
         [0124]     While post-filtering methods can reduce artifacts, they cannot guarantee good results for all images.  
         [0125]     If the real image spectrum is not limited, no over-sampling rate is great enough to avoid aliasing.  
         [0126]     An increased, over-sampling frequency reduces artifacts visibility but is computationally heavy, while super-sampling is often simple and feasible. Super-sampling is often simpler and feasible than pre-filtering and this extensively contributed to their diffusion at the commercial level.  
         [0127]     The area sampling method belongs to the class of pre-filtering techniques. Loss of details that fall between two samples suggests an approach called “un-weighted area sampling” that consists in integrating the signal over a square centered at each point of the picture grid, dividing the obtained value for the square area and using the result as the average intensity of the pixel considered. Pixels can be thought of as an array of non-overlapping squares. Each projection of an object contributes to the intensity of those pixels enclosed by its projection. This is in proportion to the area of each pixel that it covers, without considering its location within the pixel. Details cannot be lost, as is the case in simple point sampling. The main disadvantage of un-weighed area sampling lies in that a small object enclosed completely within a pixel that moves out of it causes a change on the image only when it intersects pixel edges. When an object moves far from the pixel center toward another pixel, this change will be represented in the image. In other words, the intensity of the pixel is weighed depending on the distance of the object from the pixel center. Weighed area sampling assigns different weights to the various parts of the pixels. In that way, the contribution of an object to pixel intensity depends on its distance from the pixel center. The best results are obtained in this case when the weighing function of pixels is overlapping in the middle and lower. Such overlapping allows having smoother transitions between the intensity of close pixels.  
         [0128]     The super-sampling and multi-sampling methods are the simpler anti-aliasing techniques easy to implement in hardware form. The distinction between those methods is not well defined and some implementations can be regarded as belonging to both categories. Both methods however belong to the post-filtering category.  
         [0129]     Super-sampling or FSAA (Full Scene Anti-Aliasing) renders the image at a higher resolution to be subsequently scaled to final dimension.  
         [0130]     With multisampling, each pixel is sampled at different positions and these samples can be used to reconstruct a final value.  
         [0131]     Both methods require more than one sample per pixel of the final image. For example, the 2× over-sampling perform rendering at two-fold resolution in order to obtain 2 sub-pixel samples for each output pixel; 4× FSAA does the same at four-fold resolution etc. Samples generated with over-sampling are combined with filtering techniques to mix those contributions with smoother edges.  
         [0132]     The technique known as MSAA (Multi-Sampling Anti-Aliasing) is another alternative to traditional super-sampling used in FSAA. As in super-sampling, multisampling computes the scene at a higher resolution with a small difference by using texture samples for sub-pixel. For example 4× super-sampling generates 4 texels (texture elements) for each sub-pixel, while 4× MSAA generates a single texture for each sub-pixel.  
         [0133]     Picture quality depends on the number and pattern of the samples taken for per each pixel. Unfortunately, both super-sampling and multi-sampling cannot completely remove aliasing; whatever the density of the sub-pixel grid, there can be always higher frequencies than samples can capture. If the signal is not subject to low-pass before rendering, the greater part of the images is affected by aliasing.  
         [0134]     However, especially in real time rendering, both methods produce acceptable images, considering the low cost in performance and are used in most implementations of the anti-aliasing function in commercial systems.  
         [0135]     Super-sampling can be summarized in the following steps:  
         [0136]     i. creating an image at a resolution greater than the target resolution;  
         [0137]     ii. low-pass filtering, and  
         [0138]     iii. the filtered picture is re-sampled  
         [0139]     The first step is simple but requires memory and bandwidth. A further optimization consists in combining the steps ii) and iii) by generating a filtered picture only at points corresponding to new samples. A low-pass, “boxcar” type filter is easy to implement as the simple average of samples, but results are not satisfactory.  
         [0140]     Sinc2 and Gaussian filtering produce better results. Sinc2 is the Fourier transform of a triangular function while Gaussian is a Fourier transform of itself. Both have strong attenuation far from the center and are fast to compute. A method that uses a Sinc2 filter is mostly known as bilinear filtering.  
