Matrix concatenation in a graphics display system

Abstract: To avoid the overflow problem, in a graphics display system which employs matrix concatenation for coordinate transformation and matrix element precision equal to that inherent to the system, which can cause an out-of-bounds location of a data element, number translation shift factors are introduced for the last row of the matrix which when used to operate on matrix elements. This will maintain the elements within the physical boundaries of the graphics base by preventing overflow. The disclosed modification can be implemented in microcode in a commercially availably graphics display system such as the IBM 5080 Graphics System.

Description: The present invention relates to modifications to the way a graphics display system handles matrix concatenation in order to avoid error when the precision of the matrix elements equals that inherent in the system. The IBM 5080 Graphics Display System as described in IBM publication GA23-2012-0, IBM 5080 Model 2 Principles of Operation, performs concatenation on transformation matrices in which there is a single shift factor for a number of matrix shift elements, for example, 12 elements for a three-dimensional space. The prior art IBM 5080 transformation matrix calculation does not provide a translation shift factor as does the instant invention to control overflow conditions resulting from concatenation of a number of matrices. The IBM Technical Disclosure Bulletin Vol. 27 No. 1A June 1984 pages 139-142 shows coordinate transformation and scaling in a graphics display processor. The method described comprises coordinate transformation using a transformation matrix in 2D mode or 3D mode. The present invention provides a graphics display system in which the processing of data internally in order to aggregate the display data elements collectively defining the picture to be output involves matrix transformations, single and/or in sequence and in which the precision of the matrix elements is that inherent to the system, wherein error resulting from overflow which may arise when transformation matrices are concatenated is avoided by: forming a first matrix comprising a plurality of matrix elements, at least one scale shift factor and a plurality of translation shift factors; forming a second matrix comprising a plurality of matrix elements, at least one scale shift factor and a plurality of translation shift factors; calculating results of matrix concatenation for a first n rows of each matrix, to form a third matrix; normalising a first n rows of the third matrix; and calculating translation terms comprising a last row of the third matrix to complete the third matrix which comprises translated graphics data characterised in that the step of calculating translation terms comprises multiplying a last row of the first matrix with predetermined elements of the second matrix and storing a result in an intermediate storage; calculating an exponent from the translations shift factor and the scale shift factor; and repeating the multiplying step for each element in the last row of the first matrix with elements in the second matrix and adding results of the multiplications to achieve translated data. The present invention will be described further by way of example with reference to an embodiment thereof as illustrated in the accompanying drawings in which: Fig. 1 is a block diagram of a raster graphics system which may perform the error avoidance of the present invention. Fig. 2 is a block diagram of a display processor of the system of Fig. 1 for performing the error avoidance of the present invention. Fig. 3 is a logical data flow diagram for the display system shown in Figs. 1 and 2. Fig. 4 is a schematic diagram showing the results of matrix concatenation on graphics data in a large coordinate space. Fig. 5 is a schematic diagram of the calculation of the concatenation of two matrices in accordance with the present invention. Fig. 6 is a simplified flow chart of the process of matrix concatenation in accordance with the present invention. Fig. 7, which includes Figs. 7(a), 7(b), 7(c), 7(d), and 7(e), is a flow chart of the matrix multiplication in accordance with the present invention. Fig. 8 is a flow chart of normalisation of matrix elements in the resulting third matrix. Fig. 9 is a flow chart