Vertical sampling rate decimation and line-offset decimation of colour difference signals

Signal Processing 16 (1989) 109-127 North-Holland

109

VERTICAL S A M P L I N G RATE D E C I M A T I O N AND LINE-OFFSET DECIMATION OF C O L O U R D I F F E R E N C E S I G N A L S B. GIROD* lnstitut fiir Theoretische Nachrichtentechnik und lnformationsverarbeitung, Universitiit Hannover, Callinstrafle 32, 3 Hannover 1, Fed. Rep. German),

W. GEUEN** Forschungsinstitut der Deutschen Bundespost beim Fernmeldetechnischen Zentralamt, Am Kavalleriesand 3, 61 Darmstadt, Fed. Rep. Germany Received 2 November 1987 Revised 12 April 1988

Abstract. For a digital transmission of broadcast television signals with rates of 140 MBit/s or 34 MBit/s 1:2 vertical decimation and 1:2 line-offset decimation of the colour difference signals R-Y and B-Y are considered. A covenient method for the calculation of the spectrum of a three-dimensional signal, that has been sampled with a regular periodic lattice, is given. Different anti-aliasing filters and interpolation filters for vertical decimation and for line-offset decimation are compared with respect to the resulting picture quality. For line-offset decimation and interpolation diagonally separable filters with a diamond-shaped passband are designed. Subjective tests have been carried out to assess the picture quality that is achieved with different chrominance decimation schemes. It is found that a balanced colour resolution is not a major quality criterion of the h u m a n observer. A 4: l : l - s y s t e m is preferred to other 1:2 intrafield decimation schemes for the colour difference signals. When compared to a 4:2 : 2-system, the picture quality deterioration by a 4:1 : 1-system is very small. All decimation schemes investigated provide a picture quality that is clearly better than with today's PAL television system.

Zusammenfassuug. Fiir eine digitale Ubertragung von Rundfunkfernsehsignalen mit l~lbertragungsraten von 140 MBit/s oder 34 MBit/s werden vertikale Unterabtastung und Zeilenoffsetunterabtastung der Farbdifferenzsignale um den Faktor 2 untersucht. Es wird eine besonders einfache Methode zur Berechung des Spektrums eines dreidimensionalen Signals angegeben, das a u f einem regelm~il3igen periodischen Raster abgetastet ist. Fiir die vertikale Unterabtastung und die Zeilenoffsetunterabtastung werden verschiedene Anti-Aliasing-Filter und lnterpolationsfilter im Hinblick a u f die Bildqualitiit verglichen. Fiir Zeilenoffset-Unterabtastung und-lnterpolation werden diagonal separierbare Filter mit rautenf6rmigem DurchlaBbereich entworfen. Subjektive Tests wurden durchgefiihrt, u m die Bildqualitiit zu bestimmen, die mit den verschiedenen Farbunterabtastverfahren erreicht wird. Dabei stellt es sich heraus, dab eine ausgewogene Farbaufl6sung kein wichtiges Qualitfitskriterium fiir den menschlichen Beobachter darstellt. Ein 4: l : l - S y s t e m wird niimlich anderen l:2-1ntrafield-Unterabtastverfahren vorgezogen. Verglichen zu einem 4 : 2 : 2 - S y s t e m sind die Qualit~itseinbuBen durch ein 4: l : l - S y s t e m sehr gering. Alle untersuchten Unterabtastverfahren fiihren zu einer Bildqualitiit, die der des heutigen PAL-Fernsehsystems deutlich iiberlegen ist. R6sum6. Pour la transmission num6rique des signaux de t616vision de diffusion avec des taux de 140 Mbit/s ou 34 Mbit/s, la d6cimation verticale par un rapport 1 : 2 et la d6cimation de d6calage de ligne par le m~me rapport des signaux de diff6rence de couleurs R-Y et B-Y sont consid6r6es, Une m&hode qui convient pour le calcul du spectre d ' u n signal tridimensionnel qui a 6t6 6chantillonn6 avec une trame p6riodique r6guli~re est donn6e. Diff6rents filtres anti-repliement et d'interpolation pour la d6cimation verticale et pour la d6cimation de d6calage de ligne sont compar6s sur la base de la qualit6 de l'image r6sultante. Pour le d6calage de ligne et l'interpolation des filtres diagonalement s6parables et ~t bande passante en forme de diamant sont 61abor6s. Des tests sujectifs ont 6t6 faits pour juger la qualit6 de l'image obtenue par les diff6rentes m6thode de d6cimation de la chrominance. I1 est trouv6 q u ' u n e r6solution couleur balanc6e n'est pas un crit~re important p o u r * Currently at the MIT, Media Lab., Cambridge, MA, U.S.A. ** Moved to SEL Research, Lorenzstr. 10, 7000 Stuttgart 40, Fed. Rep. Germany.

110

B. Girod, W. Geuen / Decimation of colour signals l'observateur humain. Un syst/~me 4 : 1 : 1 est prdf6r6 aux autres m6thodes de d6cimation intrachamps 1:2 pour les signaux de diff6rence de couleurs. Compar6e/~ celle d'un syst~me 4 : 2 : 2, la d6t6rioration de qualit6 de l'image par un syst~me 4 : 1 : 1 est tr~s faible. Touteg les m6thodes de d6cimation examin6es donnent une qualit6 d'image qui est nettement meilleure qu'avec le syst~me de t61~vision actuel PAL.

Keywords. Colour television, decimation/interpolation, line-offset decimation, subjective tests. I. Introduction

For a digital transmission of broadcast television signals with rates of 140 MBit/s or 34 MBit/s [3] sampling rate conversion of the video signal components is an important means of data compression. For the chrominance components of the video signal a horizontal as well as a vertical loss of resolution compared to the digital studio signal [6] might be acceptable and, accordingly, horizontal and vertical sampling rate decimation might be legitimate [25, 2]. Throughout this contribution we denote the process of periodically keeping samples of the waveform at certain spatio-temporal positions and dropping all other samples in between as decimation. The converse, i.e. a periodic generation of additional samples at new sampling positions or the reconstruction of the space-time continuous waveform, will be denoted as interpolation. Accordingly, a sampling rate conversion by a rational factor, which sometimes is considered as a general type of interpolation, in our terms consists of a cascade of interpolation and decimation. The investigations described in this paper focus on • 1 : 2 vertical decimation and interpolation of the colour difference signals R-Y and B-Y; • 1:2 line-offset decimation and interpolation of the colour difference signals R-Y and B-Y. Probably, the first author who widely applied sampling theory to television signals was Pearson in 1975 [16]. Several papers have been published since that time that focus on different aspects of video signal sampling. The fundamentals of sampling theory have been applied to television signals in [20, 24, 5, 4]. The design of filters for a sampling rate conversion of video signals has been considered in [5, 18, 15]. Maximally fiat filters with equal passband and stopband widths are given in Signal Processing