         [0141]     By increasing the number of samples, picture quality also increases but on the other hand the memory requirements also increase while overall performance effectiveness decreases. An accurate selection of sampling patterns is a crucial step in real cases. Regular sampling patterns (called OGSS, Ordered Grid Super Sampling) are not able to remove aliasing artifacts that are well visible to the human eye, which is sensitive to regular patterns. By using a regular grid, better results are obtained in terms of performance and memory requirements by selecting sampling points from a narrow subset of all possible sub-sample positions (for example by using n positions in an n×n grid).  
         [0142]     This technique, called “sparse” sampling, uses a sampling grid that uses less memory and bandwidth. An example of this sampling is RGSS (Rotating Grid Super Sampling).  
         [0143]     Stochastic sampling techniques are based on the assumption that the human visual system is more sensitive to regular aliasing patterns than to “noisy” patterns. Such a method can be used with ray-tracing, while ray-casting can be done easily everywhere inside the pixel area. Sampling is done using a random pattern. A Poisson pattern may be generated by adding random locations inside the grid. A Poisson disc is more costly to compute since limits exist on the relative distance between points. So the generation of a Poisson disc requires the storage of the location of each point. If a new point is added and the minimum distance test fails, then the point is dismissed and a new one is generated at a new position. A jitter pattern can be generated using a regular grid and moving each point by a random offset both in X and Y directions.  
         [0144]     Fourier transforms of those patterns identify the noise distribution in the frequency domain of the image. Poisson pattern generate almost equal amounts of noise at low and high frequencies. However, the results may not be satisfactory since the noise at low frequency is visible more easily by the human visual system. A Poisson disc avoids the low frequency noise (but is more “expensive” to compute), while jittering techniques seem to offer a better price/performance ratio.  
         [0145]     A-Buffer (anti-aliased, area-averaged, accumulation buffer) is a general technique for removing hidden surfaces with anti-aliasing. This technique solves the visibility problem in an arbitrary number of opaque, transparent and intersecting objects. One of the advantages of the techniques that use the A-Buffer lies in that they ignore the span of pixels covered completely by a polygon, in order to work only on edge pixels.  
         [0146]     Exemplary for these types of “edge” anti-aliasing algorithms is the Z-Buffer technique, currently used in present-day graphic hardware. This technique solves easily the problem of detecting visible surfaces, but has several disadvantages. It does not manage transparent surfaces and is affected by aliasing. The article by Catmull cited in the foregoing introduces an improved algorithm that is accurate but is computationally heavy in terms of anti-aliasing and visible surface detection in the continuous domain: the polygon fragments that fall within a pixel are considered and the visible area of each fragment is computed. The values so obtained are used as weights to compute the final pixel color.  
         [0147]     The algorithm, introduced by L. Carpenter in: “The A-Buffer, An Antialiased Hidden Surface Method”, Proceedings of the 11 th  Annual Conference On Computer Graphics And Interactive Techniques, pp. 103-108, 1984 combines the Z-Buffer with the effects of the antialiasing of the Catmull algorithm. In this A-Buffer algorithm, the exact computation of the area is approximated by using the discrete sub-pixel samples filtered with a box filter. This approach is easier to implement and quite efficient in hardware terms, since, e.g., a sub-fragment area defined with a 4×8 bit mask can be computed using simple logic operations (XOR) instead of the floating point needed by the analytical methods.  
         [0148]     The A-Buffer keeps memory-concatenated lists of internal fragments at each pixel. The fragments are ordered in a “front to back” order to manage transparency. To each fragment an object tag is associated for efficient processing: if two fragments are overlapped in the z direction, then they belong to same objects and are then merged to save memory.  
         [0149]     In addition to the minimum z value, for each fragment the method also keeps the maximum z value in order to approximate the rendering of intersecting surfaces. This approximation assumes that the fragments have opposite inclinations. Sometimes the method may thus fail and the rendering is not correct. After the fragments are processed, their contributions to the final pixel color are computed by using as weights the values of the sub-pixel mask.  
