Dr Pavan Chakraborty IIIT-Allahabad

Transcription

1 GVC-43 Lecture - 5 Ref: Donald Hearn & M. Pauline Baker, Comuter Grahics Foley, van Dam, Feiner & Hughes, Comuter Grahics Princiles & Practice Dr Pavan Chakraborty IIIT-Allahabad

2 Summary of line drawing so far. Exlicit form of line Inefficient, difficult to control. Parametric form of line. Exress line in terms of arameter t DDA algorithm Imlicit form of line Only need to test for side of line. Bresenham algorithm. Can also draw circles.

3 Summary of Aliasing. Samling theory tells us aliasing is caused by frequencies being resent above the Nyquist limit. Ideal solution : band-ass filter to remove high frequencies. Fourier transform tells us the transform of a bandass filter is a sinc function. Convolution theory tells us we can convolve with a sinc function in the satial domain instead. A sinc function is an imractical filter.

4 Helicoter Blades Why do helicoter blades sin backwards on TV? htt:// 4

5 Aliasing Aliasing: a high-frequency signal masquerading as a low frequency Actual (high-frequency) signal Samling Interval Samled (aliased) signal Caused by insufficient samling (samling rate is too small) 5

6 More Examles of Aliasing Strobe light on driing water: Temoral aliasing Sokes on a rotating wheel: Temoral aliasing Moiré atterns: Satial aliasing

7 More Examles Original Rendered Anti-aliasing Textures Aliased Original Rendered Claymation Photograh Aliased Photograh 7

8 Aliasing and Line Drawing We draw lines by samling at intervals of one ixel and drawing the closest ixels Results in stair-steing Samling Interval Samling Interval 8

9 Idea: Anti-aliasing Lines Make line thicker Fade line out (removes high frequencies) Now samle the line 9

10 Anti-aliasing Lines Solution 1 Unweighted Area Samling: Treat line as a single-ixel wide rectangle Color ixels according to the ercentage of each ixel covered by the rectangle.

11 Antialiasing Two ways of antialiasing Gather all the values into the ixels -Loo round the ixels. -Used for comlex scenes. -Cast out rays, convolve result into ixel (Pixel Grid Imulse) x line

12 Antialiasing Two ways of antialiasing Scatter values into the ixels -Loo along the line. -If line is delta function, we just swee the ixel filter along the line (Line Pixel) x imulse

13 Antialiasing lines. Obvious : Need a grey level dislay in order to remove aliasing. Convolve line with filter function for ixel Box filter area samle Convolution with conical filter function. Price to be aid : trade off satial resolution Line aears more blurred, it s exact osition is no longer as well defined. In ractice : contrast of lines much reduced. 1 ixel

14 Antialiasing by area samling. Convolve line with box filter function Draw line as thin rectangle. Calculate area of square ixel covered by line Problem : Equal areas contribute equal intensity, regardless of distance from line centre Small area in the ixels centre contributes as much as a small area at the ixels edge.

15 Weighted area filtering. Convolution with a conical filter. Easy to comute, symmetrical. Lines are same distance from ixel centre, but area of ixel covered is very different in the square case

16 Weighted area filtering. Diameter is ixels, so overla occurs Ensures all of the grid is covere Area is normalised Only need to know distance from ixel centre to line Guta-Sroull algorithm.

17 Solution 1: Unweighted Area Samling Pixel area is unit square Constant weighting function Pixel color is determined by comuting the amount of the ixel covered by the line, then shading accordingly Easy to comute, gives reasonable results One Pixel Line

18 Solution : Weighted Area Samling Treat ixel area as a circle with a radius of one ixel Use a radially symmetric weighting function (e.g., cone): Areas closer to the ixel center are weighted more heavily Better results than unweighted, slightly higher cost One Pixel Line 18

19 Guta-Sroull Algorithm Calculate ixel intensity by comuting distance from ixel center to line using the midoint line algorithm. y 1 N E Line to draw m D θ v y x E x 1

20 Guta-Sroull Algorithm (cont) d is the erendicular distance from E to the line How do we comute it? θ Δy Δx D θ v θ E cosθ cosθ D D v x x v x x y y