[20, 15, 9]. Vertical 1:2 sampling rate decimation of the colour difference signals has been studied in [11,21]. In [12] filter characteristics for the colour difference signals I and Q are proposed based on an investigation of visibility thresholds for ringing and aliasing. Offset sampling for television signals has been suggested in [20, 24, 19, 10]. A complete theory of twodimensional offset sampling has been presented by Mersereau [13]. This paper is organized as follows: Section 2 describes how the spectrum of a three-dimensional signal, which has been sampled with a regular periodic lattice, can be calculated. Sections 3 and 4 consider the special demands of anti-aliasing filtering and interpolation filtering for video signals. Section 5 summarizes results for a vertical sampling rate decimation of colour difference signals and compares different anti-aliasing filters and interpolation filters with respect to the resulting picture quality. Section 6 summarizes the corresponding results for line-offset decimation with diagonally separable filters. Section 7 reports the results of subjective tests that have been carried out to assess the picture quality achieved by different chrominance decimation-interpolation schemes. 2. Three-dimensional sampling with regular, periodic sampling lattices

Consider a space-time continuous threedimensional signal l(x, y, t). This signal is sampled with a sampling lattice grid(x, y, t): s(x, y, t ) = l(x, y, t) • grid(x, y, t).

(1)

The multiplication in the space-time domain corresponds to a convolution in the frequency domain,

S(~o~, %, ~o,) = L(oJ.., ~o.v, ~o,) * GRID(oJx, W,,, ~o,),

(2)

B. Girod, W. Geuen / Decimation of colour signals where S(tox, toy, to,) is the Fourier transform of the sampled signal:

with

L=pJX,

LLLL

s(x, y, t)

GRID(w~, w,., w,)

x e -j~x J%Y j,o,, dx dy dt.

1

(3)

As grid(x, y, t) consists of periodic repetitions of delta-functions, the same holds for its Fourier transform GRID(w~, wy, w,). Thus, the spectrum of the sampled signal contains replications of the signal baseband at higher spatial and temporal frequencies. In order to avoid aliasing, i.e. an overlap between the baseband and its replications, we have to know GRID(w~,wy, w,). In what follows, a simple method for calculation of GRID(tox, toy, to,) will be given. Let grid(x, y, t) be a regular sampling lattice, i.e. all sampling positions are elements of the set

-IPx. P,.zl x

S

S

i---oc j

~

G([i]L,[j]M,[k]N)

cc k - - 3 e

2wi to

X~

2wj w

2~rk'~

y---h-' '---if-/' -y At l

L-1

6(I,m,n)=

M

f.

~

i-O

j-O

I N---1

S ~,(i,j,k) k

0

× WE W ~ W~',

Y,t=n.

T},

(4)

(9)

for

I=O, 1 , . . . , L - 1 ; m = O , where l, m, n are integer numbers. Furthermore, let grid(x, y, t) be a periodic sampling lattice, i.e.

1,...,M-1;

n=0, 1,..., N-l, with

g r i d ( x + lP,, y + mPv, t+ nP,) = grid(x, y, t),

(8)

where [a]b denotes the operation " a modulo b", and G ( i , j , k ) is the L × M × N Discrete Fourier Transform (DFT) of the array of lattice element coefficients:

M={x,y, tlx=l'X, y=m.

M=Pv/Y , N=P,/T.

With these definitions, the Fourier transform of the sampling grid can be calculated as

S(to~, toy, to,) =

111

(5)

WL =

e

j(2Tr/L),

WM e-J(2-~/M), =

W N ~-- e - J ( 2 ~ / N ) .

for all integer numbers I, m, n. We then define the lattice element g(x, y, t) as grid(x, y, t),

forO~
g(x, y, t)= 0,

(6)

else,

and the lattice element coefficients ~,(i,j, k) such that the lattice element g(x, y, t) can be described by a superimposition of L. M . N shifted deltafunctions: L-1

g(x,y,t)= Z i

M--1 N

E 0 j-O

I

5~ ~,(i,j,k) k=t)

× 6(x - iX, y - j Y , t - kT),

(7)

Thus, GRID(wx, Wy, to,) is a regular, periodic lattice with lattice element coeffÉcients that are the DFTs of the lattice element coefficients of grid(x, y, t). The idea of expressing the Fourier transform of the sampling grid conveniently by means of a DFT of lattice element coefficients is originally due to Pirsch [ 17]. A straightforward method for calculation of GRID(to~, wv, w,) results: (a) Identify L, M, N and the array of lattice element coefficients. (b) Compute the 3D-DFT (equation (9)) of the lattice element coefficients. (c) Plug in the result in equation (8). Vol. 16, No. 2, February 1989

112

B. Girod, W. Geuen / Decimation of colour signals

In the Appendix four examples are given that compute the Fourier transform of the sampling grid for (1) an interlaced sampling grid; (2) an interlaced sampling grid after I : 2 vertical sampling rate decimation; (3) an interlaced sampling grid after 1:2 lineoffset decimation; (4) an interlaced sampling grid after 1:2 lineoffset decimation without frame reset.

3. Interpolation filtering

In order to interpolate a signal, baseband replications in the spectrum of the sampled signal have to be removed. For a multi-dimensionally sampled signal there is an infinite number of possible basebands which, together with their replications, entirely fill the frequency space. A baseband therefore has to be agreed upon, which we refer to as the agreed baseband (ABB). In order to construct the ABB, we have to distinguish different schemes of interpolation filtering: • intrafield filters that operate merely spatially; n fixed spatio-temporal filters; • motion-adaptive filters that switch between an intrafield filter for moving parts and a spatiotemporal filter with higher spatial resolution for non-moving parts of the image sequence; • motion-compensating filters that take into account the displacement of moving objects. For intrafield filtering, the ABB has to be constructed such that the baseband can be separated from its replications by a spatial filter. Baseband replications have to be considered without respect to their temporal frequency. Fixed spatio-temporal filtering leads to a considerable blur of moving objects, if temporal Iowpass-filtering is involved, and should not be used in broadcast applications [4, 7, 22]. For motion-adaptive interpolation, different ABBs for moving and non-moving parts have to be constructed. For non-moving parts, baseband replications at w, = 0 are removed by the spatial lowpass characteristic of the interpolation Signal Processing

filter, while baseband replications at higher temporal frequencies can be removed by its temporal lowpass characteristic. For motion-compensating interpolation, special ABBs have to be constructed for each velocity [7]. In Sections 5 and 6 only intrafield filters will be considered for the decimation of colour difference signals that reduce the spatial resolution only, and have the lowest complexity. The removal of baseband replications by the interpolation filter is necessary only in the visible frequency range. If the human observer cannot perceive spectral components outside the ABB, theoretically interpolation filtering is not required: the human eye takes over the task of interpolation filtering. In practice, however, problems occur if the inherent non-linearity of the display device gives rise to intermodulation effects, and "invisible" high frequency components become visible as moire patterns. With current television systems the viewing distance is usually chosen such that the ABB is well within the range of visible frequency components, and interpolation errors are visible as high frequency patterns. The high frequency attenuation of the human visual system aids interpolation filtering, such that the demands concerning transition range or stopband attenuation are low compared to anti-aliasing filter specifications (Section 4). Video signals contain strong low frequency components. Because of their large amplitude, replications of low frequency components are much more visible than replications of higher frequency components. Spectral components at these "critical frequencies" should be suppressed by zeros in the interpolation filter transfer function [18]. The "critical frequencies" are identical to those frequencies where GRID(w~, wv, oJ,) (equation (8)) is non-zero. The condition of the zeros in the transfer function of the interpolation filter at the critical frequencies can be shown to be equivalent to the condition that for a given sampling grid the sum of all coefficient subsets, which can be generated by sampling shifted versions of the interpolation filter impulse response with the given grid, is