         [0150]     A disadvantage of the A-Buffer algorithm, related to its hardware implementation, lies in that there are no limits to the dimensions of the list of the potentially visible fragments per pixel. Additionally, the computational load associated with each pixel is not known at an early stage. While hardware implementations exist, the A-Buffer technique is mostly adapted for software implementation in high quality rendering and has been used successfully by the Computer Division of Lucasfilm Ltd. (today Pixar) to produce computer-generated films.  
         [0151]     The EXACT (EXact Area CalculaTion) algorithm proposed by A. Schilling et al. in: “Exact: Algorithm And Hardware Architecture For An Improved A-Buffer”, Proceedings of the 20 th  Annual Conference On Computer Graphics And Interactive Techniques, pp. 85-91, ACM Press, 1993, solves the A-Buffer problem related to the intersecting surfaces managing by introducing priority masks, which keep track of what surfaces are visible in each part of the sub-pixel mask. Priority masks are used to modify the coverage masks if needed. This approach manages correctly anti-aliasing in all cases of intersecting surfaces.  
         [0152]     The article by S. Winner, et al.: “Hardware Accelerated Rendering Of Antialiasing Using A Modified A-Buffer Algorithm”, Proceedings of the 24 th  Annual Conference On Computer Graphics And Interactive Techniques, pp. 307-316, AMC Press/Addison-Wesley Publishing Co., 1997, offers order-independent anti-aliasing and transparency, but can require multiple rendering passes. If there are n primitives per pixel that cannot be combined, n passes are required. The advantage of this modified A-Buffer algorithm is that it does not require extra memory and typically changes by 40% performance versus non-anti-aliased cases. To compute the fragment coverage a 4×4 sub-pixel mask is used with 8 sampling positions. The screen is partitioned in blocks of 16×32 pixels that are processed one at a time. This optimization of memory usage allows the Z-buffer and the frame buffer to be stored on chip leading to a more efficient computation, while compromising performance by an acceptable value when compared to arrangements where anti-aliasing is not used.  
         [0153]     N. P. Jouppi et al. propose in: “Z3: An Economical Hardware Technique For High-Quality Antialiasing And Transparency”, Proceedings of the 1999 Eurographics/SIGGRAPH Workshop On Graphics Hardware, pp. 85-93, AMC Press, 1999 a hardware technique of the A-Buffer type that uses a fixed quantity of added memory per pixel. This effectively performs anti-aliasing and makes transparency “order-independent”. Compared to the basic A-Buffer algorithm, Z3 manages correctly the intersecting transparent surfaces. Z3 keeps track only of a fixed quantity of sub-pixel fragments for each pixel. This makes hardware design easier since the memory requirements are well known and predictable. In addition to the normal z values, also the z slopes are saved in both x and y directions for each fragment. The name Z3 derives from the fact that tri-dimensional z information is used for each fragment. Keeping track of the gradients is advantageous since there are many cases where the z information can cause error in the anti-aliasing and visibility procedure.  
         [0154]     The R-buffer (Re circulating fragment buffer) technique, introduced in C. M. Wittenbrink et al.: “R-buffer: A Pointerless A-Buffer Hardware Architecture”, Proceedings of The ACM SIGGRAPH/EUROGRAPHICS Workshop On Graphics Hardware, pp. 73-80, ACM Press, 2001, is a hardware solution for order-independent transparency with an arbitrary number of levels, that supports A-buffer anti-aliasing. The R-buffer is a buffer managed as a circular FIFO, different from the Z-buffer and frame buffer, that stores all fragments not enclosed in the Z-buffer for a given pixel. These buffers are stored in the same on-chip memory of the hardware accelerator. Circular management of the buffer simplifies hardware design, since dynamic memory allocation is not supported. By adding a R-buffer, control and compare logic, some bits and an additional z value for each pixel, it is possible solve transparency problems that are order-independent. A state machine is associated with each pixel to compose transparent and opaque fragments. Depending on the opacity and depth of the fragment and the previous state of frame buffer for the same position, a fragment can be stored in the R-buffer, composed in the frame buffer, or not used. The final composition of fragments to compute transparency and anti-aliasing requires multiple rendering passes.  