21 Guta-Sroull algorithm. Calculate distance using features of mid-oint algorithm D v cosφ dx vdx dy NE dx dy M D v Angle E

22 Guta-Sroull Algorithm (cont) Recall from the midoint algorithm: So ( ax by ) 0 f ( x, y) c y ax c b y 1 y v y x N E θ D m x 1 E Line to v draw and y v a( x 1) b c v y v y Therefore v a( x 1) b c y

23 Guta-Sroull Algorithm (cont) From revious slide: a( x 1) c v b So bv a( x 1) c y by y 1 y v y x N E θ D m x 1 E Line to v draw From the midoint comutation, b x So: 1 v x a x 1) by c f ( x 1, ( y )

24 Guta-Sroull Algorithm (cont) From the midoint algorithm, we had the decision variable 1 d f m) f ( x, y ) ( 1 Going back to our revious equation: v x f ( x 1, y ) a( x 1) by 1 1 a( x 1) b( y ) b 1 f ( x 1, y b ) f ( m) b d b d x c y 1 y v y c x N E θ D m x 1 E Line to v draw

25 Guta-Sroull Algorithm (cont) So, D x v x y d x x y y 1 y v y D N E E D u v 1-v 1v D lower The denominator is constant. x x 1 Since we are blurring the line, we also need to comute the color at the ixels above and below the E ixel D u (1 v) x x y D lower (1 v) x x y 5

26 If the NE ixel had been chosen: x d b d b m f b y x f c b y b x a c y b x a y x f x v ) ( ) 1/ 1, ( ) ( 1) ( 1) ( 1) ( 1) 1, ( 1 1 ) (1 y x x v D lower ) (1 y x x v D u y x x d y x x v D

27 Guta-Sroull Algorithm Summary Comute midoint line algorithm, with the following alterations at each iteration: At each iteration of the algorithm: If the E ixel is chosen If the NE ixel is chosen Udate d as in the regular algorithm Color the current ixel according to D Comute (1 v) x D u d x x y Color uer and lower ixels accordingly x y D D d x x y D lower (1 v) x x y

28 Guta-Sroull algorithm. Calculate distance using features of mid-oint algorithm D v cosφ dx vdx dy See Foley Van-Dam Sec D d dx dx dy d is the decision variable.

29 Filter shae. Cone filter Simly set ixel to a multile of the distance Gaussian filter Store recomuted values in look u table Thick lines Store area intersection in look-u table.

30 Solution 3: Suer-samling Divide ixel u into sub-ixels :, 3 3, 4 4, etc. Sub-ixel is colored if inside line Pixel color is the average of its sub-ixel colors Easy to imlement (in software and hardware) No anti-aliasing Anti-aliasing ( suer-samling)

31 Many Tyes of Suersamling Grid Random Poisson Disc Jittered

32 Foreground and Background Comute ercent of ixel covered by line, Line color is c l Background color is c b Pixel color is comuted as color c l (1-) c b

33 Polygon Anti-aliasing To anti-alias a line, we treat it as a rectangle Anti-aliasing a olygon is similar. Some concerns: Micro-olygons: smaller than a ixel Suer-samling: There may still be olygons that sli between the cracks 33

34 Summary of antialiasing lines Use square unweighted average filter Poor reresentation of line. Weighted average filter better Use Cone Symmetrical, only need to know distance Use decision variable calculated in Bresenham. Guta-Sroull algorithm.

35 Convolution theorem In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution is the oint-wise roduct of Fourier transforms. In other words, convolution in one domain (e.g., time domain) equals oint-wise multilication in the other domain (e.g., frequency domain). Versions of the convolution theorem are true for various Fourier-related transforms. Let and be two functions with convolution. (Note that the asterisk denotes convolution in this context, and not multilication. The tensor roduct symbol is sometimes used instead.) Let denote the Fourier transform oerator, so and are the Fourier transforms of and, resectively. Then where denotes oint-wise multilication. It also works the other way around: By alying the inverse Fourier transform, we can write: Note that the relationshis above are only valid for the form of the Fourier transform shown in the Proof section below. The transform may be normalised in other Indian ways, Institute in of which Information case Technology constant - Allahabad scaling factors (tyically