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of a particular object are not visible from a particular angle or particular viewpoint. In this scribe we will describe the object-space method and image space method. We

will also discuss Algorithm based on Z-buffer method, A-buffer method, and Scan-Line Method.

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- 1. CS 362: Graphics Scribe Hidden Surface Removal By: Rajkumar Singh (09010138) Akhilesh Shah (09010147) Course Instructors: Professor Pinaki Mitra Professor Samit Bhattacharya Department of Computer Science and Engineering Indian Institute of Technology, Guwahati 23rd April, 2012 1
- 2. Abstract In Computer Graphics, Hidden surface determination also known as Visible Surface determination or hidden surface removal is the process used to determine which surfaces of a perticular object are not visible from a perticular angle or perticular viewpoint. In this scribe we will describe the object-space method and image space method. We will also discuss Algorithm based on Z-buﬀer method, A-buﬀer method, and Scan-Line Method. Keywords: A buufer, Z-buﬀer, depth buﬀer, Refresh buﬀer, Hidden Surface Re- moval(HSR). 1 Hidden Surface removal Visible Surface detection or Hidden surface removal algorithms are mainly classiﬁed according to whether they deal with object deﬁnations directly or the images that are obtained after projections. Broadly there are two types of methods shown as follows. 1.1 Object-Space Method In object-space method we took whole object under consideration. This method com- pares objects and parts of objects to each other within the scene deﬁnation to determine that which surfaces, as a whole are not visible from a certain viewpoint and so that can mark the surfaces as visible which are visible. 1.2 Image-Space Method In Image-Space algorithm, visibility is decided by point by point at each pixel position on the projection plane. Here we consider the projected image of the object not the object itself like object-space method. Visibility of unvisibility is decided by pixel by pixel or point by point traversal. Note: Most visible surface/ Hidden Surface removal algorithms use image-space methods, although object-space methods can also be used eﬀectively to locate visible surfaces or to removing hidden surfaces. Eg. Line Diplaying algorithm uses object-space methods for displaying lines in a wire-frame. 2 Hidden Surface Removal Algorithms Algorithms are describes below. 2
- 3. 2.1 Back-Face Detection A fast and simple object-space method for identifying the back face of poly-hedron is based on the Inside-Outside (given a point is inside or outside of a plane/surface). Let Plane Parameters are A, B, C, D then a point (x, y, z) will be inside to a plane or on the plane if Ax + By + Cz + D ≤ 0 When an inside point is along the line of sight to the surface the polygon must be a back face (it means we are inside that face and can not see its front from our viewing position). We can simplify the test by taking a vector let say N which is normal to a polygon surface. Let the components of N are (A, B, C). V is a vector in the viewing direction from the camera or eye position. As shown in Figure 1. The polygon will be back face if dot product of its normal vector and viewing vector is greater than zero.means V · N ≥ 0 If the object descriptions have been converted to projection coordinates and our viewing direction is parallal direction is parallel to the viewing Zv axis, then V = (0, 0, Vz) and V · N = VzC So that we only need to consider the sign of C, the z component of the normal vector N. Figure 1: Polyhedron In a Right handed viewing system with viewing direction along the negative zv axis, the polygon is a back face C < 0. Also viewer can not see any face whose normal has z component C = 0, since the viewing direction in the given example is grazing that polygon. This in general, we can label any face if its normal vector has a z-component value C ≤ 0. Some Observation based on the z-componet value. C < 0 Back face of the polyhedron C = 0 Grazing the polygon, Not visible from Viewing position C > 0 Face Visible from viewing position. Similar methods can be used for Left handed viewing position systems. In Left 3