         [0155]     Insofar as the associated hardware is concerned, the FAA-16× technology used by Matrox Parhelia-512 is based on the assumption that the percentage of pixels that composes the scene and that really needs anti-aliasing is between 5 and 10% on the average. FAA-16× cleverly identifies fragments to be sent to the anti-aliasing unit by inspecting only triangle edges and the computing coverage with 16× accuracy. A pixel can be classified as totally, partially or absolutely not covered. This information can be used to ascertain if a pixel must be rejected (if not covered by a triangle), written into the frame buffer (if covered) or sent to the anti-aliasing unit FAA-16× (if partially covered). All pixels that are not fragmented are immediately copied into the frame buffer, while fragmented data are saved in a special fragment buffer. The fragment buffer contains also a list of information items related to all the fragmented pixels, and in particular contains the coverage and color of sub-pixels. The FAA-16× unit updates continuously the fragment buffer to determine the final representation of scene fragments. The fragments are blended at the end and the resulting pixels are written into the frame buffer to complete the image.  
         [0156]     The ATI SmoothVision technology as introduced in 2001 uses groups of 16 samples, which are divided in a variable number of pixels, depending on the antialiasing effect needed. In the 2× mode, samples cover 8 pixels, in the 4× mode they cover 4 pixels and so on. Each pixel has 8 pre-programmed alternatives of “jittered” positions of sampling points, which give to the sampling pattern a pseudo-casual shape. In that way, images look better from the point of view of the human visual system, which is more sensitive to regular patterns. Being a super-sampling technique, SmoothVision performs texturing information look-up at each sampling point rather than one time per pixel.  
         [0157]     The RealityEngine graphic architecture by SGI, as described in K. Akeley: “Reality Engine Graphics”, Proceedings of 20 th  Annual Conference On Computer Graphics And Interactive Techniques, pp. 109-116, ACM Press, 1993 uses super-sampling with a 4×4 regular grid that is not effective from the point of view of performance and quality. RealityEngine selects between two sampling techniques: point sampling and area sampling.  
         [0158]     The InfiniteReality architecture, introduced in J. S. Montrym, et al. in: “Infinitereality: A Real-Time Graphics System”, Proceedings of 24 th  Annual Conference On Computer Graphics And Interactive Techniques, pp. 293-302, ACM Press/Addison-Wesley Publishing Co., 1997, has been designed with the purpose of implementing OpenGL fully in hardware. For anti-aliasing purposes it uses a sparse 8×8 super-sampling: each primitive is sampled at 1, 4 or 8 of the 64 possible positions. The z information of all sub-samples is stored to compute accurately the visible surfaces.  
         [0159]     In the 2× multi-sampling mode of GeForce 3 and 4 of NVIDIA, the image is rendered with two separate positions in the frame buffer. The sampling points are moved by an offset equal to half a pixel between two sampling passes. Those samples are then filtered to obtain final colors for the pixel. In the normal model, only one sample per pixel is used. In a quincunx mode, during one pass a sample is filtered with four other samples (taps) taken from the previous sampling pass. In the Accuview method, implemented in GeForce 4, the sampling positions are moved slightly to obtain a more accurate sampling. This technique exists also in a 4× version.  
       BRIEF SUMMARY OF THE INVENTION  
       [0160]     One embodiment of the present invention provides a graphic engine arrangement adapted to reduce aliasing artifacts in computer graphics.  
         [0161]     One embodiment of the present invention provides an OPENGL-ES compliant arrangement for embedded computing mobile devices able to achieve full screen anti-aliasing with no negative impact in terms of i) communication bandwidth between the graphic pipeline and the memory in comparison with arrangements where no anti-aliasing is used, and ii) memory size.  
         [0162]     One embodiment of the present invention is a method having the characteristics referred to specifically in the claims that follow. One embodiment of the invention also relates to a corresponding system, as well as a computer program product loadable in the memory of at least one computer and comprising software code portions for performing the method of the invention when the product is run on a computer.  
         [0163]     A preferred embodiment of the arrangement described herein involves:  
         [0164]     the generation, for each pixel, of two sub-pixels placed on triangle edges during the rasterization step,  
         [0165]     compression of those sub-pixels to the same size of an uncompressed pixel and stored in the main memory, and  
         [0166]     when the pixels are read from memory, at least one of:  
         [0167]     feeding the graphic pipeline for further processing after the pixels have been decompressed to the original uncompressed size, and  
         [0168]     feeding the graphic pipeline for further display after the pixels have been decompressed to retrieve the two sub-pixels at the uncompressed size.  
         [0169]     Preferably, median filters are applied to the sub-pixels to create a new plurality of sub-pixels. Then by combining in a linear or non-linear way, the de-compressed sub-pixels and the newly created sub-pixels, other new sub-pixels are generated and finally averaged to give the final anti-aliased pixels to be sent to the display unit. 
     