B. Girod, W. Geuen / Decimation of colour signals

113

equal to some fixed number (usually equal to one).

5. 1:2 vertical decimation and interpolation of colour difference signals

4. Anti-aliasing filtering

In Fig. 1 the ABB is given for colour difference signals sampled at rates of 6.75 MHz or 3.375 MHz.

If after sampling or decimation of video signals the baseband and its replications overlap, the original baseband signal cannot be recovered by a linear interpolation filter. Aliasing effects are especially annoying along edges where periodic patterns, similar to a chain, appear, and at high frequency periodic structures, where moire patterns result. If the signal contains moving detail, aliasing structures will generally be moving at some other velocity in some other direction, and can visually be discriminated easily. Furthermore, aliasing errors occur in the passband region of the interpolation filter, i.e. at well perceptable low signal frequencies. In order to avoid aliasing the signal should be filtered before sampling such that the baseband and

(a)

o

+

m / , / 2 d(7;, d ~-~n(cpd)

J I

0

0

0

i

its replications can be separated by the given type of interpolation filter, i.e., the signal bandwidth must be restricted to the ABB. Note that this criterion does not postulate merely the absence of spectral overlap, as the following example shows. If a signal contains purely translatory motion, temporal sampling without temporal filtering does not result in an overlap of the baseband and its replications. Interpolation, however, requires a motioncompensating filter in order to extract the baseband [7]. Clearly, both the anti-aliasing filter and interpolation filter have to work with the same baseband, namely the "agreed baseband" (ABB). The stopband of the anti-aliasing filter has to attenuate especially those spectral components that are convolved to most annoying very low frequencies by sampling or decimation. This criterion can be met by forcing the anti-aliasing filter transfer function to zero at all those "critical frequencies" that are convolved to the origin of the frequency space. As in the case of interpolation filtering, the critical frequencies are those frequencies where GRID(wx, wy, w,) (equation (8)) is nonzero.

o

~-~ (cpd)

O

(b)

O

.

/

/ / / /

/

/ / A /

/

//I

/./ /~ ~/,/,~ 20 (.O "~-~n( CP d )

/ , •

~O/A

0

O ---~(cpd)

Fig. 1. Parameters and agreed b a s e b a n d (ABB) for a colour difference signal s c a n n e d with 625 lines/50 H z / 2 : l interlace. Sampling frequency: (a) 6.75 M H z ; (b) 3.375 MHz. Viewing distance: 6 x screen height. Horizontal period of s a m p l i n g grid: (a) P ~ = X = 2 . 2 minutes of arc; (b) P , = X = 4 . 4 m i n u t e s of arc. Vertical period of s a m p l i n g grid: P, = 2 • Y = 2.0 minutes of arc. T e m p o r a l period of s a m p l i n g grid: P, = 2 • T = 40 milliseconds. Vol. 16. No. 2, February 1989

B. Girod, W. Geuen / Decimation of colour signals

114

A 625 line/50 Hz television scan with 2 : 1 interlace is assumed that corresponds to the sampling grid in Example 1 of the Appendix with the parameters given in Fig. 1. Intrafield interpolation filtering has been assumed, which is clearly a simplification for a conventional television system. Today's interlaced displays rely on the interpolation properties of the human eye which behaves like a motioncompensating spatio-temporal lowpass filter. Since, furthermore, conventional television systems do not use vertical anti-aliasing filters, the vertical bandwidth is somewhat larger than under these assumptions. Fig. 1 shows that the vertical resolution of a colour difference signal sampled at 3.375 MHz is more than a factor of two higher than its horizontal resolution. For a balanced resolution a vertical decimation is adequate. Computer simulations have been done in order to evaluate anti-aliasing filters and interpolation filters for a 1 : 2 vertical decimation of the colour difference signals R-Y and B-Y. The resulting grid corresponds to Example 2 in the Appendix with the parameters given in Fig. 2. For a viewing distance of six times the screen height [14], the maximum vertical frequency of the ABB is

2o ~t~

X(cpa)

10

20

(cpd) Fig. 2. P a r a m e t e r s and a g r e e d b a s e b a n d (ABB) for a c o l o u r difference signal s c a n n e d with 625 l i n e s / 5 0 H z / 2 : l interlace a n d t r a n s m i t t e d after a 1 : 2 vertical intrafield d e c i m a t i o n . Sampling frequency: 3.375 MHz. Viewing distance: 6 x screen height. H o r i z o n t a l p e r i o d of s a m p l i n g grid: P~ = X = 4.4 m i n u t e s of arc. Vertical p e r i o d of s a m p l i n g grid: P, = 4 • Y = 4.0 m i n u t e s of arc. T e m p o r a l p e r i o d of s a m p l i n g grid: P, = 2 • T = 40 milliseconds. Signal Processing

toy/2~r = 7.5 cpd, while, for a sampling frequency of 3.375 MHz, the maximum horizontal frequency is tox/2rr = 6.9 cpd. The vertical band limitation to the ABB can be achieved by a one-dimensional intrafield lowpass filter with cut-off frequency f = 0.25, where f is the frequency normalized by the sampling frequency. All experiments were done with still pictures. As all filters investigated are intrafield filters, we do not expect additional filtering effects for moving picture detail. The perceptual effect of an easier aliasing pattern discrimination at moving edges or moving periodic structures (Section4) could, however, not be studied. The sampling frequency of the colour difference signals was 3.375 MHz. For natural test pictures, both R-Y and B-Y were processed by the same filters and displayed together with a luminance signal with full vertical resolution and a sampling rate of either 10.125 MHz or 13.5 MHz. In addition to natural pictures, a test chart was used that consists of a combination of a zone plate pattern and edges of different orientation. The edge patterns were inserted into the test chart, since we observed that there are ringing and aliasing effects that clearly show up at straight edges, but cannot be judged satisfactorily by means of a zone plate alone. The test chart could be displayed as an either R-Y or B-Y signal at a sampling rate of 3.375 MHz, or as both components simultaneously, together with a constant luminance of arbitrary level. The experiments showed that errors can be seen best when the test chart is displayed as R-Y at medium luminance levels. This finding corresponds well to physiological data on colour perception [13]. The zero-phase F I R filters with the coefficients given in Table 1 were studied. The output signals of the filters have to be normalized such that a dc gain of one results. All filter coefficient sets sum up to a power of 2. The filter transfer functions are shown in Fig. 3. The filter of lowest complexity, which has a zero at the normalized critical frequency f = 0.5, is the 3-tap filter F1. When used as an anti-aliasing filter, the stopband attenuation is satisfactory only for