showing calculation of translation terms for the resulting matrix. Fig. 10 is a block diagram of an add subroutine employed with the calculation flow shown in Fig. 9. A Graphics Display System uses a special form of matrix for coordinate transformation (see the section below), which is a 4x3 matrix (4 rows and 3 columns). The first three rows are 9 fractional numbers sharing one scalar factor, and the last row of 3 numbers are integers. In some applications, as the result of concatenation of several matrices, the result matrix element will overflow. To solve this precision problem, three independent shift factors are introduced below for the last row of the matrix. These three elements, together with the first three rows of the matrix, constitute a new matrix. (see the section on example below). Based upon the shift factors of the result matrix, if any of the translation terms exceed 16 bits, then the clipping boundary can be adjusted, that is, the user data can be adjusted to avoid overflow (see Figure 4). Matrix Format Elements of a Matrix M are represented in the form mij - element of i-th row and j-th column. Transformation Matrix A Transformation Matrix may be of the following format: (3D matrix is used here for discussion. A 2D matrix can be defined as a subset of a 3D matrix). The first nine elements m11, m12, .., m33 are fractional numbers of the form s.xxx xxxx xxxx xxxx The numbers m41, m42, m43 are numbers of the form sxxx xxxx xxxx xxxx The above numbers are represented in the 2's complement format. The number sf is the shift factor. The range of sf is from 0 to 12. The last three terms m41, m42, and m43 are the translation terms. Coordinate Transformation Given a point (s,y,z) in integer format, the result of the matrix multiplication by M is newx = (x*m11 + y*m21 + z*m31)*(2**sf) + m41 newy = (x*m12 + y*m22 + z*m32)*(2**sf) + m42 newz = (x*m13 + y*m32 + z*m33)*(2**sf) + m43 Matrix Concatenation Given two Matrices M, N: The resultant matrix Q = M*N is given by q11 = m11*n11 + m12*n21 + m13*n31 q12 = m11*n12 + m12*n22 + m13*n32 q13 = M11*n13 + m12*n23 + m13*n33 q21 = m21*n11 + m22*n21 + m23*n31 q22 = m21*n12 + m22*n22 + m23*n32 q23 = m21*n13 + m22*n23 + m23*n33 q31 = m31*n11 + m32*n21 + m33*n31 q32 = m31*n12 + m32*n22 + m33*n32 q33 = m31*n13 + m32*n23 + m33*n33 sf = sf1 + sf2 q41 = (m41*n11 + m42*n21 + m43*n31)(2**sf2) + n41 q42 = (m41*n12 + m42*n22 + m43*n32)(2**sf2) + n42 q43 = (m41*n13 + m42*n23 + m43*n33)(2**sf2) + n43 Note: The shift factor sf3 must be adjusted if the number of leading zeros of all the nine terms are greater than zero. PHIGS/Matrix The PHIGS Matrix, is on the following format: (3D matrix is used here for discussion. A 2D matrix can be defined as a subset of a 3D matrix) The first nine elements (m11, m12, .., m33) are fractional numbers of the form s.xxx xxxx xxxx xxxx The numbers (m41, m42, m43) are numbers of the form sxxx xxxx xxxx xxxx The number (s)f is the shift factor and the range of sf is from -16 to 16. To simplify implementation, the shift factor = -512 (X'FE00) is used to indicate the Identity Matrix. The numbers (t1f, t2f, t3f) are the shift factors for the three matrix elements (m41, m42, m43). That is, the translation terms are m41*(2**t1f) m42*(2**t2f) m43*(2**t3f) The range of each of (t1f, t2f, t3f) is from -16 to 16. Coordinate Transformation Given a point (x,y,z) in integer format, the result of the matrix multiplication by M is newx = (x*m11 + y*m21 + z*m31)*(2**sf) + m41*(2**t1f) newy = (x*m12 + y*m22 + z*m32)*(2**sf) + m42*(2**t2f) newz = (x*m13 + y*m23 + z*m33)*(2**sf) + m43*(2**t3f) Matrix Concatenation Given two PHIGS Matrices M, N: The resultant matrix Q = M*N is given by q11 = m11*n11 + m12*n21 + m13*n31 q12 = m11*n12 + m12*n22 + m13*n32 q13 = m11*n13 + m12*n23 + m13*n33 q21 = m21*n11 + m22*n21 + m23*n31 q22 = m21*n12 + m22*n22 + m23*n32 q23 = m21*n13 + m22*n23 + m23*n33 q31 = m31*n11 + m32*n21 + m33*n31 q32 = m31*n12 + m32*n22 + m33*n32 q33 = m31*n13 + m32*n23 + m33*n33 sf3 = sf1 + sf2 (See note 1) The above calculation is the same as that done for the matrix. The other three terms of the matrix elements q41*(2**t1f3) = m41*n11*(2**t1f1)*(2**sf2) + m42*n21*(2**t2f1)*(2**sf2) + m43*n31*(2**t3f1)*(2**sf2) + n41*(2**t1f2) q42*(2**t2f3) = m41*n12*(2**t1f1)*(2**sf2) + m42*n22*(2**t2f1)*(2**sf2) + m43*n32*(2**t3f1)*(2**sf2) + n42*(2**t2f2) q43*(2**t3f3) = m41*n13*(2**t1f1)*(2**sf2) + m42*n23*(2**t2f1)*(2**sf2) + m43*n33*(2**t3f1)*(2**sf2) + n43*(2**t3f2) The