• 4.1.1-PICTURE SYNTHESIS:
• 4.1.2-THE TEMPORAL FACTOR:
• 4.1.3-SPACES OF RESULTS:
• 4.1.4-SPACE OF DISPLAY:

• 4.2.1-PROJECTION:
• 4.2.2-TRANSFORMATIONS:
• 4.2.3-REDUCTIONS:
• 4.2.4-THE CHAIN OF PRIMARY OPERATIONS:
• 4.2.5-THE CHAIN OF SECONDARY OPERATIONS:
• 4.2.5.1-The problem of color selection:
• 4.2.5.2-Various representation modes:
• 4.2.5.3-Dynamic display:

• 4.3.1-A PICTURE IS ONLY A SINGLE VIEWPOINT:
• 4.3.2-TRANSFORMATIONS:
• 4.3.3-ARTEFACTS:
• 4.3.3.1-Geometric illusions:
• 4.3.3.2-Luminance illusions:
• 4.3.3.3-Chrominance illusions:
• 4.3.3.4-"Computing" illusions:
• 4.3.4-UNKNOWN OBJECTS:
• 4.3.5-THE QUALITY OF THE DISPLAY SYSTEM:

4.1.1-PICTURE SYNTHESIS:

In its generally-accepted form, synthesis is defined as being an assembly process in which elements pertaining to a given "object" are gathered into a coherent whole, thereby providing a general oversight. In our context, the object is the system being measured or simulated and the elements are the results obtained (by measurement or simulation). Synthesis will therefore be an application that can be used to pass from the space of results to the space of display.




4.1.2-THE TEMPORAL FACTOR:

the dimensions of the spaces of results and display include the time factor (except in certain specific cases such as static problems or the problem of representation of "mathematical" objects, for example). We shall start from a hypothesis which, although it is not systematically realised at present, can be approximated or simulated i.e. that the picture display system is capable of producing dynamic displays. Working from this basis, it becomes clear that "three different time factors" will be useful:


• Tp: the inherent model time, i.e. the variable t which is included in equations,
• Tm: the time corresponding to an observer movement around the "objects" represented (as we shall see in paragraph 4.2.1, this is enormously helpful during the observation of complex systems with a spatial dimension greater than 2), and
• Td: the time that could be used to represent a dimension which cannot otherwise be displayed. For example, and for the purposes of simplification, the graph of a function y=f(x) can be represented either in the traditional manner within a cartesian coordinate system in which a curve shows the value of the ordinate y for every value of the abscissa x, or in a less conventional manner by a mobile point on an axis for which the instantaneous abscissa has the value f(t).

It quickly becomes obvious that these three different times can be used simultaneously and it is clear, as we shall see later, that a number of precautions must be taken inasmuch as all three time factors refer to the same physical time (see figure "Tridimensional display of the dynamics of a linear superposition of 6 eigenstates of the Hydrogen atom (bidimensional computation)").




4.1.3-SPACES OF RESULTS:

spaces in simulated systems, whether "physical" or "abstract", are generally subsets of Rn as regards the mathematical formulation and subsets E of Qn as regards computer programming (real integers being approximated by "almost" rational numbers, using floating point representation -see for a review of problems inherent to their use-). The results themselves will generally be considered as functions defined in each point of E. Depending on the situation, they may be:


• scalar, i.e. having a rational value at each point (e.g. a pressure field -see figure V14-),
• "multi-scalar", i.e. having several rational values of different types at each point (e.g. a pressure field and a temperature field),
• vectorial, i.e. having a vector at each point (e.g. a motion field -see figure F3-). Finally, it is possible to imagine more "exotic" functions (e.g. by "overlaying" a curl field with a velocity field).

In fact, it is clear that the above list is far from complete and is given solely in order to show the complexity of the problem. It is easy to convince oneself, as we shall illustrate below, that scalar functions in one- and bidimensional spaces are the easiest to represent, and are highly instructive, leading the way to a wide range of possibilities.




4.1.4-SPACE OF DISPLAY:

in this case, the situation is simpler. In a realistic, pragmatic context, far removed from all hypothetical resources (e.g. using real time numerical holograms for example...), the only way of displaying information is by means of a "video" system. In simple terms, it consists of four basic components:


• an "interface" linking the simulation resources (in this context, the word "interface" has the widest possible meaning);
• a picture memory containing at least as many words (of one or more bytes) as there are points to be represented;
• a video generator (see "computing illusions" in paragraph 4.3.3.4) which provides the correspondance between numerical values and colors and which produces television signals;
• and, finally, a monitor which will display the pictures that stimulate our visual senses (reproduction resources such as a camera, VCR, videodisk, etc... can be added).

Thus a display system brings us back to a bidimensional scalar field which is memorized in the picture memory in the form of a matrix, each of its elements appearing in the form of a colored pixel. It is then easy to understand how the visual synthesis of results usually reduces the quantity (and quality) of the data contained in the results, and to what extent visual synthesis (except in a few cases such as 2D scalar fields, although this statement must be tempered for reasons relating to the choice of representation modes as we shall see in paragraph 4.2.5) can present only one point of view at a given moment in time (this remark does not ignore the notion of "windowing" but underlines the bidimensional aspect of the display medium).

4.2.1-PROJECTION:

this operation is used to pass from the n dimensions of the space of results to the two dimensions of a plane (e.g. the space of display). Unfortunately, it does not provide an overview of the results and it is there that the inherent movement of the observer according to time Tm can be very useful, on condition that times Tp and Td are frozen. The development of techniques of the flight simulator type, although they are constricting (inasmuch as they demand the computation of pictures in real time, i.e. in a fraction of a second), is to be highly recommended and will be looked at in detail below. In particular when applied to objects that are, at present, unknown to us, for example, tridimensional fractal aggregates (see figure "Black and white display of a tridimensional function") or certain works of art, they would render apprehension possible or easier.