B. Girod, W. Geuen / Decimation of colour signals Table 1 Sets of filter coefficients for 1 : 2 decimation or interpolation Filter

Degree Filter coefficients

F1 F2 F3 F4 F5 F6

3 7 7 7 7 21

F7

21

1,2,1 - 1 , 0, 9, 16, 9, 0, - 1 - 5 , 3, 37, 58, 37, 3, - 5 - 5 , 7, 37, 50, 37, 7, - 5 1, 4, 7, 8, 7, 4, 1 -1, 11, 0, -10, - 2 , 15, - 1 , -27, - 1 , 80, 128,... 1, 13, 7, -22, - 5 , 32, 1, -55, -3, 160, 254,...

less critical natural test pictures. For "steep" colour edges, and especially for periodic structures, aliasing effects are clearly visible. A 3-tap filter is not sufficient to suppress the aliasing components due to vertical 1:2 decimation. However, F1 yields good results as an interpolation filter and is sufficient for natural test pictures as well as for the test chart. Accordingly, all further anti-aliasing filters were investigated with F1 as interpolation filter. Four 7-tap anti-aliasing filters were tested, which represent different trade-offs between passband width and stopband attenuation. F2 is a maximally flat 7-tap filter with coefficients that are especially suited for a digital hardware realization. The coefficients of F2 have been published, e.g. in [20, 15, 9]. F2 has recently been integrated as a prototype NMOS circuit for video applications [8]. F3 has been proposed by the Independent Broadcasting Authority (IBA) for a 1:2 decimation [21]. F4 has been designed for this application by a mixed-integer optimization, as proposed in [18]. The coefficients of filter F5 are samples of a squared cosine function with its first zero at f = 0.25, which is the desired bandedge of an antialiasing or interpolation filter for a 1 : 2 decimation. The stopband attenuation at the "critical frequency" f = 0.5 is sufficient for each of F2, F3, F4, or F5. While for F2 the stopband attenuation is not satisfactory for the zone plate pattern, and F5 leads to a considerable blur of colour detail,

115

both F3 and F4 are good compromises between stopband attenuation and passband width. With F3 or F4, for many natural test pictures there is no visible difference between the full resolution original and the image with vertical chrominance decimation and interpolation at standard viewing conditions [14] with a viewing distance of six times the screen height. In order to find out whether an even better picture quality can be achieved using high order anti-aliasing filters, two 21-tap filters were investigated. F6 has been designed for a 1 : 2 horizontal sampling rate conversion by the method given in [18], and has successfully been used for this application. For a vertical 1 : 2 decimation this filter does not result in a picture quality that is satisfactory in all respects. Although the resolution for fine periodic structures is superior to all other filters discussed so far, moire patterns are visible for periodic patterns of high vertical frequency. The aliasing suppression at the critical frequency f = 0.5 is not sufficient, as also can be seen from the frequency response (Fig. 3). For a horizontal 1:2 decimation the critical frequency is at the bandedge of the original signal, and there are no strong signal components at this frequency as these usually are attenuated by an analog filter prior A / D conversion. Thus, a zero at f = 0 . 5 is not crucial for an anti-aliasing filter for horizontal 1 : 2 sampling rate decimation. For vertical decimation the interlaced TV signal can contain strong signal components at f = 0.5, and a zero at this critical frequency is important. Filter F7 is another 21-tap anti-aliasing filter that combines the good resolution properties of F6 with a sufficient attenuation at the critical frequency. F7 has the best antialiasing performance of all filters discussed. When compared to F3 and F4, the resolution for narrow horizontal stripes is improved. Under standard viewing conditions [14], this could, however, only be observed for the zone plate pattern. For natural test pictures, there was no visible difference. On the other hand, at artificially generated colour edges, F6 and F7 produce noticably more ringing than, for example F3 or F4. Vol. 16+ No. 2. February 1989

B. Girod, W. Geuen / Decimation of colour signals

116

0 10" 2030~0"

f

a

50

i

0.1

o.o

03

0.2

Normalized

0

10-

==

0.5

frequency

0

._8

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10-

20-

oc

20'

30-

c

30"

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b

50 0.0

f f

,

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0.3

0./.

05

C 50 0.0

frequency

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Normalized

o'.~

0.3

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frequency

0 rn "o w

10-

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c

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50

i

0,0

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1~3 10"0 c o

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0.4

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frequency

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frequency

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g o.o

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r

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0.2

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i

0.3

o.~

~

f 0.5

frequency

Fig. 3. Normalized transfer functions of the filters used for anti-aliasing filtering or interpolation filtering. The frequency axis is normalized with respect to the sampling frequency, attenuation is in dB.

Signal

Processing

B. Girod, W. Geuen / Decimation of colour signals

117 (a)

The results of an i n f o r m a l subjective e v a l u a t i o n are s u m m a r i z e d in T a b l e 2. In c o n c l u s i o n , there is

©

(3

little difference b e t w e e n the a n t i - a l i a s i n g filters F2, F3, F4, F6, a n d F7 for n a t u r a l pictures. For the zone plate, the 21-tap filter F7 is best. T a k i n g into a c c o u n t the filter degree, F3 a n d F4 seem to be the best c o m p r o m i s e s b e t w e e n complexity a n d performance. For i n t e r p o l a t i o n filtering, the 3-tap filter F1 is always sufficient. Table 2

•

Evaluation of anti-aliasing filters for a vertical 1: 2 decimation of colour difference signals

Filter no.

Passband width

Stopband attenuation

FI F2 F3 F4 F5 F6 F7

+ + + + -++ ++

0 + + ++ + ++

Ringing at artificial colour edges

°

t~o

•

I

°

w

~Y (cpd)

++ + + + ++ -

©

(b)

©

-

+ +, excellent; +, good; 0, fair; - , objectionable; - - , not acceptable.