- 4. Handed Viewing systems back faces are identiﬁed by C ≥ 0 when the viewing dorec- tion is along the positive zv axis. By examining parameter C for the diﬀerent planes deﬁning an object, all the back faces can be identiﬁed. For a single convex polyhedron this test identiﬁes all the hidden surfaces because each surface in polyhedron is either completely visible or completely hidden. And also if a scene contains only nonoverlapping convex polyhedra, then again all hidden surfaces are identiﬁed with the back method. For concave polyhedron more tests need to be carried out to determine whether there are additional faces that are totally or partially obscured by other faces. 2.2 Depth-Buﬀer Method This approach is based on image-Space method.In this method For detecting visible surfaces we compares surface depth at each pixel position on the projection plane. This procedure is also called as Z-buﬀer method, Since object depth is measured from the view plane along the z axisof a viewing system. Here Each surface of the scene is pro- cessed separately, one point time across the surface. The method is usually applied to scenes containing only polygon surfaces, because depth values can be computed very quickly and the method is easy to implement. But the method can be applied to non- planar surfaces as well. Below is the ﬁgure in which comparison between actual scene and z-buﬀer scene is shown. Figure 2: Comparison between Z-buﬀer and Actual Scene With the object descriptions converted to projection coordinates, each (x, y, z) position on a polygon surface corresponds to the orthographic projection point (x, y) on the view plane. Therefore, for each pixel position (x, y) on the view plane, object depths can be compared by comparing z values. Figure 3 shows the comparison between depths. this ﬁgure shows surfaces at varying distance along the orthographic projection line from postion (x, y) in a view plane taken as the xvyv plane. Surface S1 is the closest at this position, so its surface intensity at (x, y) is saved. We can implement the depth-buﬀer algorithm in normalized coordinates, so that z val- ues range from 0 at the black clipping plane to zmax at the front clipping plane. The plane of zmax can be set either to 1 (for unit cube) or to the largest value that can be 4
- 5. stored on the system. As implied by the name of this method, two buﬀer areas are required. A depth buﬀer is used to store depth values for each (x, y) position as surfaces are processed, and the refresh buﬀer stores the intensity values for each position.Initially, all positions in the depth buﬀer are set to 0 (minimum depth), and the refresh buﬀer is initialized to the background intensity. Each surface listed in the polygon tables is then processed, one scan line at a time, calculating the depth (z depth) at each (x, y) pixel. The calculated depth is then compared with the value stored in the depth buﬀer. If it is greater than the value in depth buﬀer then this new calculated value is stored in the depth buﬀer. And the surfce intensity is calculated at the same position and stored in the same lo- cation in the (x, y) in the refresh buﬀer. Figure 3: Polygons at diﬀerent depth In short whatever algorithm we discussed above is discussed below in small points. 1. Initialize the Depth buﬀer and refresh buﬀer with minus iniﬁnty and background intensity respectively for all buﬀer positions (x, y). depth(x, y) = −∞ refresh(x, y) = Ibackground 2. For each position on each polygon surface, compare depth values to previouslystored valuesin the depth buﬀer to determine visibility. • Calculate the depth z for each position (x, y) on the polygon. • if z > depth(x, y), then set depth(x, y) = z, refresh(x, y) = Isurface(x, y) where Ibackground is the value for the background intensity, and Isurface(x, y) is the projectedd intensity value for the surface at pixel position (x, y). 5
- 6. Equation of plane is: Ax + By + Cz + D = 0 (1) Depth values for a surface position (x, y) are calculated from the plane equation for each surface: z = −Ax − By − D C (2) For any scan line, adjacent hrizontal positions across the line diﬀer by 1, and a vertical y value on an adjacent scan line diﬀers by 1. if the depth of position (x, y) has been determined to be z, then the depth z of the next position (x + 1, y) along the scan line is obtained from given equation. z = −A(x + 1) − By − D C (3) ⇒ z = z − −A C (4) Figure 4: Depth at adjacent position The Ratio −A C is constant for each surface, so succeding depth values across a scan line are obtained from preceding values with a single addition. We ﬁrst determine the y-coordinate the y-coordinate extends of each polygon, and process the surface from tha topmost scan line. Starting at a top vertex, we can recrsively calculate x position down a left edge of the polygon as x = x − 1 m , where m is the slope of the edge in the ﬁgure shown below. Figure 5: scan line and a polygon 6
