Clustered dot and line multilevel halftoning for electrographic colour printing

A method is disclosed for rendering monochrome or colored continuous tone images by a system having restricted continuous tone rendering capabilities, such as electrographic printers capable of printing more than two density levels on each addressable micro dot. A method for preferred halftone dot growth is described, starting from isolated dots arranged along base lines and auxiliary lines, evolving to high density lines along the base lines and approximating full continuous tone rendition for high density regions. Preferred arrangements for the orientation, spacing and absolute location of the base lines and auxiliary lines are disclosed and a method to generate preferred arrangements. A moire free combination of three halftone images is described for the reproduction of color images. A method to select a restricted set of energy levels to obtain linear reflectance response is disclosed. Halftone cells are arranged in supercells to improve the density resolution of the system.

FIELD OF THE INVENTION 
The present invention relates to image rendering systems having the 
capability to render consistently only a restricted amount of density 
levels in a black and white or colour reproduction of a continuous tone 
image. More particularly the methods of the invention can be used in an 
electrographic printer having multilevel capabilities. 
BACKGROUND OF THE INVENTION 
The reproduction of continuous tone images is traditionally done by full 
contone reproduction such as colour photo prints or by binary halftoning 
techniques such as colour offset printing. Electrographic printing, where 
a latent image is formed by static electricity that is locally discharged 
to form graphical representations, has an important member called 
electrophotographic printing. In electrophotographic printing, the static 
energy is partially removed by a directed light beam. Electrophotographic 
printers have traditionally a capability of binary printing. The illusion 
of continuous tone images is reached by binary halftoning techniques. 
Every addressable spot on the output, further called micro dot, can get a 
high density corresponding with full toner coverage or a low density 
corresponding with the absence of toner. 
Recently, electrographic printers have also limited contone capabilities. 
That means that the amount of toner per micro dot can be modulated more 
continuously, such that the micro dot--after rendering--can have apart 
from a low density and a high density also some mid densities. The density 
level can be regulated by an energy level that is applied to the micro dot 
by an output device. Agfa-Gevaert N.V. from Mortsel Belgium markets such 
an electrophotographic printer under the trade name Chromapress. This is a 
duplex colour printer (cyan, magenta, yellow, black) having a resolution 
of 600 micro dots per inch producing 1000 A3 pages per hour. Per micro 
dot, 64 energy levels can be selected. The output device can be also a 
thermographic printer, inkjet printer, more generally an electrographic 
printer etc. The problem with the mid densities is that these are not 
stable as a consequence of the physics of the electrographic process. By 
instability is meant that there is not a one to one relation between the 
energy level applied to the device and the density level obtained on the 
reproduction. The density level of a first micro dot is strongly dependent 
on the energy level applied to the micro dots in the direct neighbourhood 
of the first micro dot. Therefore several methods have been proposed to 
enhance the stability of the micro dots. This can be done up to a certain 
limit dependent on the density level. An important aspect of the remaining 
instability is that not enough density levels per micro dot can be 
rendered. Therefore a technique related to binary halftoning must be used, 
which is called multilevel halftoning. A problem with halftoning is that 
the spatial resolution is decreased to improve the density resolution. 
Another problem is that internal moire can show up due to the interaction 
between the micro dots and the halftoning pattern. These problems have 
been addressed in WO-A-93 26116, for multilevel halftoning of images 
having one colour component. FIG. 7 of that application discloses a 3-bit 
grey halftone dot layout according to a mixed dot type embodiment. For low 
output densities, isolated halftone dots appear on a background. The 
halftone dots comprise microdots having two different density levels. For 
higher output density levels, isolated bands appear and for the highest 
densities, maximum two different density levels are present in each 
halftone cell. 
However, if different colour components are printed on top of each other to 
get colour reproductions, colour moire can occur between the different 
components. This problem is not addressed in the above mentioned 
application. Colour moire or inter image moire is different from internal 
moire, as will be described below. 
EP-A-0 370 271 discloses the formation of halftone dots to prevent rosette 
moire and colour shift, but is related to binary halftoning. Problems of 
stability of micro dots and aliasing of line structures by use of multiple 
levels are not addressed in this application. 
OBJECTS OF THE INVENTION 
It is a first object of the invention to render images with a consistent 
and predictable density on the reproduction. 
It is a second object of the invention to optimise the spatial resolution 
while keeping the density resolution as high as is necessary to guarantee 
the impression of continuous tone image reproduction. 
It is a third object of the invention to eliminate internal moire and inter 
image moire. 
It is a fourth object of the invention to make the reproduction obtained 
from the combination of several colour components less registration 
dependent thereby avoiding line structure aliasing. 
SUMMARY OF THE INVENTION 
In accordance with the present invention, a colour or multi tone 
reproduction is disclosed comprising at least three halftone images 
printed on top of each other, wherein each of said images is rendered on a 
recorder grid of micro dots and comprises isolated halftone dots wherein: 
the area outside the halftone dots has a minimum background density 
D.sub.min ; 
the halftone dots include at least one micro dot having a density level 
D.sub.1 and at least one micro dot having a different density level 
D.sub.2, D.sub.1 and D.sub.2 both higher than D.sub.min ; 
the centers of the halftone dots are arranged along a first set of parallel 
equidistant base lines having a first orientation and along a second set 
of parallel equidistant auxiliary lines having a second orientation 
different from the first orientation; 
the points of intersection of any base line with any auxiliary line have an 
identical relative position with respect to the micro dot closest to said 
point of intersection. 
The requirement that the halftone dots are isolated and composed of two 
different densities makes that unstable micro dots can be stabilised by 
stable micro dots with higher density. The requirement of identical 
relative position reduces or eliminates the internal moire. Other 
advantages will become clear from the detailed description below. 