translation shift factors (t1f3, t2f3, t3f3: see Note 2) are calculated in the next section. (*) Note 1. The shift factor sf3 must be adjusted if the number of leading zeros of all the nine terms are greater than zero. Note 2. The translation shift factors must be calculated from the three 32-bit interim numbers q41, q42, q43; by using the leading zeros or ones counts. (see the section on Calculating Matrix Elements) Comparison with a Prior Art Matrix The difference between this matrix and a prior art matrix such as the IBM 5080 matrix can be summarised in the following: An Example Considering an integer in 16-bit 2's complement format, ranging from -32768 to 32767, and the following concatenation of 2 matrices: Because of the 16-bit format, using cos45 = sin45 = .7071 in the prior art procedure, the product will become which is, since (M*N) will be computed incorrectly with this precision instead of, as one would get if one had more than 16-bit precision, the correct M*N: Applying the procedure disclosed herein to the above two matrices M and N, the matrix will have the new format The matrix will have the new format and the disclosed procedure will yield the product M*N which represents the correct result - Adjusting the Number Ranges of Graphics Data In case the terms of the result matrix of the concatenation would cause overflow, because the shift factors are too large, then the system will adjust the number range of the graphics data: In the above example, the system could automatically divide the graphics data by 2, thus avoiding the overflow in the processing. (see Fig. 4). Implementation The IBM 5080 is used here to illustrate an implementation. (see IBM Graphics System 5080 Mod2 Principles of Operation GA23-2012-0). A Raster Graphics System The raster graphics system illustrated in Fig. 1 consists of the following major components: 1. System Control Processor 2. Host Communication Interface Processor 3. Display Processor 4. Hardware Rasteriser - Vector Generator 5. Video Pixel Memory 6. System Memory Brief Overview of the Functions of the Major Components. 1. System Control Processor The System Control Processor is a general purpose processor that has master control of the System. the System Control Processor is responsible for servicing all attached Graphics I/O devices (except the light pen and Display Monitor). Coordinating the associated processing with the Display Processor. The System Control Processor interfaces with the host via Host Communication Interface. 2. Host Communication Interface The Host Communication Interface provides the serial interface of the System to the host. 3. Display Processor The DP is responsible for executing the graphics orders in the Display Storage Program, residing in the system memory and is concerned mainly with the generation of the image that will appear on the Display Monitor. It has the following functions: Decoding graphics orders and executing non-drawing order; e.g. book keeping and control. Performs the transformation and clipping function to the geometric primitives: lines, characters, polygons, etc. Preparing the following geometric objects for display: lines, characters, markers, filled polygons, by preprocessing and feeding the data to the Vector generator and Video Pixel Memory 4. Vector Generator Vector generator is a Hardware Implementation of the Bresenham Line Generating Algorithm, which takes the end points of a vector (line) as input, and generates pixels in the Video Pixel Memory as output for display. 5. Video Pixel Memory Video Pixel Memory consists of 8 1k by 1k bit planes, which supports 256 colours simultaneously via colour look-up tables. The image stored here will be displayed in the Monitor. The Logical Data Flow For the logical data flow of the graphics system, see Fig. 3. 1. The Application Program is loaded from the host via the Host communication Interface to the System Memory; 2. The System Control Processor preprocesses the data (depending on the work required), then interrupt the DP; 3. The Display Processor then processes the data; 4. The data is then passed to the VPM for display directly or via the Vector generator. Transformation and Clipping Transformation and Clipping are controlled by the contents of Attribute Register 19, which consist of 5 bits 00MP TDC0 (For the bit definition and the control flow, see Fig. 4). In the following, a vector (line) will be used as an example and there are three stages to the operation: 1. Transformation For the end points of a vector, which is in the 16-bit fixed number format, the matrix multiplication is done in the (-32k,32k-1), 16-bit x, y, and z space. 