- 7. Depth values of the edge are obtained recursively as z = Ax −B(y−1)−D C x = y m − b m y − 1 = mx + b x = y−b−1 m x = x − 1 m ⇒ z = −(Ax+By+D) C + A m +B C ⇒ z = z + A m + B C (5) For Vertical left edge since m → ∞ hence we calculate z using given equation z = z + B C (6) Note: This Depth-buﬀer alias Z-buﬀer method works only for Opague surfaces, but for Transparent surfaces we have to store the vertical scan line as well as modifying z-buﬀer method and hence this method(modiﬁcationof z-buﬀer method) is known as A-buﬀer method. 2.3 A-Buﬀer Method The A-Buﬀer method represents an anti-aliased, area-averaged, accumulation buﬀer method. It was developed for implementation in the surface-rendering system called REYES (”Ren- ders Everything You Ever Saw”). A drawback of Depth-buﬀer method is that it deals only with opaque surfaces and cannot accumulate intensity values for more than one surface as required for displaying transparent surfaces. In A-buﬀer method, each position in buﬀer can reference a linked list of surfaces. Thus, multiple surface intensity can be considered at each pixel position and object edges can be anti-aliased. Each position in A-buﬀer has two ﬁelds: • Depth ﬁeld - stores positive or negative real number. • Intensity ﬁeld - stores surface intensity information or a pointer value. Figure 6: surface overlap 7
- 8. Positive depth ﬁeld signiﬁes that the number stored at that position is the depth of single surface overlapping the corresponding pixel area. The intensity ﬁeld stores the RGB com- ponent of surface colour at that point and the percent of pixel coverage. Negative depth ﬁeld signiﬁes multiple surface contribution to the intensity. The intensity ﬁeld stores a pointer to linked list of surface data. Figure 7: Organization of A-buﬀer pixel position Data for each surface in the linked list includes: • RGB intensity component • opacity parameter (percent of transparency) • depth • percent of area covered • surface identiﬁer • other surface rendering parameters • pointer to next surface Scan lines are processed to determine surface overlaps of pixel across the individual scan lines. Surfaces are subdivided into polygon mesh and clipped against the pixel boundaries. Opacity factors and percent of surface overlaps can be used to calculate the intensity of pixel as an average of the contributions from the overlapping surfaces. 2.4 Scan Line Method This is image-space based method.This is used for removing hidden surface and this is an extension of the scan-line algorithm for ﬁlling polygon interiors and here in this algorithm we deal with multiple surfaces, not just a single surface. To determine the visible surfaces, all polygon surfaces intersecting the scan line are examined. Across each scan line, surface nearest to the view plane is determined by making depth calculations. After that, intensity 8
- 9. value for that position is entered into refresh buﬀer. Edge Table and Polygon Table are set up for various surfaces. Edge table contains:- • The x-coordinate of the end with the smaller y-coordinate. • The y-coordinate of the edges other end. • The x-increment . • The polygon identiﬁcation number indicating the polygon to which the edge belongs. Figure 8: Edge Table Entry Polygon table contains:- • The coeﬃcients of the plane equation. • Shading or colour information for the polygon. • An in-out boolean ﬂag initialized to FALSE and used during the scan line processing. Figure 9: Polygon Table Entry In the below shown ﬁgure it is shown that how we actually locate the visible portions of surfaces for pixel position along the line. The active list for scan line 1 contains information from the edge table for edges AB, BD, EH and FG. For positions along this scan line between edges AB and BD, only the ﬂag for surface S2 is on. Therefore, no depth calculation is required, and intensity information for surfaces S2. Figure 10: Scan Line Method for Hidden Surface Removal 9