In a lot of printing or image rendering devices, the density on the carrier 
is obtained via a process in which the energy is modulated spatially to 
obtain spatially varying densities on the carrier. For thermography, 
thermosublimation, thermal transfer processes etc., energy is applied to a 
thermal head or the like. Usually, the more energy is applied, the higher 
the density on the carrier. For electrophotography, a semiconductor drum 
is loaded by a negative voltage and illuminated by a light source to 
diffuse the charge, where toner particles must be attracted. Also here, 
the energy level of the light source impinging on the drum is proportional 
to the density on the carrier. It is possible that the light source always 
gives a constant light power, but that the amount of energy on the 
semiconductor drum is modulated by deflection of the light source towards 
or away from the drum. It is possible to think of systems where an 
increasing energy level gives a lower density level on the carrier. The 
embodiments set out for this invention can be used also for this type of 
devices.

It has been found that the stability of isolated micro dots increases as 
their density increases. If no energy is applied to the micro dot, 
generally no toner will be attracted to the location of this micro dot. If 
the maximal energy is applied, toner particles will totally cover the 
micro dot. If a low energy level is applied, the amount of toner particles 
can fluctuate randomly between broad limits. The more toner particles are 
deposit on the micro dot, the higher the density reached on this 
particular micro dot. Moreover, it has been found that micro dots, getting 
a low energy level, behave differently depending on the energy levels 
applied to micro dots in their neighbourhood. Micro dots getting a high 
energy level render more consistently the same density, independent from 
the neighbouring micro dots. Further research revealed that micro dots, 
getting a low energy level, become more stable when they are adjacent to a 
micro dot with high energy level. By adjacency is meant that the micro 
dots touch each other side by side or by a corner point. Therefore it is 
advantageous to divide the printable area in halftone cells, each cell 
comprising the same amount of micro dots and redistribute the density of 
the individual micro dots in one halftone cell such that the average 
density--averaged over the micro dots of one halftone cell--approaches the 
average density required for the reproduction of the image. A low average 
density can be reached by two extreme arrangements: 
(1) one or more micro dots in the halftone cell get the highest possible 
density; the other micro dots in the halftone cell get the lowest possible 
density, except for one micro dot that gets an intermediate density so 
that the average density approximates the required density. 
(2) all micro dots get essentially the same density such that the required 
density is equally distributed over all micro dots. 
The first method creates stable density levels but reduces the spatial 
resolution to the size of the halftone cells. The second method preserves 
the spatial resolution, but low densities will randomly fluctuate between 
broad limits. Method (1) is absolutely necessary for low density levels, 
while method (2) can be used for high densities without quality 
degradation. Therefore, in low densities a first one micro dot of a 
halftone cell is raised to a stable density, before a second micro dot in 
the halftone cell can get a small density to increase the average density 
of the halftone cell. From a certain average density level, the micro dots 
surrounding the first stable micro dot can get an increasing density. This 
means that in low densities, only the first micro dot in each halftone 
cell will be visible. In higher densities, this first micro dot and micro 
dots clustered around this first micro dot will be visible. Because all 
halftone cells are arranged in the same way and repeated periodically in 
two dimensions over the whole reproduction, one will notice in low density 
regions a regular mesh of spotlike zones. 
As the average density increases, the area of the micro dots contributing 
to the average density increases. The shape of these clustered micro dots 
can be round, square, elliptical, elongated etc. We have selected an 
elongated shape evolving towards a line screen, because a line screen 
makes the combination of colour components in a colour reproduction less 
registration dependent. This means that in medium densities equally spaced 
parallel lines of micro dots having a high density are visible. Adjacent 
to these lines, the micro dots have a lower density. The stability of the 
latter micro dots is secured by the high density micro dots along the 
parallel lines. 
When the density gets higher, the average density can be more and more 
equally distributed over all micro dots constituting the halftone cell. 
This has the effect that the space between the parallel lines gets filled 
with micro dots having a higher density and that the lines seem to 
disappear. 
The above described arrangement of the micro dots in a halftone cell can be 
obtained by the combination of two line screens. A line screen is a screen 
for which the density along a fixed direction, called the screen angle, 
remains the same for all points along a line in that direction. The 
density varies from low density to high density along every line in a 
direction orthogonal to the screen angle. For photomechanical 
reproduction, the light reflected or transmitted by the original 
continuous tone image is directed to the line screen. The light is 
attenuated proportionally to the local density of the line screen, and the 
attenuated light is directed to a photosensitive material. The combination 
of the two line screens can be realised by putting them on top of each 
other with different screen angles or by applying the first line screen, 
developing the photosensitive material and then modulating the light from 
this photosensitive material by the second line screen under a different 
angle. This will render small dots in the low density regions, line 
structures in the medium densities and a continuum in the highest 
densities. 
The same principles can be applied to characterise formation of the micro 
dots in a halftone cell. In electronic imaging this formation can be 
controlled by a set of N-1 threshold matrices, each matrix having M 
elements, or by a set of M pixel tone curves, each pixel tone curve having 
L entries. The value N represents the number of energy levels. For a 
binary system, N equals 2 and one threshold matrix can describe the 
evolution of the micro dots within one halftone cell. For a system with 16 
energy levels, N=16. The value M represents the number of micro dots in 
the halftone cell. For a square halftone cell with 4 pixels and 4 lines of 
micro dots, M=16. The value L represents the number of digital intensity 
levels I.sub.x,y by which the continuous tone image information for 
location (x,y) on the carrier or one component for colour images is 
represented. For eight bit systems, L=256, i.e. the grey value of a pixel 
of the continuous tone image can be represented by digital values from 0 
to 255. The representation of the halftone cell by pixel tone curves gives 
more flexibility than the representation by threshold matrices, because 
threshold matrices force that the energy level for any specific micro dot 
does not decrease when the average density level of the halftone cell 
increases. Moreover, pixel tone curves are faster to convert a pixel level 
to an energy level. This operation can be done by one look up table 
operation once the location of the micro dot relative to the halftone cell 
is known. It is obvious that a set of threshold matrices can be converted 
to the corresponding set of pixel tone curves and that pixel tone curves 
that are never descending can be transformed to a set of corresponding 
threshold matrices. 