    
     BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS  
       [0170]     The invention will now be described, purely by way of non-limiting example, with reference to the annexed figures of drawing, wherein:  
         [0171]      FIG. 1  has already been described in the foregoing;  
         [0172]      FIG. 2  shows sample of image processed according to the arrangement described herein;  
         [0173]      FIGS. 3   a  and  3   b  show two possible alternatives in the definition of sub-pixels in the arrangement described herein;  
         [0174]      FIG. 4  is a block diagram of a write operation into a frame buffer at the output of a rasterizer in the arrangement described herein;  
         [0175]      FIG. 5  is a block diagram of a read operation in the arrangement described herein;  
         [0176]      FIG. 6  shows operation of a filtering unit in the arrangement described herein;  
         [0177]      FIG. 7  shows a grid with selected pixels, in the arrangement described herein; and  
         [0178]      FIG. 8  shows the process of composing the grid of  FIG. 7 . 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0179]     As an example, the rendering process of a set of triangles received in a generic order as submitted by an application will be considered.  
         [0180]     For each triangle, the rendering algorithm will compute the color of a set of pixels of the image that represents the shape and the colors of the original triangle.  
         [0181]     The image is assumed to be totally comprised of width-by-height pixels. Each triangle can color a higher or lower amount of these pixels. Each pixel in the image is a set of Red, Green, Blue and Alpha (RGBA) components each including n (in a known manner) bits (typically n=8).  
         [0182]     For each triangle the rasterization stage  113  of  FIG. 1  will compute from its 3 vertices the RGBA values of each pixel that fall inside the triangle (i.e., primitive) area.  
         [0183]     The arrangement described herein is based on the following steps.  
         [0184]     First of all, for each triangle received from the application and during the rasterization step, all the pixels that fall inside the triangle area are located as shown in  FIG. 2  with grey color;  
         [0185]     Then for each pixel (D 1  E 1  F 1  A 2  B 2  C 2  D 2  E 2  F 2  B 3  C 3  D 3  E 3  F 3  F 4  in example) two new sub-pixels are computed.  
         [0186]     These two sub-pixels are placed on the triangle edges and their positions are shown in  FIG. 3   a  as a preferred example defined by their (x,y) positions (0.5,0), and (1,0.5) respectively.  
         [0187]     The RGBA components of the two sub-pixels are computed by the rasterization component, and then the two sub-pixels are compressed with a compression method. Exemplary of such compression is the arrangement described in U.S. Patent Application Publication No. 2004/120594. This arrangement performs compression/decompression as a conversion between a first format and a second format, the second format being a format compressed via vector quantization. Vector quantization results from repeated application of a scalar quantizer with a quantization step determined in an adaptive way. The arrangement in question takes for each sub-pixel one of the colors (the same applies to the R, G, B and alpha components) coded at 8 bits and compresses two of them to a total of 8 bits.  
         [0188]     The compression of two RGBA sub-pixels will produce the same amount of bits of an uncompressed pixel. For example, if two sub-pixels (RGBA) are coded with 32 bits each (64 bits in total), then the compressed word will include 32 bits in total.  
         [0189]     A simple block diagram during the writing operation is shown in  FIG. 4 , where  200  indicates the frame memory block,  210  indicates the rasterizer block,  220  indicates the codec block.  
         [0190]     The rasterizer block  210  outputs:  
         [0191]     the address location  230  where the sub-pixels are written in the frame buffer  200 ;  
         [0192]     the pixel information  240  that can be comprised of two sub-pixels.  
         [0193]     The address  230  is used to find the location in the frame memory  200  where the two sub-pixels compressed by the codec block  220  are written. The compressed word is stored in the frame buffer at a location depending on the pixel coordinate on the image.  
         [0194]     When read, any location of the frame buffer is de-compressed in order to generate two sub-pixels each comprised of R, G, B and alpha components. Access to the frame buffer can occur basically for two reasons: in order to use such information for further display or for further processing by the graphic pipeline, such as, e.g., blending with semi-transparent objects. The word read is decompressed in order to generate two sub-pixel each made of R, G, B and alpha components.  
         [0195]      FIG. 5  shows a block diagram during the read operation from the frame. Read events can be dictated by further processing of sub-pixels in the graphic pipeline or for related display.  
         [0196]     The graphic pipeline  310  issues the address  330  where the pixel  340  is read from the frame buffer  300 . The decoder  320  decompresses the information read from the frame buffer and produces at the output two decompressed sub-pixels  340  and  345 . That information can be sent to the graphic pipeline  310  and/or the filtering unit  350 .  
         [0197]     The filtering unit  350  operates as shown in  FIG. 6 . For each pixel to be produced at the output will:  
         [0198]     borrow two sub-pixels from neighboring pixels,  
         [0199]     create, by using adaptive filters, new sub-pixels, named AA, BB, CC, DD, and  
         [0200]     apply a (median) filter to the sub-pixels above in order to compute the final pixel to be sent to the display.  
         [0201]     The latter step mentioned above is detailed below in pseudo-code language.  
         [0202]     Compute AA:  
                                                   If (distance(A,D)&lt;=distance(C,B)) and            (distance(A,D)&lt;=distance(A,C))           Then AA=median(E,N,C)           Else           If (distance(C,B)&lt;=distance(A,D)) and            (distance(C,B)&lt;=distance(A,C))           Then AA=median(E,N,A)           Else           If (([distance(A,C)+distance(B,D)]/2&lt;=distance(A,D)) and            ([distance(A,C)+distance(B,D)]/2 &lt;= distance(C,B))) or            (([distance(A,B)+distance(C,D)]/2&lt;=distance(A,D)) and            ([distance(A,B)+distance(C,D)]/2 &lt;= distance(C,B)))           Then AA=(A+C+N+E)/4                      
 