2. Clipping Using the two end points of a vector (line) and clip to the clipping box specified by the users. (The computation is done is 16-bit space). 3. Mapping Mapping the contents inside the clipping box (in 3D) or clipping window (in 2D) to a viewport in the screen specified by the user. (The screen coordinate is (0,4k-1) by (0,4k-1), which then mapped to the 1k by 1k screen). Display Processor The Display Processor is a microprogrammed system. It fetches the data from the memory and sends the data out to the raster display via the Vector Generator, which is a rasteriser. It takes the line segment end points coordinates as input, and generates pixels in the video pixel memory. The main ingredients of the system (see Fig. 2) are: 1. Sequencer, e.g. AMD2910A; 2. 72-bit wide writable control store; 3. 16-bit ALU, e.g. 4 bit-slice AMD2903; 4. a 16×16 multiplier with 32-bit accumulator, e.g. WTL2010; 5. the barrel shifter is a custom-made chip, which does arithmetic multi-bit shifts for 32-bit data in one cycle. It also detects the leading 0's/1's count - the number of consecutive leading bits which are equal in the high 16-bit data register. 6. a clipper (for checking rival accept/reject); 7. 4k x 16 scratch ram; 8. Logic for microcode next address coming from the content of scratchpad ram registers - indexed addressing. The microprogram is stored in the writable control store. Calculating Matrix Elements This section covers the calculation of the translation terms and their shift factors by using the multiplier/accumulator, barrel shifter, leading 0's/1's counter in the Display Processor. (see Fig. 3). Calculating the Three New Matrix Elements in Matrix Concatenation The item numbers in the following subroutines are referred to the operational components of Fig. 3. Here we cover the calculation of: q41 = m41*n11*(2**t1f1)*(2**sf2) + m42*n21*(2**t2f1)*(2**sf2) + m43*n31*(2**t3f1)*(2**sf2) + n41*(2**t1f2) (The terms q42 and q43 can be handled in the same way). Subroutine to compute the new translation number: 1. Use the Multiplier/Accumulator (item #4) to calculate the 32-bit number m41*n11 and store it in two registers aH, aL; (item #3) 2. Store the number t1f1 + sf2 + 1 in a register expa; (item #3) (see note 1) 3. Use the Multiplier/Accumulator (item #4) to calculate the 32-bit number m42*n21 and store it in two registers bH, bL, (item #3) 4. Store the number t2f1 + sf2 + 1 in a register expb; (item#3) (see note 1) 5. Call subroutine ADD; 6. Use the Multiplier/Accumulator (item #4) to calculate the 32-bit number m43*n31 and store it in two registers bH, bL; (item #3) 7. Store the number t3f1 + sf2 + 1 in a register expb; (item #3) (see note 1) 8. Call subroutine ADD; 9. Store n41 in a register bH; (item #3) and 10. Store 0 in bL; (item #3) 11. Store the number tsf2 in a register expb; (item #3) 12. Call subroutine ADD; 13. Store the leading 0's/1's count (item #6) of aH in a register leada; (item#3) 14. Shift aH and aL to the left by (leada - 1); (item #6) 15. expa <-- expa - (leada - 1); (item #3) (the count leada - 1 is either 1 or 0) 16. q41 <-- aH; (items #5,#3) 17. t1f3 <-- expa. (items #5,#3) Subroutine ADD Subroutine to Add two numbers represented by 32-bit mantissa and 16-bit exponent. (ah aL)*(2**expa) + (bH bL)*(2**expb) Subroutine ADD 1. Store the leading 0's/1's count (item #6) of aH in a register leada; (item #3) 2. Shift aH and aL to the left by (leada - 2); (item #6) 3. expa <-- expa - (leada - 2); (item 33) 4. Store the leading 0's/1's count (item #6) of bH in a register leadb; (item #) 5. Shift bH and bL to the left by the (leadb - 2); (item #6) 6. expb <-- expb - (leadb - 2); (item #3) (see Note 2) 7. Check whether expa > = expb ? (item #3) 8. If expa > = expb a. Shift bH, bL to the right by expa - expb: 9item 36) b. Add the two 32-bit numbers aH, aL and bH, bL together; (item #3) c. Store the result in aH and aL. (item #3) 9. Else (expa < expb) a. Shift aH, aL