- ~X-X(cpd)

6. 1:2 line-offset decimation and interpolation of colour difference signals After 1 : 2 line-offset d e c i m a t i o n of a n interlaced television scan, a s a m p l i n g grid, as in E x a m p l e 3 of the A p p e n d i x , results. D i a m o n d - s h a p e d ABBs for a line-offset s a m p l i n g grid are s h o w n in Fig. 4 for s a m p l i n g frequencies of 3 . 3 7 5 M H z a n d 6.75 M H z before d e c i m a t i o n . A s s u m i n g a viewing distance o f six times the screen height [14], the m a x i m u m vertical f r e q u e n c y of the ABB is to>,/2~r = 15.0 cpd. The maximum horizontal f r e q u e n c y is w x / 2 ~ r = 1 3 . 8 c p d

for a s a m p l i n g

f r e q u e n c y of 6.75 MHz, a n d tox/2Tr= 6.9 cpd for 3.375 MHz. For a b a l a n c e d resolution of colour detail a d i a m o n d - s h a p e d ABB should be used with a s a m p l i n g f r e q u e n c y o f 6.75 MHz. The d i a m o n d - s h a p e d b a n d l i m i t a t i o n can be d o n e by a cascade o f two identical d i a g o n a l l y

2o • °

° w ~(cpd )

Fig. 4. Parameters and agreed baseband (ABB) for a colour difference signal scanned with 625 lines/50 Hz/2:l interlace and transmitted after a 1:2 line-offset intrafield decimation. Sampling frequency: (a) 6.75 MHz; (b) 3.375 MHz. Viewing distance: 6 x screen height. Horizontal period of sampling grid: (a) P, =2 • X = 4.4 minutes of arc; (b) P~ =2 • X =8.8 minutes of arc. Vertical period of sampling grid: P, = 4. Y = 4.0 minutes of arc. Temporal period of sampling grid: P, = 2 • T = 40 milliseconds. o p e r a t i n g o n e - d i m e n s i o n a l Iowpass filters with cutoff frequencies f = 0.25, using the samples of one field. As the d i a g o n a l l y separable filtering

Vol. 16, No, 2, February1989

118

B. Girod, W. Geuen / Decimation of colour signals

algorithm is not straightforward and to our knowledge has not been published yet, Fig. 5 shows its conceptual stages in the spatial domain and in the f r e q u e n c y domain. First, samples of value "zero"

inserted at offset positions, as shown in Fig. 5(b). A one-dimensional filtering on this denser sampling lattice in the NWSE direction, combined with 1:2 NWSE decimation, follows are

Spatialdomain

Frequencydomain

a) . . . . . . .

~

•

0

•

0

0

• 0

•

0 • 0 • 0 0 0 0

• 0

0

•

0

<>

X \

?

V

x

~y

b)

Iooo • • oo oo • o • o 9 o • o ° OeOoO ° o ° o ! OlOlOeOlOoOoOe

y

\ /

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~'x "G /

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w

y

c) "

.7.

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¢

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I°! °°°°°°° °°°°° °o°o°o°o°o°o°o o o o o • o o o o o o o o

°°°

~X

]°o

stopband~

/2"x x

Fig. 5. 1 : 2 offset-decimation with a cascade of two diagonal filters in the spatial domain and in the frequency domain. (a) Orthogonal sampling grid; (b) insertion of zeros at offset positions; (c) NWSE filtering; (d) result of NWSE decimation; (e) NESW filtering; (f) result of NESW decimation, line-offset grid. Frequencies, where replications of the baseband necessarily occur due to the sampling structure, are marked by (.). Signal Processing

119

B. Girod, W. Geuen / Decimation of colour signals _Spatial domain

Frequency domain

d ) 0

0

0

0

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0

0

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0

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,

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x

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x

o o

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/?.. "<4~

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W:,(

Wy

Fig. 5--continued. (Fig. 5(c, d)). This step results in a sampling grid that is twice as dense in the N E S W direction as in the NWSE direction. On this grid, a onedimensional NESW filtering is performed in combination with a N E S W decimation, which results in the desired overall 1:2 line-offset lattice

(Fig. 5(e, f)). Interpolation from the line-offset lattice back to the original orthogonal lattice can be done by corresponding steps in reverse order. A cascade of two diagonal filters can also be implemented in a straightforward manner as a planar filter. The computational demands of the Vol. 16, No. 2, February 1989

B. Girod, W. Geuen / Decimation of colour signals

120

approaches can be compared roughly considering the number of multiplications. The planar filter requires 1(N2+ 1) multiplications for each output sample, if the region of support of its impulse response covers N lines and N columns (Fig. 6). The realization as a cascade of two diagonal filters requires 2 N multiplications. For N > 3, the computational load for the diagonally separated filters is less. For large N the saving is approximately N/4 in terms of multiplications. 0 0

o

0 0 0

o o

Table 3 E v a l u a t i o n o f a n t i - a l i a s i n g filters for 1 : 2 line-offset d e c i m a t i o n of colour difference signals

Filter no.

Passband width

Stopband attenuation

R i n g i n g at artificial c o l o u r edges

F1 F2 F3 F4 F5 F6 F7

0 0 0 0 -+ +

0 + + ++ + + +

++ + + + ++ 0 0

0

+ + , excellent; + , g o o d ; 0, fair; - , o b j e c t i o n a b l e ; - - , not acceptable.

0 0 0 0 0 o 0 0 0 0 0 0 0 0 0 o 0 o 0 0 o o 0 o 0 o 0 o 0 o o o 0 0 0 0 o 0 o o 0 0 o 0 o 0 0 o

o 0 0 0

I._ r

N

=

11

.d -J

Fig. 6. R e g i o n of s u p p o r t of the e q u i v a l e n t p l a n a r i m p u l s e r e s p o n s e for a c a s c a d e of two d i a g o n a l filters with N = 11 taps for 1 : 2 line-offset d e c i m a t i o n or 2 : 1 line-offset i n t e r p o l a t i o n .

As for the vertical decimation, computer simulations have been used to evaluate different antialiasing filters and interpolation filters for 1 : 2 lineoffset sampling of the colour difference signals R-Y and B-Y. Again, an informal subjective evaluation was done with still pictures. For the anti-aliasing filters similar observations were made as for the case of 1:2 vertical subsampling. For line-offset decimation a flat frequency response in the passband is more important than for vertical decimation, since for vertical and horizontal structures the attenuations at the bandedge of the two diagonal filters multiply. For the 7-tap filters F2, F3, and F4, the resulting passband widths are fair. The stopband attenuation of the 21-tap filter F6 at the critical frequency again is not sufficient (see Section 5), while the 21-tap filter F7 was found to be adequate for diagonal anti-aliasing filtering. Table 3 summarizes the evaluation of the antialiasing filters. Signal Processing

With interpolation from the line-offset lattice back to the original orthogonal lattice a severe problem can occur at vertical edges. Vertical edges contain the highest horizontal frequencies. As shown in Fig. 7, with offset sampling the horizontal high frequency components are repeated at frequencies that also correspond to the highest vertical frequencies. The highest horizontal and vertical frequencies cannot be separated sufficiently by a poor interpolation filter. This effect causes a raggedness of interpolated vertical edges. Vertical edges look like the edge of a stamp. We found that in order to avoid the "stamp effect", an attenuation of at least 20 dB at the "corners" of the diamond-shaped frequency response is required. Accordingly, interpolation with filters F1, F2, F3, F6, and F7 does not result in good picture quality (Table 4). The 7-tap filter F4 has been designed especially for diagonal interpolation without the "stamp effect". F4 has sufficient attenuation at f = 0 . 2 5 . The resulting picture quality is improved. Experiments with interpolation filters with fewer taps did not yield satisfactory results. Another way to overcome the "stamp effect" might be a further restriction of the ABB. With this solution a good interpolation filter with less taps seems to be possible.