In FIG. 1 is shown how the transformation of a continuous tone image 26 to 
a multilevel halftone image 30 is realised. The carrier 21 is divided into 
micro dots 22. Every micro dot 22 on the carrier 21 is individually 
addressable by the rendering device 23 by an address (x,y). Dependent on 
the required size and orientation of the reproduction, the resolution of 
the rendering device and the original resolution of the image data 
representing the original image, the pixels of the continuous tone image 
are geometrically rearranged by techniques well known in the art to obtain 
an image 26 of pixel data 29, each having a location (x,y) and an 
intensity value I.sub.x,y. This geometric rearrangement can happen before 
the conversion to a multilevel halftone image or during the conversion, as 
the intensity values I.sub.x,y are required. This operation makes that for 
every micro dot 22, there is one input pixel 29. 
All micro dots on the carrier are organised in halftone cells 31. In FIG. 1 
each halftone cell is composed of sixteen micro dots 22. A clock device 28 
generates a clock pulse for every micro dot. At every clock pulse, an 
address generating device 27, coupled to the clock device 28, generates 
the coordinates (x,y) for the next micro dot 22 to be imaged. This address 
(x,y) addresses a pixel 29 of the continuous tone image information 26 and 
sends its intensity level I.sub.x,y to the tone curve transformation unit 
25. This unit also receives the address (x,y) and relates this address 
(x,y) to a micro dot element number i within a halftone cell. For this 
example, i ranges from 1 to 16, because there are sixteen micro dots per 
halftone cell. Each micro dot i has a pixel tone curve L.sub.i associated 
that transforms the intensity level I.sub.x,y to an energy index number j, 
corresponding to energy level E.sub.j. As will be discussed later, the 
energy index j can take a reduced amount of numbers, to address a 
restricted set of energy levels E.sub.j. A table 36 containing these 
energy levels can be indexed by index j to give energy level E.sub.j. That 
energy level E.sub.j together with the address (x,y) is sent to the 
rendering device 23, which results in a density written on micro dot 22 on 
the carrier 21 at location (x,y). 
The way the pixel tone curves are filled with values will determine the 
look of the reproduction. We make use of the concept of line screens to 
fill up values in the pixel tone curve elements. The procedure is 
described here in accordance with FIG. 2. For each constant intensity 
level I.sub.x,y, we generate one value for each pixel tone curve L.sub.i 
within a halftone cell. We select a set of equidistant parallel lines e.g. 
41, 42 that gives an identical line pattern in every halftone cell 45. The 
distance, measured along a line orthogonal to the parallel lines, between 
every two adjacent lines is D. For a given intensity I.sub.x,y, we define 
bands 43, parallel to and centered along the parallel lines, having a 
width W dependent on the intensity level I.sub.x,y. 
The width W is D for the lowest intensity level; 
The width W is 0 for the highest intensity level. 
The width W of each band is further a decreasing function of the intensity 
level I.sub.x,y, i.e. W is proportional to (1-I) if I is the linearly 
normalised value for I.sub.x,y such that I has values between 0 and 1. The 
clause "y varies proportional to x" means that y is an ascending function 
of x, i.e. whenever x increases, y does not decrease. For this specific 
case, for the highest density the bands touch each other and for the 
lowest density there are no bands at all. As can be seen from FIG. 2, each 
micro dot 44 is covered by one or two bands, depending on the width of the 
band. It is now the area of this micro dot 44 covered by the bands 43 that 
gives the amount of density allotted to the specific micro dot. Is the 
micro dot fully covered by one or more bands, then this micro dot will get 
the highest possible density. If the micro dot is not covered by any band, 
then the micro dot will get the lowest possible density. If the micro dot 
is covered for 50% of its area by the bands, then it will get an 
intermediate density between the minimum and maximum possible density. 
This way we can get line patterns as shown in FIG. 3. On a density scale 
from 0 to 255, resulting line patterns are shown for a density of 
respectively 15, 36 and 98. On the left side such a line pattern is shown 
for a line orientation of about 166.degree., on the right side the line 
orientation is 45.degree.. This type of one-dimensional modulation gives 
improved density stability of images. Moreover, the aliasing that can be 
expected to happen when a line with a certain orientation is imaged on a 
discrete grid as the one formed by the micro dots, is smoothed by the 
electrographic process itself, that tends to smear out the toner particles 
along a band along the line. The toner particles are concentrated along 
this line and the background deposition further away from the line center 
is diminished. This has the effect of edge enhancement, originating from 
the electrical field in the finite developing gap. This method has the 
effect that the required density is "concentrated" along the parallel 
lines, giving the lowest density to micro dots more distant from said 
lines. 
A further improvement can be reached by repeating the above step with lines 
having a second direction not coinciding with the first direction of the 
first lines. The density values assigned by the previous method to the 
micro dots are called V.sub.1 and are now used to be modulated by the 
second set of lines. The width of the second set of bands is now 
proportional to V.sub.1. If V.sub.1 is zero, then the bands will have 
width zero; if V.sub.1 is maximal--i.e. the area of the micro dot was 
totally covered by the first bands--then the second bands have a width 
equal to the spacing of their center lines. Again for the micro dot having 
value V.sub.1 assigned, the area covered by the second bands is measured, 
giving a value V.sub.3. This value V.sub.3 is zero if the second bands do 
not cover any part of the micro dot. This value V.sub.3 is maximal if the 
micro dot is totally covered by one or more of these second bands. This 
second modulation "concentrates" the previous line-wise densities towards 
the points where both lines cross each other. This can be seen in FIG. 4, 
where every figure corresponds to the one of FIG. 3, where the extra line 
modulation is added. It is obvious from this figure that for low densities 
(e.g. 15) a periodical repetition of spots appears. For a higher density 
(e.g. 36) these spots get elongated and start connection to lines. For 
even higher densities (e.g. 92), the lines are clearly visible and start 
to fade out to a continuous density all over the reproduction. 
