[...]... signals that the value P0123 is ready It receives dx and dy from the ALU and outputs a bilinearly interpolated pixel after reading every fourth pixel P0 and P1 are first linearly interpolated using dx to compute P01 P2 and P3 are then interpolated using dx to compute P23 Finally, the vertical fraction dy is used to linearly interpolate P01 and P23 and obtain the bilinearly interpolated pixel P P is... cases, simulations were conducted varying g and h over {−.1, −.01, −.001, −.0001, 0, 0001, 001, 01, 1} Parameters a and e were set to 1, and the remaining parameters were set to 0 An error image was calculated for each method using the direct warping image as a reference, and the mean squared error (MSE) was computed from each error image The mean, median, and maximum values of mean squared error for... prevent overflow If a is the number of overflow bits and the reference sprite is P × Q pixels, then j ≥ log2 (MAX[P , Q]) + a (2.22) For example, if the reference and warped sprite are both 720 × 576 pixels and four overflow bits are used, then a = 4, k = 10, j = 10, and 24 total bits are required for calculating xn and yn 2.3.2 Memory Bandwidth Reduction Memory bandwidth is a concern for high-resolution sprites... , and c to obtain the coefficients in equations (2.11) and (2.12) x = ra x + rb y + rc y = rd x + re y + rf (2.11) (2.12) The horizontal position x is incremented from 0 to M − 1, where M is the width of the image For each value of x , y is obtained from the line table The current value of the x and y coordinates, xn and yn , are calculated from the previous values of the x and y coordinates, xn−1 and. .. = 0, d = 0, e = 1, f = 0, g = −0.1, and h = 0.002 case using a coastguard image The MSE for the constant denominator method was 0.00043 The error is so small that it can hardly be picked up by the eyes Table 2.2 shows a performance comparison ©2001 CRC Press LLC between the various approximation methods as g and h are varied between −0.1 and 0.1 for the coastguard image Table 2.2 Coastguard Mean Squared... divisions and only 14 multiplications per frame are therefore required for the affine transform 2.2.4 Simulation Results To compare the visual quality of the warping approximations, five methods were implemented in C++: direct warping, constant denominator, quadratic, quadratic Chebyshev, and cubic Chebyshev The methods were then used to warp the checkerboard image, which is a standard test image for... coefficients in equations (2.11) and (2.12) For each pixel, the coordinates of the corresponding pixel in the original frame are calculated and partitioned into an integer part (xl , yl ) and a fractional part (dx, dy) The integer part is output to the pixel cache while the fractional part is passed to the interpolation unit The pixel cache outputs pixels P0 , P1 , P2 , and P3 These are the four pixels... line ©2001 CRC Press LLC Because xn and yn are generally not integers, bilinear interpolation is used to calculate the value of the warped pixel using the four pixels nearest to (xn , yn ) in the original sprite The warped pixel P is calculated using the following three equations, as shown in Figure 2.2: P01 = P0 + (P1 − P0 ) ∗ dx P23 = P2 + (P3 − P2 ) ∗ dx P = P01 + (P23 − P01 ) ∗ dy (2.18) (2.19) (2.20)... Steps 1, 2, and 4 are best suited for floating point operations However, since steps 1 and 2 are performed once per frame, and step 4 is performed once per constant denominator line, they consume only a small fraction of the computational power Step 3 is also performed once per frame The additions in step 3 can be performed in fixed point Most of the computations are performed in steps 5 and 6, since... for steps 5 and 6, the number of floating point multiplications is reduced to at most 12M + 12N + 196 and the number of floating point additions becomes 4.5M + 4.5N + 199 The number of floating point divisions remains 1.5M + 1.5N + 35 Almost all of the operations are now fixed point 4.5MN fixed point multiplications and 11.5MN + 1.5M fixed point additions at most are required for steps 3, 5, and 6 Table