B. Girod, W. Geuen / Decimation of colour signals

121

annoying vertical y pattern

// /~

//

/

/ \ ~ ~

/

bandedge of poor interpolation filter tOx

//

tOy Fig. 7. The c a u s e of the " s t a m p effect" at vertical edges is s h o w n in the f r e q u e n c y d o m a i n . For an i n t e r p o l a t i o n from the offset s a m p l i n g grid the r e p l i c a t i o n s o f the vertical edge s p e c t r u m have to be removed. A p o o r i n t e r p o l a t i o n filter d o e s not a t t e n u a t e the a n n o y i n g vertical f r e q u e n c y pattern sufficiently.

In conclusion, a good picture quality can be obtained for line-offset decimation with a cascade of two diagonal 21-tap filters F7 for aliasing suppression and a cascade of two diagonal 7-tap filters F4 for interpolation.

Table 4 Evaluation of filters for an interpolation of colour difference signals from a 1: 2 line-offset grid to an orthogonal grid

Filter no.

Passband width

Stopband attenuation

Ringing at artificial colour edges

F1 F2 F3 F4 F5 F6 F7

0 0 0 0 - + +

0 + ++ 0 0

+ + + + + ++ 0 0

+ + , excellent; + , g o o d ; 0, fair; - , o b j e c t i o n a b l e ; - - , not acceptable.

7. A subjective comparison of decimation schemes for the colour difference signals So far, schemes have been optimized that allow a vertical decimation or a line-offset decimation of the colour difference signals. However, we do not know how the different schemes perform compared with one another. This section reports a subjective assessment of the picture quality achieved with different chrominance decimationinterpolation schemes. In subjective tests we investigated the following systems: (a) 4:2:2 system. This provides the full colour resolution according to the digital studio standard [6]. The colour difference signals R-Y and B-Y of a 625 line/50 Hz television scan with 2 : 1 interlace are sampled at 6.75 MHz. The corresponding ABB is shown in Fig. l(a). The luminance signal is sampled at 13.5 MHz. (b) 4:1 : 1 system. This differs from the 4 : 2 : 2 system only by the sampling frequencies for R-Y Vol. 16, No. 2, February 1989

B. Girod, W. Geuen / Decimation of colour signals

122

and B-Y, which are reduced to 3.375 MHz. The corresponding ABB for the colour difference signals is shown in Fig. l(b). (c) 4:(1:1)o system. This results after a 1:2 line-offset intrafield decimation of the colour difference signals of the 4 : 2 : 2 system. The corresponding ABB is shown in Fig. 4(a). A cascade of two diagonal filters F7 is used for aliasing suppression and a cascade of two diagonal filters F4 is used for interpolation (Table 1). (d) 4:2:0 system. This results after a 1:2 vertical decimation of the colour difference signals of the 4 : 2 : 2 system. Filter F4 was used as an antialiasing filter and filter F1 as an interpolation filter (Table 1). (e) 4:1:0 system. This results after a 1:2 vertical decimation of the colour difference signals of the 4:1 : 1 system. The ABB is shown in Fig. 2. As for the 4:2:0-system, filter F4 was used as an anti-

aliasing filter and filter F1 as an interpolation filter (Table 1). (f) PAL system. This is the composite system that is used today in many countries for TV program distribution. For comparison, we passed the components of the 4 : 2 : 2 signal through a PAL coder/decoder combination. Both PAL coder and decoder (studio quality) were carefully adjusted to yield the best picture quality. The subjective tests were carried out with four different still colour pictures (Fig. 8). The pictures "Boy with toys", "Boats with lighthouse", and "Locomotive" have been digitized from slides by a flying spot scanner. "Boy with toys" and "Boats with lighthouse" were chosen from a set of pictures, recommended by the EBU for subjective testing. The " F T Z picture" has been generated by a computer. The pictures were judged by 12 television

Fig. 8. Pictures used in the subjective tests. (a) Boy with toys; (b) boats with lighthouse; (c) locomotive; (d) FTZ picture. Signal Processing

B. Girod, W. Geuen / Decimation of colour signals

engineers with normal visual acuity and colour sensitivity. The viewing conditions and the test procedure utilized were in accordance with C C I R Recommendation 500 [14] and with the method recommended by the EBU [1]. A viewing distance of four times the screen height was chosen. We have also tried to obtain results for larger viewing distances, e.g. for six times the screen height, but in this case the picture quality of all systems is too close such that a ranking of the methods becomes very unreliable. During a preparatory phase, the impairments to be judged were pointed out to the observers. The 4: 2 : 2 system served as a reference. In the assessment phase, the six different systems (a) to (f) were presented in random order. Each presentation consisted of four phases: (a) 10s--reference picture; (b) 3 s--mid-grey as separation between displays; (c) 10 s--picture to be judged with respect to the reference; (d) 3 s - - m i d - g r e y as separation and voting period. The total duration of a test session was about 25 min. Four observers took part simultaneously in a session. Picture quality was rated on the five-grade quality scale (Table 5) [14]. The authors think that the grades are not meaningful as an absolute rating for the class of impairments investigated. Picture quality of the six systems (a)-(f) is quite close, and differences are subtle. Most observers chose to use grades 3 to 5 to express the differences that they perceived. Some of them remarked, however, that they would have judged differently if more severe impairments had been included in the tests. Table 5 Quality scale used for subjective tests 5--excellent 4----good 3- - f air 2--poor l~bad