- 10. For the given example we will show the Active edge table entries. The example is as follows. Figure 11: Scan Line Method for Hidden Surface Removal Scan Line Entries a AB AC [In out ﬂag of ABCisonBetweenABandAC b ABAC ABCon DEEF DEFon c ABD ABCon EBC ABC, DEFon EF DEFon c+1 ABD ABCon EBC ABC, DEFon EF DEFon c+2 ABBC ABCon DEEF DEFon Table 1: Active Edge Table for Scan line Method in Aove Example In the above example Between Edge DF and BC since Flags for both ABC and DEF are On, So we do not have to perform the depth calculation for these two. The above described algorithms(Scan line method does not work for the scenes in which there are cyclically overlapping of surfaces. One of such case is shown below. Figure 12: Cyclically Overlapping Surface 10
- 11. 2.5 Depth-Sorting Method or Painter’s Algorithm This method uses both Image-space method and objects-space method. This method for solving the hidden-surface removal is also known as Painter’s Algorithm because of the similarity between the Painting creation and this algorithm execution. In this Algorithm we perform these given basic tasks. 1. All the surfaces are sorted in the order of decresing depths. This sorting is carried out in both image-space and Object-space. For this we use the deepest point on each surface for comparison. 2. Surfaces are scan converted in order, First we start with the surface of largest depth. This scan conversion of the polygon surfaces is performed in image-space or the pro- jected space. By refering the ﬁgure shown below. We perform certain tests that are as follows for each surface that overlaps with S (Shown in ﬁgure). Figure 13: Cyclically Overlapping Surface On performing following tests. if any of starting three tests is true no reasoning is necessary for that surface. The tests are listed in order of increasing diﬃculty. Test 1: The bounding rectangle of the two surfaces on xy−plane do not overlap, Test 2: Surface S us Completely behind the overlapping surface relative to the viewing position. 11
- 12. In the above shown ﬁgure it can be checked by the sign of the endpoints. Test 3: The overlapping surface is completely in front of S relative to the viewing positions. Test 4: The projection of the two surfaces on the view plane do not overlap. 3 Quetions Question 1 In what case is scan line algorithm not applicable? Scan line Algorithm is not applicable in case of Intersecting surfaces or the cases of cyclic overlap as shown in ﬁgure below. Figure 14: Cyclically Overlapping Surface Question 2 Give an example of back face detection alogrithm. Consider an example of a cube as shown in ﬁgure below. n and V are the unit normal vector to any surface and unit vector in viewing direction respectively. In this example, 12
- 13. V = -k, nA = k, nC = -k Figure 15: Example for Back Face Detection Algorithm If dot product of n and V is greater than or equal to zero, the surface is a back face. (n ·V) ≥ 0 In given example, nA · V = −1,nC · V = 1,nx · V = 0,for surfaces x = B, D, EandF. Hence, in the con- sidered example, only surface B is visible. Question 3 Give an example of Z-buﬀer algorithm (show the calculated depth and color). Consider the ﬁgure as shown below. Surface A (Red coloured) overlaps surface B (Blue coloured). Consider that surface A is at depth 3 and surface B at depth 5. Now, we prepare a table containing depth and intensity of any point for a surface determined by polygon EFGH. Figure 16: Example for Z-buﬀer Algorithm 13
- 14. Figure 17: Table showing Depth and Intensity of each point Question 4 Give an example of A buﬀer Algorithm. Consider the ﬁgure as shown below. Transparent surface A (Red coloured) overlaps trans- parent surface B (Blue coloured) and surface C (Green coloured). Consider that surface A is at depth 3, surface B at depth 5 and surface C at depth 7. Now, we determine intensity at various points of considered polygon EFGH. Figure 18: Example for Z-buﬀer Algorithm For point A, +ve real number is stored in depth buﬀer ﬁeld to signify one surface overlap. Intensity stored in intensity buﬀer ﬁeld. However, for point G that has multiple surface overlaps, -ve real number is stored in depth ﬁeld. The intensity ﬁeld points to next node of the linked list that stores depth value and intensity of Surface A. Next pointer is to surface B and so forth. 14
- 15. Figure 19: Example for A-buﬀer Algorithm Question 5 Explain Depth sorting method. • Both image and object space operations. • Perform the basic functions. • Surface are sorted in order of decreasing depth. • Surface are scan converted in order, starting with the surface of greatest depth. • Often referred as painters algorithm. • Test listed in the order of increasing diﬃculty. • Surfaces do not overlap. • Projections of the two surfaces on to the view plane do nor overlap. 15

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