to the right by expb - expa; (item #6) b. Add the two 32-bit numbers aH, aL and bH, bL together; (item #3) c. Store the result in aH and aL. (item #3) d. expa <-- expb (item #3) (End of Subroutine Add) After the calculation, q41, which is generated by (aH aL)*(2**expa), is contained in three registers - Note Note 1. The output of the multiplier/accumulator is of the format sxxx xxxx xxxx xxxx x.xxx xxxx xxxx xxxx to adjust to the format sxxx xxxx xxxx xxxx .xxxx xxxx xxxx xxxx since the exponent of the first number is greater than that of the second number by 1, the exponent incremented by 1. Note 2. (count - 2) is used to avoid possible overflow in the addition when expa = expb. Mapping to 5080 Matrix Format In order to use the 5080 transformation capability, the new Matrix must be mapped to the 5080 matrix format. For a Matrix M a new variable tsf is defined as tsf = max (t1f, t2f, t3f) When a PHIGS matrix is mapped to the 5080 to the 5080 matrix element format, both the transformation matrix and the clipping boundaries are changed to prevent possible data overflow caused by the concatenation of matrices. We first adjust the shift factors, translation terms of the matrix and the clipping boundaries. The decision table is listed below: After the adjustment, if the new shift factor sf is negative, then each of the first nine terms m11, m12, m13, m21, m22, m23, m31, m32, m33 is multiplied by 2**sf, and sf is set to zero. PHIGS Matrix Manipulation There are two applications of the new Matrix: 1. As a utility function used by the application program; 2. As a utility function used in computing the PHIGS transformation environment. Utility Function There are 256 graphics program registers defined for the 5080 display program. Each register is of 16-bits, and denoted by GRn, where n is an integer from 0 to 255. There is a 5080 graphics order for matrix concatenation of the following format: opcode a - address b - address c - address which concatenate two matrices A and B - A*B, and the result is in matrix C. All these matrices are located in the graphics purpose registers - a-address is the beginning address (register address) of the first matrix element of A - b-address is the beginning address (register address) of the first matrix element of B - c-address is the beginning address (register address) of the first matrix element of C - in the following manner - Computing the PHIGS Transformation Environment Associated with each drawing primitive are the following matrices in the PHIGS/5080 interface: 1. View Matrix 2. Global Matrix 3. Local Matrix 4. Normalisation Matrix Each of the first three matrices consists of 16 elements while the fourth one consists of 7 elements - shift factor, 3 translation terms, and three translation shift factors. This is expanded to a matrix in the 16 element format before it is used in the computation so that the input data string ff, m41, m42, m43, tf1, tf2, tf3 will result in the following matrix Order of Concatenation The order of concatenation of the above 4 matrices is (Normal) -> (Local) -> (Global) -> (View) The result is saved in 16 registers called TraMatrix Furthermore, Global and View Matrices do not change very oftenso that it is advantageous to retain the concatenation of (Global)(View) in 16 registers called TemMatrix Two bits are correspondingly defined in the control registers - Tra-bit (to indicate a change has occurred in TraMatrix) Tem-bit (to indicate a change has occurred in TemMatrix) and the use of these improves the performance in the traversal time. TraMatrix is the matrix mapped to the 5080 matrix format. Data Flow for PHIGS Matrices Manipulation For each PHIGS drawing primitive - polylines, polymarkers, annotation text, etc; the matrix must be (re-) computed before the processing of the draw data. The data flow of the matrix manipulation is as follows: 1. If Tra-bit is off, then exit (the current matrix is not changed); 2. If Tra-bit is on, then check the Tem-bit; and reset Tra-bit; a. If Tem-bit is off, then skip this step; b. If Tem-bit is on, then multiply the two matrices Global and View, and put the result in TemMatrix; and reset Tem-bit; 3. Multiply the three matrices Normal, Local, and TemMatrix and put the result in TraMatrix; 4. Map TraMatrix to the 5080 matrix format, adjusting the clipping boundaries and viewpoint if necessary.