Because for higher densities the micro dots are more stable, the densities 
can be distributed more equally over the micro dots. Therefore the method 
can be modified in a sense that a homogeneous density distribution is 
added to the line-wise or dot-wise distribution. The homogeneous 
distribution gains more importance for higher densities. Therefore, from 
the value V.sub.1 we can derive a value V.sub.2 that is a linear 
combination of the line-wise modulated density V.sub.1 and the homogeneous 
density distribution: 
EQU V.sub.2 =(1-W.sub.1)*V.sub.1 +W.sub.1 *(1-I) 
W.sub.1 is a positive weight factor, not bigger than one. I in (1-I) 
corresponds to the normalised form of the intensity I.sub.x,y. This value 
V.sub.2 is now used to define the width of the second bands. Another value 
for V.sub.3 will now be obtained whenever W.sub.1 is different from zero. 
The same method can be applied to the value V.sub.3, obtained from the 
second set of bands for the specific micro dot, giving a modified value 
EQU V=(1=W.sub.2)*V.sub.3 +W.sub.2 *V.sub.2 
It is obvious that the first method is a special case of this modified 
method, if W.sub.1 and W.sub.2 are selected to be zero. 
In FIG. 5, the first row shows what happens if the first method is applied 
with both weights zero. This figure is effectively correct for low 
densities. For higher densities, the band like structure becomes more 
apparent. The diamond common to the two bands is imaged at high density by 
this operation. The second row shows what happens if the first weight 
W.sub.1 is taken zero and the second taken one. A line screen results. The 
last line shows what happens if both weights have an intermediate value. 
It has been found that the best choice for the weights W.sub.1 and W.sub.2 
is increasing with the density. The higher the density, the more the 
distribution over the halftone cell is allowed to be homogeneous. 
The value obtained for V can now be mapped to the available energy levels 
E.sub.j. Preferentially, a limited set of energy levels is selected and 
fixed ranges of V-values V.sub.j, V.sub.j+1 ! are mapped to one energy 
level E.sub.j. If the number of selected energy levels is 16, then four 
bits suffice to index these energy levels. The pixel tone curves can then 
transform the input levels I.sub.x,y to a four bit value. Usually 
I.sub.x,y is represented by eight bits. This choice gives a saving of 50% 
memory. In electrographic devices where the energy levels are less stable, 
it can be advantageous to allocate only four energy levels, reducing the 
number of bits to represent one micro dot to 2. Other devices having more 
contone capabilities can output reproductions with increased quality if 
the number of energy levels is taken 64, requiring six bits per micro dot. 
The choice of the angle of the parallel lines and the distance between 
these lines is first of all restricted by the requirement that these lines 
have to cover all the halftone cells in the same way. Moreover, it has 
been found that more restrictions are necessary to avoid internal moire. 
Internal moire is due to the interaction of the screen--in this case the 
line screen or the combination of both line screens--with the recorder 
grid or the micro dots. The preferred arrangement of the isolated spotlike 
zones--called halftone dots on the analogy of binary halftoning--for this 
invention is such that they form a periodical structure. More precisely, 
they are arranged along two sets of equidistant parallel lines. Since for 
the highlights, one single small and isolated halftone dot comes up per 
halftone cell, the relative position of the halftone dot with respect to 
the recorder grid or the micro dots should be equal for all halftone 
cells. The first set of equidistant parallel lines is called the base 
lines, the second set of parallel lines, having a different orientation 
and thus intersecting the base lines, are called the auxiliary lines. 
Because the base lines and auxiliary lines go through the midpoints of the 
halftone dots, the center of the halftone dots is situated at the point of 
intersection of a base line and an auxiliary line. It has been found 
that--in order to avoid internal moire--these points of intersection or 
the centers of the halftone dots must be situated on points that have 
always the same relative position or "spatial offset" with respect to the 
closest micro dot. In other words, if all points of intersection are 
translated over exactly the same distance such that one point of 
intersection coincides with the center of a micro dot, then all points of 
intersection coincide with the center of a micro dot. We make here the 
difference between base lines and auxiliary lines, because at medium image 
densities the micro dots with highest density are arranged along these 
base lines. Among the micro dots along these base lines there can be also 
be a density difference. In that case, the micro dots with highest density 
are situated closest to the auxiliary lines. 
This type of arrangement of halftone dots, gradually evolving to line 
patterns can be used for monochrome images. A combination of these 
arrangements can be used for multi tone reproductions. In that case one 
halftone image having one colour will be superimposed on one or more such 
images having a different colour. It is known from binary halftoning that 
two halftone images can interfere with each other and produce moire. When 
a third halftone image is superimposed on the set of two, secondary moire 
can arise. It has been found that these types of moire can also be 
produced by multilevel halftone images if the dot or line arrangement for 
each individual halftone image and the combination is not selected 
adequately. Specially for colour reproductions composed of three halftone 
images: a cyan, magenta and yellow component or reproductions composed of 
four halftone images: a cyan, magenta, yellow and black component, it has 
been found that at least three of the individual components preferably 
have the properties for the monochrome halftone image as discussed above. 
In that case, there are at least three sets of base and auxiliary lines. 
Like in binary halftoning such as offset printing, only three of the four 
separations of cyan, magenta, yellow and black are given high weight in 
the optimisation of the screen. Therefore we emphasize on the role of 
three separations with their accompanying screen geometry. Each set of 
base and auxiliary lines can be selected from a set of three or more 
generic lines. Every set of base and auxiliary lines will produce a set of 
points of intersection. The relative position of these points of 
intersection relative to the closest micro dots can be established for 
every set. In a preferred embodiment, this relative position must be equal 
for each of at least three colour components. 
Also the orientation of the base lines plays an important role when three 
multilevel halftone images are combined to produce multi tone or more 
specifically colour images. This is especially true for the mid tone areas 
of the image, where line-wise structures along the base line show up. 
First of all it is important that the base line of one colour component is 
not parallel to any of the base lines of the other two components. Three 
base lines selected from the three halftone images, such that they have no 
common point of intersection, form a triangle. In order to have quite 
similar screen rulings or line rulings and thus also a comparable spatial 
resolution for the three components of the colour image, it is preferred 
that this triangle has no obtuse angle. Implementations where the 
deviations between the spatial screen rulings of the different colour 
separations are minimal, have the advantage that physical processes such 
as dot gain are similar for the different separations. Especially in 
electrography with finite gap magnetic brush development, the development 
response will be different for screens of different spatial rulings. 
