quantity definition symbol -------- ------------------------------------------ --------- physical dimensions Height and width of screen image in cm. ScreenX,Y logical dimensions " " " " " " " pixels PixelsX,Y pixel size Measured in cm/pixel. PixelSizeX,YClearly,

ScreenX = PixelSizeX * PixelsX [Eq. 1] (cm) = (cm/pix) * (pix)(and likewise in the y-direction). Taking the ratio of y to x:

ScreenY PixelSizeY PixelsY ------- = ---------- * ------- [Eq. 2] ScreenX PixelSizeX PixelsX physical pixel logical (display) (aspect) (data) ratio ratio ratio (often 0.75)

Note that one may model pixels as points or rectangles. If they are points, then what we call "pixel size" should properly be called "pixel spacing." In addition, Eq. 1 should read

Screen = PixelSpacing * (Pixels - 1)

since the image does not extend to the left of the first pixel or to the right of the last. (See Fig. 1)

In this document we will be thinking of pixels as rectangles with non-zero height and width. The alteration needed, for pixels as points, in subsequent equations is trivial, and in fact the difference it makes becomes negligible for large images.

How do we arrive at a good-looking CD-i display? Before we develop a general equation, we can work backward from the desired result to the starting image that will yield that result. In Figure 3, we have a CD-i display with a circle that looks properly round. (Artwork limitations make it look a bit tall. Consider it round.)

Moving back to the computer image that generates it, we see that the "circle" must look short. In fact, everything in the artwork must look vertically squashed. Well, this is not a convenient way for artists to work. How can the artist work in a domain where things look to his or her eye the way they will appear on the final CD-i display? This is where resampling comes in.

The original artwork can be created in a taller size and then squeezed to the proper size. In the process of squeezing, the image will get pre-distorted in such a way that it will look right again when displayed on the CD-i TV screen. The logical height of the original image should be the height of the final image multiplied by the pixel ratio of the final image. (The alert reader will note that the product has been rounded up to an even number, out of sheer obsessive-compulsion.) The general equation that gives the required starting height is Eq. 3b below.

How is this squeezing accomplished? Resampling can be described as the process of reconstructing a smooth curve that describes the variation of brightness & color from pixel to pixel and then finding values on that curve in between the original ones. This is illustrated in the figure below.

It is important to recognize that resampling and pixel ratio change are identical. One cannot resample an image (say to convert from a logical size of 200x100 to a logical size of 100x200) and then do aspect ratio correction to make circles look round again. Once you have resampled, the only remaining factor that bears on the proportion of the final image displayed is the pixel ratio of the playback device.

For example, imagine somebody asks us to convert a 6x4 image into a 12x4 image. We may do this by sampling the source image at twice the resolution horizontally, as shown in Fig. 5.

We obtain the desired 12-horizontal-pixel format, but when the image is displayed, the colored square has become elongated. Resampling has caused/is identical with pixel ratio distortion. To fix this we would have to change the pixel ratio of the destination display, or grab from a shorter rectangle in the source image, or display to a narrower rectangle in the destination display. In the real world the first option will not be available. The second option involves cropping some of the original image - which may not be acceptable, for example in a feature film. The third option will not use all the available display area ("letterboxing").

quantity definition symbol -------- ---------- ------ source pixel ratio Ratio of pixel size dy/dx Rs source logical grab rect Height & width in pixels of GrabX,Y region to be resampled result logical rect Height & width in pixels of ResultX,Y resampled image playback pixel ratio Pixel ratio on playback device Rp pixel ratio distortion > 1 if playback is too skinny; D < 1 if too fat.

Consider what the physical display dimensions of the grab rectangle are. From Eq. 2 the ratio of its height to width must be

GrabY grab rect physical ratio = Rs * -----. [Eq. 3a] GrabXThis rectangle may have to be squeezed or stretched (resampled) to make it fit into the playback rectangle, whose proportion is

ResultY playback rect physical ratio = Rp * -------. [Eq. 3b] ResultXIt is the

GrabY ResultY Rs * ----- = Rp * ------- [Eq. 4a] GrabX ResultXwhich is the same as

Rp * ResultY GrabY = ------------ * GrabX [Eq. 4b] Rs * ResultXwhich is the same as

GrabY / GrabX Rp = Rs * -----------------. [Eq. 4c] ResultY / ResultXNotice that Eq. 4b gives the starting (grab) height of an image to be converted as illustrated in Fig. 3 and described above. Eq. 4c is often described by saying that the pixel ratio is being

pixel height pixel ratio = ------------ pixel width pixel clock frequency * scan line period 3 = ---------------------------------------- * - [Eq. 5] number of active lines 4How do we understand this in terms of the foregoing? What's the 3/4 all about? Observe that

PixelsX = pixel clock frequency * scan line period (approximately) PixelsY = number of active lines (approximately)Substituting that (and pretending the approximate equalities are exact) yields

PixelsY 3 pixel ratio = ------- * - PixelsY 4or

3/4 = pixel ratio * logical ratioThis is the same as Eq. 2. Thus Eq. 5 says that the CD-i player draws a picture on the screen whose proportions are 3:4 in all cases.