Claims: A graphics display system in which, the processing of data internally in order to aggregate the display data elements collectively defining the picture to be output involves matrix transformations, single and/or in sequence and in which the precision of the matrix elements is that inherent to the system, wherein error resulting from overflow which may arise when transformation matrices are concatenated is avoided by: forming a first matrix comprising a plurality of matrix elements, at least one scale shift factor and a plurality of translation shift factors; forming a second matrix comprising a plurality of matrix elements, at least one scale shift factor and a plurality of translation shift factors; calculating results of matrix concatenation for a first n rows of each matrix, to form a third matrix; normalising a first n rows of the third matrix; and calculating translation terms comprising a last row of the third matrix to complete the third matrix which comprises translated graphics data characterised in that the step of calculating translation terms comprises multiplying a last row of the first matrix with predetermined elements of the second matrix and storing a result in an intermediate storage; calculating an exponent from the translations shift factor and the scale shift factor; and repeating the multiplying step for each element in the last row of the first matrix with elements in the second matrix and adding results of the multiplications to achieve translated data. A system as claimed in claim 1, wherein the plurality of matrix elements comprises rotation elements. A system as claimed in claim 1, wherein the plurality of matrix elements comprises scaling elements. A system as claimed in claim 1, wherein the plurality of matrix elements comprises shearing elements. A system as claimed in any preceding claim, wherein the step of calculating results of matrix concatenation for first n rows of each matrix, further comprises the steps of: multiplying a first row of a first matrix by a first column of a second matrix. testing for an overflow result; dividing result by a predetermined constant if an overflow occurs; shifting the result to multiply by a second predetermined factor and load a higher order predetermined number of bits of the result to a first element in the third matrix; multiply the first row of the first matrix by the second column of the second matrix; repeating the overflow test step, the divide step for an overflow and the shift step as above; loading the high order predetermined number of bits to a second element in the third matrix; repeating the steps as above on a first row of the first matrix and a third column of the second matrix to calculate a third element of the third matrix; repeating the above steps on a second row of the first matrix and a first column of the second matrix to calculate a fourth element of the third matrix; repeating the above steps on a second row of the first matrix and a second column of the second matrix to calculate a fifth element of the third matrix; repeating the above steps for each succeeding row of the first matrix with each succeeding column of the second matrix to calculate each succeeding element of the third matrix; and calculating a scale factor for the third matrix as a function of scale factors of the first and second matrix. A system as claimed in any preceding claim, wherein the normalising step further comprises the steps of: determining a minimum of leading zeros in any element of the third matrix; and shifting each element in the third matrix by the minimum number of leading zeros. A system as claimed in any preceding claim, further comprising the step of adjusting clipping boundaries based on values of a shift factor and translation shift factors of the third scale matrix.

Inventors: FIORE ANTHONY MICHAEL, HEMPEL BRUCE CARLTON, LAIB GREGORY DONALD, LIANG BOB CHAO-CHU, FIORE, ANTHONY MICHAEL, HEMPEL, BRUCE CARLTON, LAIB, GREGORY DONALD, LIANG, BOB CHAO-CHU
Applicants: IBM, INTERNATIONAL BUSINESS MACHINES CORPORATION
Application year: 1987
Main classification: G06F7
IPC codes: G06T11, G06F17, G06F7