Geometries as presented here based on triangles without obtuse angle, will 
therefore benefit from equal development response. Even more preferable, 
the triangle should approach an equilateral triangle. These conditions 
restrict the relative orientation of the base lines for different 
multilevel halftone images with respect to each other. 
Furthermore, also the absolute orientation of the base line and the 
auxiliary line plays an important role in the quality of a single 
monochrome multilevel halftone image, and thus also in the quality of a 
multi tone image composed of a set of halftone images. It is preferred 
that the base line is not horizontal nor vertical. If the base line is 
horizontal, then the tone curves of micro dots along horizontal lines are 
equivalent in the first step of the production process of them. This means 
that the micro dots on the same horizontal line get the same density 
allotted, equally distributed over these micro dots, and micro dots not on 
the same horizontal line get densities allotted for another range of image 
intensity levels. The use of contone according to the method described 
above is especially beneficial for sloped lines. Therefore, to benefit 
from the method and in order to keep the characteristics of the different 
colour separations the same, it is preferred to chose screen angles 
different from 0.degree. and 90.degree.. 
Normally, if four colours are used for printing, then the cyan, magenta and 
black component have to obey the rules sketched above for optimal 
rendering. The yellow component is less critical, first because it looks 
less dense to the human observer and because it has less side absorptions 
in the visual band. We have found that for the yellow component the 
halftone cell structure for the black component can be taken and be 
mirrored along a horizontal axis or a vertical axis or one fixed point or 
any sloping line at 45.degree.. 
A method has been devised to find optimal combinations of angles and 
distances for a combination of three multilevel halftone images, such that 
they obey the restrictions sketched above. The method gives distances 
expressed in micro dot units. The effective screen ruling can be obtained 
by dividing the recorder grid resolution by the distance between the base 
lines expressed in micro dot units. 
The method finds three generic lines L1, L2 and L3 as shown in FIG. 16. 
L1 is selected as the base line B1 for the first halftone image. 
L2 is selected as auxiliary line A1 for the first halftone image. 
L2 is selected as the base line B2 for the second halftone image. 
L3 is selected as auxiliary line A2 for the second halftone image. 
L3 is selected as the base line B3 for the third halftone image. 
L1 is selected as auxiliary line A3 for the third halftone image. 
As shown in FIG. 6, the orientation of L1 is given by a first vector V1 
from the origin (0,0) to point P1 (X.sub.1,Y.sub.1). If the origin is 
situated on the center of a micro dot, the point P1 must be situated also 
on the center of another micro dot. This means that X.sub.1 and Y.sub.1 
have integer values. This restricts the choice for X.sub.1 and Y.sub.1 
dramatically. This choice is further restricted by the screen ruling that 
is aimed for. Next all possible positions for a second point P2 with 
coordinates (X.sub.2,Y.sub.2) are tested. The set of points P2 is 
restricted by the fact that again both X.sub.2 and Y.sub.2 must have 
integer values and that the length L.sub.2 of the vector P2 given by 
EQU L.sub.2 =sqrt(X.sub.2.sup.2 +Y.sub.2.sup.2) 
must be not too different from length L.sub.1 of the vector P1, given by: 
EQU L.sub.1 =sqrt(X.sub.1.sup.2 +Y.sub.1.sup.2) 
From these two vectors P1 and P2, we can derive a third vector P3=P2-P1, 
with coordinates (X.sub.2 -X.sub.1,Y.sub.2 -Y.sub.1). We take now the 
first generic line parallel to P1, the second generic line parallel to P2 
and the third generic line parallel to P3. The distance between generic 
lines L1 is the orthogonal distance of P2 to vector P1. The second generic 
line is taken parallel to P2. The spacing between generic lines L2 is the 
orthogonal distance of point P1 to vector P2. The third generic line L3 is 
taken parallel to vector P3. The spacing between lines L3 is the 
orthogonal distance of point P1 or P2 to vector P3 through the origin. It 
is obvious that the spacing between the generic lines is dependent on the 
length of P1, P2 and P3. If we want equal spacings and thus equal line 
rulings, the length of these vectors must be equal. In other words, the 
differences L.sub.2 -L.sub.1, L.sub.3 -L.sub.1 and L.sub.2 -L.sub.3, where 
L.sub.3 is: 
EQU L.sub.3 =sqrt(X.sub.3.sup.2 +Y.sub.3.sup.2) 
must be minimal. Therefore, for every possible vector P2, the following 
metric is computed: 
EQU M=(L.sub.1 -L.sub.2).sup.2 +(L.sub.2 -L.sub.3).sup.2 +(L.sub.3 
-L.sub.1).sup.2 !/A.sup.2 
with A=area of the triangle (0.0). P1, P2. The vector P2 that minimizes 
this metric M is taken as the best candidate for the vector P2. Another 
preferred way of selecting the best base lines is by listing all 
combinations (X.sub.1,Y.sub.1), (X.sub.2,Y.sub.2), computing the area for 
the corresponding parallelogram built upon those two vectors and selecting 
for each area one combination that gives the best metric. We give here a 
table of some vectors P1, P2, P3 found by this method, along with the 
number of micro dots or "cell area" in the halftone cell they define. 
It is possible to restrict the (X.sub.1,Y.sub.1) combinations to those 
where X.sub.1 and Y.sub.1 are both non-negative and X.sub.1 
.gtoreq.Y.sub.1. After processing, equivalent cell structures can be 
obtained by interchanging the role of X and Y, meaning mirroring about a 
line of 45.degree., by making X.sub.1 and/or Y.sub.1 negative, or any 
combination. The corresponding vectors P2 and P3 follow the transformation 
accordingly. 