         [0203]     Where the distance between two pixels can be: 
 
square root of ((Red pixel1-Red pixel2)*(Red pixel1-Red pixel2)+(Green pixel1-Green pixel2)*(Green pixel1-Green pixel2)+(Blue pixel1-Blue pixel2)*(Blue pixel1-Blue pixel2)) 
 
or (absolute value(Red pixel1-Red pixel2)+absolute value (Green pixel1-Green pixel2)+absolute value (Blue pixel1-Blue pixel2)) 
 
         [0204]     The median value of three pixels is a new pixel composed by R, G, B components where:  
         [0205]     Red component is the median value of (Red pixel1, red pixel 2, red pixel3)  
         [0206]     Green component is the median value of (Green pixel1, green pixel 2, green pixel3)  
         [0207]     Blue component is the median value of (Blue pixel1, blue pixel 2, blue pixel3)  
         [0208]     Compute BB:  
                                                   If (distance(A,D)&lt;=distance(C,B)) and            (distance(A,D)&lt;=distance(A,B))           Then BB=median(F,G,B)           Else           If (distance(C,B)&lt;=distance(A,D)) and            (distance(C,B)&lt;=distance(A,B))           Then BB=median(F,G,A)           Else           If [distance(A,C)+distance(B,D)]/2&lt;=distance(A,D) and            [distance(A,C)+distance(B,D)]/2 &lt;= distance(C,B) or            [distance(A,B)+distance(C,D)]/2&lt;=distance(A,D) and            [distance(A,B)+distance(C,D)]/2 &lt;= distance(C,B)           Then BB=(A+F+B+G)/4                      
 
         [0209]     Compute CC:  
                                                   If distance(A,D)&lt;=distance(C,B) and distance(A,D)&lt;=distance(C,D)           Then CC=median(C,M,L)           Else           If distance(C,B)&lt;=distance(A,D) and distance(C,B)&lt;=distance(C,D)           Then CC=median(M,L,D)           Else           If [distance(A,C)+distance(B,D)]/2&lt;=distance(A,D) and            [distance(A,C)+distance(B,D)]/2 &lt;= distance(C,B) or            [distance(A,B)+distance(C,D)]/2&lt;=distance(A,D) and            [distance(A,B)+distance(C,D)]/2 &lt;= distance(C,B)           Then CC=(C+M+L+D)/4                      
 