The equalities are not exact for reasons unknown to this author. The Green Book says to use 242.5 as the active line count for NTSC and 287.5 for PAL, instead of 240 and 280 respectively. Similarly, the product of pixel clock rate and active scan line period gives a pixel count larger than 384, or its high-res double value of 768.

To further confuse the issue, the numbers for pixel clock, scan line period, adn pixel ratio in § V.2.4 of the Green Book are *incorrect*. The correct numbers are:

format pixel clock freq scan line period pixel ratio ------ ---------------- ---------------- ----------- 525 TV 15.1049 MHz 52.6555 µ s 1.230 [Green Book: 1.19] 625 15.0 MHz 52.0 µ s 1.017 [Green Book: 1.05]We have empirically measured the NTSC ratio to be 1.235 ± 0.003 and the PAL/SECAM ratio to be 1.019 ± 0.003.

Before encoding, correct the (pixel) aspect ratio of the digital image to be one of the following ratios, and set the pel aspect ratio index field of the Sequence header correspondingly.In fact, this is what the Green Book intends the passage to mean; unfortunately the pixel ratios given in the Green Book are wrong. The recommended ratios as written are:

1.19 [1.2015] if no aspect ratio distortion is to occur in decoders producing a 525-line output; in this case, 625-line decoders will have approximately 13% distortion.(The MPEG standard does not provide for these exact aspect ratios, so the values in brackets should be used.)1.05 [1.0695] if no aspect ratio distortion is to occur in decoders producing a 625-line output; in this case, 525-line decoders will have approximately 12% distortion.

1.12 [1.0950] if aspect ratio are to be equalized over both types of decoders.

We see that the given ratios are the same as the (incorrect) base case ratios. The correct ratios - the pixel ratios produced by the CD-i DV decoder board - are the same as the correct base case ratios. This is obvious (after the fact) when one realizes that the DV cartridge and the base-case video both use the same pixel clock. Thus video that is being prepared for DV should be converted to pixel ratios of 1.23 for NTSC, 1.017 for PAL, or the geometric mean, 1.118, for a compromise.

The pixel ratios for Video CD (White Book) are precisely 0.9 times those for CD-i. Hence the values are 1.1069 for NTSC and 0.9157 for PAL/SECAM. These numbers are supposed to be identical to the D1/CCIR-601 pixel ratios (see below); the White Book specifies them as such. The CD-i NTSC value is a slight deviation from the standard.

The format data for D1 are as follows:

full used format logical dim. logical dim. pixel clock pixel ratio ------ ------------ ------------ ----------- ----------- NTSC 720x486 712x486 13.5 MHz 1.095 PAL/SECAM 720x576 703x576 13.5 MHz 0.9157Following is a list of recommended resampling choices for various combinations of D1 source and CD-i playback formats. The horizontal and vertical resampling ratios are given for each example, as well as the resulting pixel ratio. The numbers that correspond to digital filter complexity are the denominators (number following the colon) in the resampling ratio. Picture distortions of less than 2% are typically not noticeable. Distortions of 4% will be noticed by film professionals. Distortions of >6% will be noticed by a lot of people.

**NTSC/24 D1 -> NTSC CD-i**

691 9:5 384 x --> x Rp: 1.217 (1.1% distortion) 480 2:1 240 713 13:7 384 x --> x Rp: 1.253 (1.9% distortion) 476 17:8 224

690 15:8 368 x --> x Rp: 1.251 (1.7% distortion) 480 15:7 224 690 15:8 368 x --> x Rp: 1.233 (0.2% distortion) 472 19:9 224

690 15:8 368 x --> x Rp: 1.031 (1.3% distortion) 574 19:9 272 699 19:10 368 x --> x Rp: 1.017 (0% distortion) 574 19:9 272

702 2:1 352 x --> x Rp: 1.095 (1.0% distortion) 480 2:1 240

702 2:1 352 x --> x Rp: 0.9157 (0% distortion) 576 2:1 288

713 13:7 384 x --> x Rp: 1.244 (1.2% distortion) 371 19:9 176

713 13:7 384 x --> x Rp: 1.035 (1.8% distortion) 436 21:10 208