______________________________________ 
X.sub.1 
Y.sub.1 
X.sub.2 Y.sub.2 
X.sub.3 
Y.sub.3 
Area 
______________________________________ 
3 1 -2 1 1 2 5 
3 -1 -2 -2 1 -3 8 
4 -1 -3 -2 1 -3 11 
4 -2 -3 -2 1 -4 14 
4 -1 -3 -3 1 -4 15 
3 3 2 -4 5 -1 18 
5 -1 -4 -3 1 -4 19 
3 4 2 -5 5 -1 23 
4 4 1 -5 5 -1 24 
6 1 -3 -5 3 -4 27 
5 -2 -1 6 4 4 28 
6 -2 -2 6 4 4 32 
6 -2 -1 6 5 4 34 
7 -2 -2 6 5 4 38 
6 -3 -1 7 5 4 39 
______________________________________ 
The halftone cell is exactly the same for the three halftone images and is 
the parallelogram built upon the vectors P1 and P2. The vector P3 is the 
shortest diagonal line for this parallelogram. In FIG. 6 the example from 
the above table is shown where P1=(7,-2), P2=(-2,6) and P3=(5,4). Each 
elementary halftone cell covers 38 micro dots. These elementary halftone 
cells can be arranged in a supercell of 19.times.19 micro dots. 
The method described above gives an example to generate pixel tone curves, 
and results for every micro dot R.sub.i (i=1..M, M=number of micro dots 
per halftone cell) in a value V per entry I.sub.x,y (I=1..L, L=number of 
possible intensity levels in the input image) in the pixel tone curve 
L.sub.i. More generally, we can state that V=g(x,y,I). Other functions 
g(x,y,I) can be established using other methods. As said, among all 
possible energy levels E offered by the rendering device, only a 
restricted set of N energy levels E.sub.j are really distinct enough to 
render consistently different densities on the output. Suppose that the 
halftone cell covers M=11 micro dots, and the number of selected energy 
levels N=16, then it is theoretically possible to generate M*N=176 
different output density levels. Even if the energy levels E.sub.j are 
optimally chosen, these output density levels will not be equally spaced. 
If we have 256 input intensity levels, we must map mostly two input 
intensity levels to one output density level, and due to the non-linear 
spacing of these output densities, sometimes three or even four input 
levels must be mapped to one output density level. This can result in 
contouring and quality loss. The object is to guarantee a correct grey 
rendering with a restricted amount of energy levels E.sub.j. 
Therefore it is advantageous to take an amount of S halftone cells together 
to form a supercell. The number of micro dots in the supercell equals M*S. 
The higher S, the more different output density levels--averaged over the 
supercell--can be generated. The number S is selected such that the micro 
dots can be rearranged to form a square supercell. Each micro dot in the 
supercell has now a pixel tone curve defined by the function g(x,y,I), 
that is periodical over all supercells. It is obvious that every pixel 
tone curve with identical g(x,y,I) values will be present S times in the 
supercell. Such identical pixel tone curves are called equivalent pixel 
tone curves. Due to symmetry of the halftone cell, it is possible that the 
halftone cell itself already contained equivalent pixel tone curves. The 
function V=g(x,y,I) gives an order in which halftone cell types are formed 
to give increasing densities. A halftone cell type is the distribution of 
energy levels over the micro dots of a halftone cell. Each halftone cell 
type will give a specific average density, averaged over the halftone 
cell. The halftone cell can be an elementary halftone cell or a supercell 
composed of elementary halftone cells. To render the lowest density, all 
micro dots will get an energy level E.sub.0. For the next higher density 
for the supercell, one micro dot must get an energy level E.sub.1, while 
all the other micro dots keep energy level E.sub.0. The micro dot having 
the largest value for V=g(x,y,I) will be the candidate to increase its 
energy level. Because the supercell is composed of S elementary halftone 
cells, there will be at least S candidates to increase the energy level 
from E.sub.0 to E.sub.1, or more generally from E.sub.j to E.sub.j+1. 
Thus, the function V=g(x,y,I) gives only a coarse indication or primary 
sequence on which micro dot gets an energy increment. When converting the 
values V=g(x,y,I) just by a kind of truncation to indexes j for E.sub.j, 
which is a type of quantisation of the merely continuous values for V, all 
micro dots belonging to equivalent pixel tone curves are candidates. If no 
finer ordering is imposed, this quantisation results in an error in output 
density. One way of assigning the next higher energy level to a pixel tone 
curve, is to list them in a sequential order according to the elementary 
halftone cell where they belong to. But this will always introduce the 
same pattern in the supercell, which will give visual artifacts in the 
image. The error function due to quantisation will contain low frequency 
components that are visually perceptible. It is better to generate an 
error function that has high frequency components. This error function can 
be created by superposition of a pattern e(x,y,I) on the function g(x,y,I) 
: 
EQU g'(x,y,I)=g(x,y,I)+e(x,y,I) 
The function e(x,y,I) is preferentially selected such that it varies 
between zero and the minimum difference between different values of the 
function g(x,y,I) in any different points (x.sub.1,y.sub.1) 
(x.sub.2,y.sub.2). This means that the error function imposes a sequence 
on equivalent tone curves or identical V=g(x,y,I) values, but the error 
function does not change the order imposed by the function g(x,y,I). 
In a preferred embodiment, the error function e(x,y,I) is not dependent on 
the intensity level I and as such e(x,y) is only dependent on the location 
(x,y) of the micro dot within the supercell. This way the same subordinate 
sequence is imposed on all identical g(x,y,I) values, whatever the 
intensity value might be. 
In a more preferred embodiment, the error function e(x,y) varies according 
to the sequential numbering in a Bayer matrix. A Bayer matrix is well 
known in the art (see e.g. "An optimum method for two-level rendition of 
continuous-tone pictures" by B. E. Bayer in Proceedings IEEE, 
International Conference on Communications, Volume 26, pages 11-15, 1973) 
for cell sizes that have a width and height that is a power of two. If the 
supercell size is not a power of two, we can define a generalised Bayer 
matrix as one with a bigger size being the next power of two and with a 
subsection taken that covers the supercell. It is also possible to define 
a function over integer values 1..8 that returns the traditional Bayer 
matrix for values 1, 2, 4 and 8, and that gives a generalised Bayer matrix 
for other values. This function can be used to generate a generalised 
Bayer matrix if the supercell size can be decomposed in prime numbers 
lower than eight. 