         [0210]     Compute DD:  
                                                   If distance(A,D)&lt;=distance(C,B) and distance(A,D)&lt;=distance(B,D)           Then DD=median(B,H,I)           Else           If distance(C,B)&lt;=distance(A,D) and distance(C,B)&lt;=distance(B,D)           Then DD=median(H,I,D)           Else           If [distance(A,C)+distance(B,D)]/2&lt;=distance(A,D) and            [distance(A,C)+distance(B,D)]/2 &lt;= distance(C,B) or            [distance(A,B)+distance(C,D)]/2&lt;=distance(A,D) and            [distance(A,B)+distance(C,D)]/2 &lt;= distance(C,B)           Then DD=(B+D+I+H)/4                      
 
         [0211]     The third step mentioned above will now be detailed as follows, where two cases are envisaged:  
         [0212]     Case 1:  
         [0213]     the following sub-pixels A, B, C, D, AA, BB, CC, DD are each multiplied by a different coefficient, added together and the sum is divided by the sum of all coefficients used  
         [0214]     Case 2:  
         [0215]     the maximum (or minimum) distance between (A, D), (C, B), (AA, DD), (BB, CC) is found and then the final pixel if the average of the above sub-pixel pair found as maximum (or minimum). Now more details are given about Case 1.  
         [0216]     Referring to  FIG. 7 , P 1 , P 2 , P 3 , P 4  are the sub-samples located at the same place as were generated by the RGSS 4× algorithm.  
         [0217]     Step 3, case 1 will create them as linear combination of the sub-sample generated during step 2.  
         [0218]     For instance: 
 
 P 1=1/ d 1* A+ 1/ d 2* AA+ 1/ d 3* C /(1/ d 1+1/ d 2+1/ d 3) 
 
 P 2=1/ d 1* B+ 1/ d 2* BB+ 1/ d 3* A /(1/ d 1+1/ d 2+1/ d 3) 
 
 P 3=1/ d 1* D+ 1/ d 2* DD+ 1/ d 3* B /(1/ d 1+1/ d 2+1/ d 3) 
 
 P 4=1/ d 1* C+ 1/ d 2* CC+ 1/ d 3* D /(1/ d 1+1/ d 2+1/ d 3) 
 
 and the pixels sent to the display will be the average of P 1 ,P 2 ,P 3 ,P 4 . 
 
         [0219]      FIG. 8  shows how for each pixel position (i, j) the grid shown in  FIG. 7  is “composed by sub-pixel”.  
         [0220]     Another option to interpolate AA, BB, CC, DD pixels is to use for each of them, four very close and surrounding sub-pixels, and see what image surface shape they represent. There are fourteen of these, in example, corresponding to fourteen possible surface shapes.  
         [0221]     For example for AA interpolation, four values A=a, N=b, C=c, E=d can be considered in the following situations.  
                                                             Number   Shape   type   AA is equal to                                0   Flat   a a   a               a a       1   High   a a   a               a b       2   Line   a b   a or b               b a       3   Aligned   a a   (a + b)/2               b b       4   Low   a b   b               b b       5   Twisted   a a   (a + b)/2           edge   b c       6   Valley   a b   a or (a + b)/2               c a       7   Edge   a b   b               b c       8   Double   a b   (a + b)/2 or (b + c)/2           twisted   c b   b often better           edge       9   Twisted   a b   (b + c)/2           edge   c c       10   Ridge   a c   c or (b + c)/2               c b       11   Edge   a b   (b + c)/2               c d       12   Double   a b   (a + c)/2 or (b + d)/2 or           twisted   d c   (b + c)/2 is better           edge       13   Line   a c   (a + b)/2 or (c + d)/2               d b                  
 
         [0222]      FIG. 3   b  shows, by way of further example, a possible alternative to locate the two sub-sample positions per each pixel in any position of the pixel edge.  
         [0223]     Therefore, without prejudice to the underlying principles of the invention, the details and embodiments may vary, also significantly, with respect to what has been described, by way of example only, without departing from the scope of the invention as defined in the claims that follow.  
         [0224]     All of the above U.S. patents, U.S. patent application publications, U.S. patent applications, foreign patents, foreign patent applications and non-patent publications referred to in this specification and/or listed in the Application Data Sheet, are incorporated herein by reference, in their entirety.