123

Thus, the results are meaningful for a comparison of the methods investigated, but one should not relate the ratings presented below to ratings reported in other investigations. The results of the subjective assessment are shown in Fig. 9 separately for the four test pictures used. It can be seen that the picture contents had a significant impact on the picture quality. The following observations hold for natural test pictures (Fig. 8(a, b, c)): • All investigated decimation schemes for the colour difference signals yield a picture quality that is better than with PAL. • The loss of colour resolution in a 4: 1 : 0 system leads to a noticeable loss in picture quality. The balanced resolution of a 4:(1 : 1)o system is • preferred to the unbalanced resolution of the 4: 2 : 0 system. • Even better than the balanced 4:(1 : 1)o system is the 4 : 1 : 1 system that does not affect the vertical resolution at all. For still pictures, the 4:1 : 1 system benefits from the 2 : 1 interlace. This is not true for the 4 : 2 : 0 system and the 4: (1 : 1)o system. • The loss in picture quality by a reduction of the sampling frequency for R-Y and B-Y from 6.75 M H z to 3.375 MHz is very small. The 4: 1 : 1 system is judged typically 0.2 quality grades below the 4 : 2 : 2 system. The results are quite different for the artificial " F T Z picture". There, a reduction of vertical resolution does not lead to a severe loss in quality. This observation can be explained by the vertical roll-off filter that was applied during the generation of the original " F T Z picture" for a pleasant interlaced display without flickering edges. Remarkably, the horizontal/vertical balance of colour resolution is not a major quality criterion for the human observer. Our results show that the 4 : 1 : 1 system is the best solution if a 1 : 2 decimation of the colour difference signals of a 4 : 2 : 2 system is desired. The 4: 1 : 1 system not only yields the best picture quality of the systems compared, but also is least complex and thus least expensive with respect to decimation and interpolation. Vol. 16, No. 2, February 1989

B. Girod, W. Geuen / Decimation of colour signals

124

5_ Boy with

toys

Boats --

.,-t

with

lighthouse

3-

O"

2.

1 ~0 7.

--

•~

3

2

I

"

~

5"

5

4

."2.

Locomotive

__

F

•~

I

~u

.~.

3-

o

o

~o o

FTZ-picture

IF o

Fig. 9. Mean quality grades for six different systems and four different test pictures. 8. Summary This paper considers decimation and interpolation of the colour difference signals R-Y and B-Y for a digital transmission o f broadcast television signals at rates of 140 MBit/s or 34 MBit/s. In a brief review of three-dimensional sampling a convenient method for the calculation of the three-dimensional spectrum of a television signal sampled with a regular periodic grid is derived. It is based on a Discrete Fourier Transform o f the lattice element coefficients that describe the sampling grid. From the three-dimensional spectrum, desirable properties of anti-aliasing filters and interpolation filters for TV signals are derived. A 1:2 vertical decimation and a 1:2 line-offset decimation of colour difference signals are investigated in detail. From an informal subjective evaluation, antialiasing filters and interpolation filters for both vertical sampling rate decimation and line-offset decimation are proposed. A balanced resolution of colour detail is provided by 1 : 2 line-offset decimation in conjunction Signal Processing

with a sampling frequency of 6.75 MHz for each colour difference signal, or, for a lower spatial resolution, a vertical 1 : 2 decimation with a sampling frequency of 3.375 MHz. The picture quality that is achieved with four different decimation schemes has been assessed in subjective tests. Surprisingly, a balanced resolution o f the colour difference signals is not a major subjective quality criterion. The human observer prefers a 4:1 : 1 system, which does not touch the vertical resolution, to a 4 : ( 1 : 1 ) o system. When compared to a 4 : 2 : 2 system, the picture quality deterioration by a 4:1 : 1 system is very small. All decimation schemes investigated provide a picture quality that is clearly better than with today's PAL television system. References [1] K. Bernath, F. Kretz and D. Wood, "The EBU method for organising subjective tests of television picture quality", EBU Review--Technical, No. 186, April 1981, pp. 66-75. [2] H. Buley and L. Stenger, "Inter-/intraframe coding of color TV signals for transmission at the third level of the

B. Girod, W. Geuen / Decimation of colour signals

[3]

[4]

[5]

[6] [7]

[8]

[9]

[10]

[ 11]

[12]

[13]

[14] [15]

[16] [17] [18]

[19]

digital hierarchy", Proc. IEEE, Vol. 73, No. 4, 1985, pp. 765-772. "Digital multiplex equipments operating at the third order bit rate of 34 368 kbit/s and the fourth order bit rate of 139 264 kbit/s and using positive justification", CCIR Rec. G.751, 1980. E. Dubois, "The sampling and reconstruction of timevarying imagery with application in video systems", Proc. IEEE, Vol. 73, No. 4, 1985, pp. 502-522. E. Dubois, M.S. Sabri and J.-Y. Ouellet, "Threedimensional spectrum and processing of digital NTSC color signals", SMPTE J., Vol. 91, No. 4, 1982, pp. 372378. "'Encoding parameters of digital television for studios", CCIR Rec. 601, 1982. B. Girod and R. Thoma, "Motion compensating field interpolation from Interlaced and Non-Interlaced Grids", 2nd International Technical Symposium on Optical and Electro-Optical Applied Science and Engineering, SPIE Conf. B 594: Image Coding, Cannes, France, Dec. 1985, pp. 186-193. H.-J. Grallert, F. Matthiesen and B. Zehner, "An integrated digital filter for the component encoding of color TV signals", Siemens Forsch.-u. Entwickl.-Ber., Vol. 13, No. 5, 1984, pp. 240-245. C. Gumacos, "Weighting coefficients for certain maximally flat non-recursive digital filters", IEEE Trans. Circuits, Syst., Vol. CAS-25, No. 4, April 1978, pp. 234235. S. Henschke, "Aufl6sungsgiinstige digitale Chromafilterung zur PCM-0bertragung von Videosignalen", N T Z Archly, Vol. 5, No. 9, 1983, pp. 249-255. G. Holach, "Spezielle Betrachungen zur Farbiibertragung yon zeitkomprimierten Komponentensignalen', Proc. of the 11. Jahrestagung der FKTG Hamburg, 1984, pp. 456471 (in German). K. Kubota and J. Ishida, "Proposed characteristics of a shaping filter for colour difference signals used in component coding", NHK Laboratories Note, No. 273, Tokyo, March 1982. R.M. Mersereau, "The processing ofhexagonally sampled two-dimensional signals", Proc. IEEE, Vol. 67, No. 6, June 1979, pp. 930-949. "'Method for the subjective assessment of the quality of television pictures", CCIR Rec. 500-2, 1982. K.H. M6hrmann, "Zur Dimensionierung phasenlinearer digitaler Filter", Frequenz, Vol. 37, No. 7, 1983, pp. 166173 (in German). D.E. Pearson, Transmission and Display of Pictorial Information, Pentech Press, London, 1975. P. Pirsch, personal communication, 1984. P. Pirsch and M. Bierling, "Changing the sampling rate of video signals by rational factors", in: Signal Processing II: Theories and Applications, Proc. EUSIPCO, 1983, pp. 171-174. H. Schr6der and H. Elsler, "Planare Vor- und Nachfilterung fiir Fernsehsignale", N T Z Archiv, Vol. 4, No. 10, 1982, pp. 303-311 (in German).