Another quality improvement can be reached when the Bayer matrix is 
randomized in the following way. Each smallest Bayer submatrix--this is a 
2.times.2 matrix if the supercell size is a multiple of 2, this is a 
3.times.3 generalised Bayer matrix if the first prime factor for the 
supercell size is 3, etc..--is randomized. That means that the sequence 
numbers are randomly permuted. For a 2.times.2 submatrix, there exist 
4|=24 permutations. A traditional white noise random generator can be used 
to generate numbers from 1 to 24 to select randomly one of the possible 
permutations for every sub-matrix in the Bayer matrix. This has the 
advantage that no aliasing effects can be observed due to the screen 
ruling imposed. 
The process for establishing the pixel tone curves can be summarized as 
follows: 
(1) Define supercells having micro dots R.sub.x,y 
(2) Compute for every micro dot R.sub.x,y a pixel tone curve G.sub.x,y 
=g(x,y,I) according to a function that maps an input intensity level I to 
an output density g(x,y,I) for the micro dot at location (x,y). The value 
g(x,y,I) indicates the order in which the energy level E.sub.j must be 
increased to energy level E.sub.j+1. 
(3) Add an error function e(x,y,I) to each pixel tone curve G.sub.x,y to 
obtain G'.sub.x,y. This error function imposes a subordinate order for 
which micro dot increases its energy level from E.sub.j to E.sub.j+1. 
(4) Assign energy levels E.sub.j or indexes j to the pixel tone curves 
L.sub.x,y in the order as indicated by the function G'.sub.x,y. 
As stated above, a restricted set of M energy levels must be selected from 
all possible energy levels. It has no sense to use all possible energy 
levels because the variation of densities produced by one energy level 
overlaps too much with the variation of the next possible energy level. By 
selecting a restricted set of energy levels, the number of bits in the 
bitmap representing the halftone image can be reduced. The number of 
energy levels is chosen in function of the contone capabilities of the 
output device. If the variation in density for different energy levels is 
small, then the restricted set can contain a big amount of energy levels, 
typically 64. Normally 16 energy levels are selected. This has the 
advantage that the energy level for two micro dots can be stored in one 
byte of eight bits. In systems with poor contone capabilities, typically 4 
energy levels will be selected. 
Preferably the energy levels must be selected such that the next energy 
level gives the same decrement in reflectance, for every energy level 
E.sub.j. This can be achieved by the following method. First of all, a 
representative subset--e.g. 16--of all possible energy levels--e.g. 
400--is selected. In this example energy level 1, 25, 50, etc. could be 
selected. It would be also possible to select all 400 energy levels to 
accomplish the following procedure. In that case, the subset is the full 
set of available energy levels. As shown in FIG. 7, with every energy 
level from the subset a patch 52 or 53 is output on the image carrier 51 
or on the unit where the latent image is converted to a visible image as 
will be discussed further. The size of each patch is preferably 10 mm high 
and 10 mm wide. As such it can be easily measured by an integrating 
densitometer. Every patch consists of small isolated zones 54. The zones 
are spaced apart from each other such that they don't influence each 
other. Preferably, there is a spacing of at least two micro dots between 
each zone 54. The zone itself consists of a high density kernel 55, and a 
halo 56. The high density kernel is preferably one micro dot, imaged by 
the maximum energy level available. Also lower energy levels can be used 
for this kernel 55, as long as it is a stable energy level that gives a 
stable density on its own and stabilises the density of its neighbouring 
micro dots whatever energy level they have. The halo 56 is imaged by one 
energy level from the subset, e.g. E.sub.1, E.sub.25. Preferably, the 
width of the halo is one micro dot. As such, each zone 54 has nine micro 
dots, the center of which is imaged at maximum energy level and the other 
eight micro dots are imaged with the energy level from the subset. Each 
patch 52 is thus imaged by three energy levels: the lowest energy level in 
the background between the zones 54, a stable energy level and an energy 
level E.sub.j from the subset. On one carrier, different patches 52, 53 
with different energy levels selected from the subset can be imaged. After 
imaging, the reflectance R.sub.j of each patch 52, 53, . . . having energy 
level E.sub.j (e.g. E.sub.1, E.sub.25, . . . ) is measured. This can be 
done by an integrating densitometer that will take the average density of 
a large amount of zones 54 together with the low density background. As 
shown in FIG. 8, the reflectance R.sub.j can be plotted against the energy 
level E.sub.j. This curve has been obtained by performing the above 
described measurements on the Chromapress system marketed by Agfa-Gevaert 
N.V. Both the reflectance and the energy level are normalised to 0,1!, by 
linear scaling and subtracting an offset. Moreover, the direction of the 
reflectance axis Norm.sub.-- ref1 is inverted, giving an ascending 
function R=f(E). The measurement points are interpolated or approximated 
by a piecewise linear, quadratic or cubic curve, giving a continuous 
function R=f(E). Because we are interested to find the energy levels that 
give equal decrements in reflectance. the interval R.sub.min,R.sub.max ! 
is divided in subintervals R.sub.k,R.sub.k+1 !, all having the same 
length on the reflectance axis. The number of subintervals is the number 
of selected energy levels E.sub.k minus one. If we want to select sixteen 
energy levels, the number of intervals will be fifteen. Via inverse 
evaluation of the function R=f(E) in the points R.sub.k delineating the 
subintervals, we find the selected energy levels E.sub.k. 
It has been surprisingly found that the same method applied to an 
arrangement as sketched in FIG. 9 gives the same results. Here, the patch 
61 on the image carrier 62 consists of parallel bands 63, spaced from each 
other at a distance such that there is no influence of the bands on each 
other. Each band 63 consists of a narrow central band 64 and two narrow 
side bands 65 and 66. As in the previous embodiment, the central band 64 
is imaged by a high energy level that gives a stable density, stabilising 
also the density of the micro dots in the side bands. The side bands 65, 
66 are imaged by energy levels E.sub.j from a subset of the available 
energy levels. The rest of the procedure is essentially the same, as 
described in relation with FIG. 7 and FIG. 8. The same energy levels 
E.sub.k are found. Moreover, if the bands are arranged such that on a full 
patch 61 the percentage of high density micro dots 64, the percentage of 
intermediate density micro dots 65 and the percentage of lowest density 
micro dots are the same as for the spotlike zones 54 in FIG. 7, then the 
reflectances R.sub.j measured are the same. This proofs that all energy 
levels produce consistently one density level as long as they are 
stabilised by a neighbouring micro dot. 
It was found also that the reflectances R.sub.j are dependent on 
environmental parameters like temperature, humidity, further dependent on 
the type of toner and paper used, the age of the drum etc. Therefor it is 
advantageous to repeat these measurements for different parameters and 
store the results E.sub.k as a function of these parameters. If e.g. 
sixteen energy levels E.sub.k are selected, then it is very easy to store 
sixteen sets of selected energy levels. The parameters named above can be 
measured or kept track off, and whenever necessary, the energy level table 
36 shown in FIG. 1 can be reloaded. This is a very fast operation that can 
improve dramatically the reproducibility of the output for different 
environments. This method to obtain a restricted set of energy levels 
E.sub.j summarises the whole printing engine in just the sequence of 
energy levels. The non linearities of the output device are corrected by 
this sequence. This method can preferentially be implemented in an 
automatic image density control system. The patches described above can be 
imaged on a specific location on the drum where the latent image is 
formed, toner is applied to this location, the location is illuminated by 
a LED or laserdiode and the reflected light is measured by a 
photosensitive sensor, that integrates over the patch. 
Once the restricted set of energy levels has been selected and the order 
for increasing the energy levels in the different pixel tone curves is 
fixed, we must select exactly L energy distributions over the supercell or 
L halftone cell types from all the possible distributions offered. L 
represents the number of intensity levels in the input image, and can be 
typically 256. Aspects due to the cell structure, such as dot gain can be 
linearised by the next method. If the number of selected energy levels N 
is sixteen, and the number of micro dots S*M in the supercell is 225, then 
there are N*S*M=16*225=3600 halftone cell types numbered N.sub.j in a 
sequence to bring the supercell from lowest density to highest density. 
The order in which the energy levels are incremented for each micro dot 
can be given by the perturbated function G'(x,y,I) as described above. It 
is now possible to output patches of 10 mm by 10 mm for all 3600 halftone 
cell types in the sequence. The patch is overlaid with supercells, and 
each supercell within the patch has exactly the same distribution of 
energy levels, given by the sequence number N.sub.j. The reflectance of 
the patch composed of supercells imaged by halftone cell type N.sub.j can 
be measured, giving a reflectance value R.sub.j. The measured reflectances 
R.sub.j can be plotted against N.sub.j as shown in FIG. 10. Also this 
curve has been obtained by performing the above described measurements on 
the Chromapress system. Here again both axes are normalised to the 
interval 0,1! and the reflection axis is oriented inversely. This gives 
on the R.sub.j axis a point with minimum reflectance R.sub.min and a point 
with maximum reflectance R.sub.max. The interval R.sub.min,R.sub.max ! is 
divided in L-1 equal subintervals, because it is an object to map the L 
intensity levels Ix,y linearly to a reflectance level R. This subdivision 
gives L-I subintervals R.sub.k,R.sub.k+1 !. For each value R.sub.k the 
value R.sub.j that is closest to it is searched and the corresponding 
sequence number N.sub.j is selected as a candidate for defining the energy 
distribution over the supercell for intensity level I.sub.k that will be 
mapped to reflectance R.sub.k. 
As the supercell gets larger, it becomes unpractical to list all 
arrangements N.sub.j found by the function G'(x,y,I). Therefore. a 
preselection of a subset of all arrangements N.sub.j, preferably 
containing 16 elements, is made. It is possible to define as the subset 
the full set of all available halftone cell types N.sub.j. Again the plot 
as shown in FIG. 10 is made, but the number of measurement points is less 
dense. For every required reflectance R.sub.k, the index j is sought such 
that R.sub.j .ltoreq.R.sub.k &lt;R.sub.j+1. R.sub.j corresponds with sequence 
number N.sub.j and R.sub.j+1 with sequence number N.sub.j+1. Because in 
the previous step we made that an increment in energy level gives a 
constant decrement in reflectance level, and for every next element in the 
sequence one energy level of the complete supercell is incremented, we can 
linearly interpolate between N.sub.j and N.sub.j+1 to find the 
sequence--number N.sub.k that gives the reflectance R.sub.k : 
EQU N.sub.k =N.sub.j +(N.sub.j+1 -N.sub.j)*(R.sub.k -R.sub.j)/(R.sub.j+1 
-R.sub.j) 
The function R.sub.j =f(N.sub.j) is highly non-linear. Therefore, if only 
sixteen samples of this function are measured, it is advantageous to make 
a continuous interpolation or approximation by a piecewise non-linear 
function, such as a cubic spline function. Approximation will further 
offer the ability to smooth out measurement errors. 
If the selected subset of combinations is such that it lists all possible 
combinations wherein the equivalent micro dots of different halftone cells 
in the same supercell always have the same energy level, then the 
difference in reflectance between the required reflectance R.sub.k and the 
obtained reflectance R.sub.j+1 can be used to determine the number of 
equivalent micro dots that must get an energy increment. Which micro dots 
will get the increment is determined by the error function or the 
subordinate sequence given to the micro dots in the supercell. 
It is obvious that this method can also be applied to elementary halftone 
cells, but the reflectance levels that can be really reached will be much 
coarser. 
Although the present invention has been described with reference to 
preferred embodiments, those skilled in the art will recognise that 
changes may be made in form and detail without departing from the spirit 
and scope of the invention.