125

[20] G.J. Tonge, The Sampling of Television Images, IBA Experimental and Development Report 112/81, 1981, pp. 1-34. [21] G. Tonge and M.D. Windram, Line Sequential Colour Transmission and Vertical Filtering in ,~AC, IBA Experimental and Development Report, No. 123/83, 1983. [22] D. Uhlenkamp and E. Giittner, "Verbesserte Wiedergabe von Norm-Fersehsignalen", N T Z Archly, Vol. 4, No. 10, 1982, pp. 313-321. [23] A. Watanabe, H. Sakata and H. lsonono, "Chromatic spatial sine-wave responses of the human visual system", NHK Laboratories Notes No. 198, Tokyo, March 1976. [24] B. Wendland, "Zur theorie der Bildabtastung", N T Z Archiv, Vol. 4, No. 10, 1982, pp. 293-301 (in German). [25] D. Westerkamp, "Adaptive intra-/interframe DPCMcoding for transmission of colour TV-signals with 34 M Bit/s", Proc. of the 1984 International Zurich Seminar, pp. 39-45.

Appendix Spatial

sampling

positions

of ~ l r i d

-

(x,7,t)

(~)

x

sampled (even

)®®@®@

at

( • s a m) p l e d

) ® ® (D @ (i)... )@@®®@ t

-

t : 2nT

fields

)

at

-

t = ( 2 n - 1)'1" (odd fields )

Y

) q) ® @ © ~,_-~-

L

Lattice

element

i

I

k

O(i,j,k)

0

0

1

2

0

0

1

0

0

0

1

0

0

0

0

1

1

1

2

•L

frequencies

A 11w

Px = X Py :

N =

2

Pt = 2T

2Y

of

~(i,j,k)

non-zero

A

mL

A

w

w

~

GRID ((*~.,~.,,.,Jt)^ :~

O:

~a~x 2n

~)

..-

1 2

coefficients

O

Spatia~

:

M :

0 0 0 0..0 0 0 0 0 ~1

0 0 O 0 0 ~

Py

non-zero cJt =--

~"

at

2n

fi

0 .......... t wt

~

= (2n-

~,

1)

Example 1. Interlaced sampling grid. Vol, 16, No. 2, February 1989

B. Girod, W. Geuen / Decimation of colour signals

126

Spatial

samp'ftng

1

1

positions

1

1

of

grid

1

(x,;¢, t

X

)

(9

Spatial

-

sampled ( even

@@@@@ (~

-

sampled t=

positions

of g r i d

(x,y,t

)

at t : 2 n T fields

(9 ®

)

at

@ ®

(~)

"

X

(9

fields

(9 ®

)

1

(9 ®

-

sampled (even

®

(~)

(2n-1)T

( odd

@@(9@(9

sampling

-

at t= 2nT fields

sampled

(D (2)

)

at

t = (2n-1)T (odd

fields

)

Y

®®¢o¢-

La ttic e

i

element

j

k

T

L=I

Px:X

M=4

Py : ,~Y

N=2

Pt : 2T

~ (i,j,k}

~

(9

element

x

L=2

Px~2X

M:4

~ = 4Y

N=2

Pt = 2T

coefficients

j

k

(~(i , j , k )

~(i,j,k)

i

j

k

0

0

1

4

0

0

1

0

1

0

1

:O(i,j,k)

0

0

0

1

2

0

0

1

0

0

1-

1

0

0

0

0

2

0

0

0

0

1

0

0

0

0

1

1

0

3

0

0

1.

1

1

0

0

2-2 i

1

1

1

0

0

1

0

0

0

2

0

0

0

0

2

1

0

I

1

1

1+

1

2

0

1

0

1

2

1

0

2

1

0

2

0

3

0

0

0

0

3

1

0 0 1 0 0 0 0

0

3

1

0

1-

1

3

0

0

2*2 i

1

3

1

1

of n o n - z e r o

GRID

,

( w x,W,/,~t

• -

Spatial

) .

non-zero

•

•

Wx

•

~)

(~

~

•

•

©

n o n - z e r o at temporal freq ....

of n o n - z e r o

inct u d i n g

~t = 0

2~O-

C)

frequencies

GRID

C,(i , j . k )

0 0 0 2*21 4 0 0 2-2j

( Wx, toy, tot )

at t e m p o r a l

frequencies ~

®

(9

Lattice

coefficients (i,j,k)

Spatial f r e q u e n c i e s

--

®

ul x

2~-

©

-- n o n - z e r o at t e m p o r a l frequencies including c~ = 0 t -- n o n - z e r o

ies

al temporal

frequencies

•

•.

•

•

..-

,g

ont,

•

0...

onl,

i

0

•

0

•

o

•

~±

Py

t

._L 2n

Z

Px

2n

Example 2. Interlaced sampling grid after 1 : 2 vertical sampling rate decimation.

Signal Processing

Example 3. Interlaced sampling grid after 1:2 line-offset decimation.

B. Girod, W. Geuen / Decimation of colour signals

127

Spatial samplincj positions of grid ( x ~ y , t )

3

i

1

3

3

(~) - sampled at t=4nT

x

)®®®(D®

(~) - sampled at t:(4n+l)T

)@@@@@... )®©®®® )@©@@®~ ) ® ® ® ® ® ~

(~

- sampled at t:(4n+2)T

(~

-

y

sampled at t :(4n+3)T

x L.._ L:2

Px:2X

M:4

I~:4Y

N:4

Pt:4T

Lattice element coefhcients j

k

(~(~ , j , k )

I

J

k

0 (i, j,k)

,

J

k

~(i.j.k)

=

}

k

~(i,j,k)

0 1

0 0

0 0

1

8

0

0

1

1

0

1

0 1

0 O

2 2

0 1

0 0

0 1

0 O

3 3

O

0

0 0

0

0

O 0

0

0

0

1

0

O

0

0

1

1

1

1

0

0

0

1

1

1

1 0

0 0

0 t

1 1

2 2

0 0

0 0

0 1

t 1

3 3

0 1

2 2

0 0

0 1

0 0

0 1

2 2

1 1

O

0

0

2

2

1

8

0

2

3

0 1 0

0 8 0

0

0

1

2

2

0

0

1

2

3

0

0

0 1

3 3

0 0

0

0

O

3

1

3

1

O

1

2 2

3

3

3 3

0

1

0 1

0

O

0 8

0

0

0 1

O

0

t

3

3

O

0

Spatial

G(i,j,k)

~(i.j.k)

Eo(i,j,k)

frequencies of non - zero GRID (~x' ~ ' wt )

i

0

0

• ..... z.... t temporal

0

0

0

0

0

0

0

0

0 ...

O

0

0

0

•

freq . . . . i es including ('Jr = 0 WX 2n

0

0

0

.......... freq . . . . I~__l~'

t temporal fes 1

only

0 ....

•

0 0

2n

~ (i, j,k)

0 0

1_

~ Example 4. Interlaced sampling grid after 1 : 2 line-offset decimation without frame-reset.

VoI, 16, No. 2, February 1989