Projection is mandatory when the dimension n is greater than 2 but it has to be said that several such operations might be used in succession. For example, a tridimensional surface could be projected onto the three planes of a cartesian coordinate system, which would itself have to be projected onto the display space plane (or monitor).




4.2.2-TRANSFORMATIONS:

transformations concern the manipulation of results without changing the dimension of the spaces. They could be used, for example, to reduce the dynamics of a field or, on the contrary, to partially enlarge the dynamics in certain important or interesting areas. It will be possible to differentiate between the operations that transform the space of results (e.g. translations, rotations, similarities, etc...) and those which transform the results themselves (e.g. the changes in "dynamics" mentioned above). In all cases, and this is important despite its general character, the picture produced in this way must contain data recalling the type of non trivial operations performed on the results!




4.2.3-REDUCTIONS:

The above operations reduce the results mandatorily and implicitly; there are others that act explicitly, for example averaging, thresholding and filtering. The latter is particularly useful in extracting the main structures from a field (see figure "The simple case of a bidimensional scalar (fractal) field presented in four widely differing color palettes", top right). Finally, cutting is used to extract a sub-space of results, e.g. a plane, and will be useful in the "internal" comprehension of spaces with a dimension greater than 2.




4.2.4-THE CHAIN OF PRIMARY OPERATIONS:

this provides the changeover from the space of "unprocessed" or "primary" results (i.e. results provided directly by simulation) to the space of bidimensional "secondary" results.




4.2.5-THE CHAIN OF SECONDARY OPERATIONS:

this final chain is used to produce the picture, and consists of the selection of representation modes, in particular the color palette. This final operation, which is often considered as no more than a puerile form of coloring, is frequently neglected and completed in an anarchic fashion yet it is a vital part of the process as a whole and it requires the definition of, and compliance with, a certain methodology.



4.2.5.1-The problem of color selection:

without contesting the notion of color and RGB additive synthesis as used at the present time in television (although it must be remembered that it is based on the reproduction of all "subjective" colors, or colored visual sensation, with the use of three basic colors i.e. Red, Green, and Blue), it should be observed that, generally speaking, there are no specific colors associated intrinsically with a given space of results. With the exception of so-called "realistic" picture synthesis, coloring will be an arbitrary process (for example: "what color is a pressure field or a quark?"). In spite of this, two general remarks can be made on the basis of correlations that the human mind comprehends more or less unconsciously and that are based as much on innate knowledge as on acquired information:


• R1: we associate "cold" colors (green, blue, etc...) with low, negative values and "warm" colors (red, yellow, etc...) with strong, positive, or even dangerous values.
• R2: we see a correlation between scales of increasing luminance (levels of gray between black and white) and sets of numerical values that progress in the same way.

These two observations may not provide an automatic definition of the colors required, but they provide a guideline for the selection process and avoid a number of frequent errors. It is, for example, fairly common to see pictures published in scientific revues in which an incoherent code confusing the "aesthetic" with the "informative" is used for arbitrary associations such as:


• values 0 - 9 yellow (warm, very light color)
• values 10 - 19 blue (cold, very dark color)
• values 20 - 29 red (warm, dark color)
• values 30 - 39 cyan (cold, light color)
• etc...

A picture using this type of incoherent order cannot be "read" without a color look-up table. Moreover, even with the table, the picture is difficult to read because the code contradicts the two "cultural" observations indicated above (R1 and R2). The mind switches between two possible interpretations (e.g. values 0 - 9 are low, yet they are coded using yellow which is naturally associated with high values). This example also reminds us that there is no one natural order between colors but rather a number of orders, in particular the order of colors in the rainbow and in television (viz. black, blue, red, magenta, green, cyan, yellow, white, in which what we wrongly term "colors", such as white, are classified in increasing order of luminance). This being so, we shall distinguish between:


• monochromatic palettes (cf. R2) which are useful for "single-tone" scalar fields,
• and polychromatic palettes (cf. R1) which are required when distinguishing between various zones, for example in scalar fields (the zones represent positive and negative values, values between two given thresholds, etc...). Within a given color, luminance can vary monotonously, periodically, or in peaks.

Let us give a simple example of this, a 2D fractal field produced by research into natural phenomenon synthesis. In this case, the space of results bears a strong similarity to the space of display and, therefore, to the picture. Each pixel in the picture has a corresponding point in the field (defined on a 780x575 grid) and the numeric segment of possible values is made to correspond with the [0,255] segment of the numbers of utilizable colors. Beneath each of the pictures in figure "The simple case of a bidimensional scalar (fractal) field presented in four widely differing color palettes", which all illustrate the same field, there is a color scale indicating the color associated with [0,255], if read from left to right. This type of indication must be systematically included with every picture in order to define or recall the code used. On figure "The simple case of a bidimensional scalar (fractal) field presented in four widely differing color palettes" (top left), a monochromatic palette is used: the high negative values are represented in dark luminance, while the high positive values are represented in light luminance. On figure 14 (top right), an "all or nothing" two-colored palette is used (note that the warm color, yellow, has been linked to negative values while the cold color, blue, is linked to positive values, in order to create the "uncomfortable sensation" described above, see observation R1). Although both pictures concern the same field, the visual impression is very different and only the main spatial structures stand out. In this case, low-frequency filtering is in operation. On figure "The simple case of a bidimensional scalar (fractal) field presented in four widely differing color palettes" (bottom left), the previous palette is periodical, and the high spatial frequencies become obvious. Finally, in figure 14 (bottom right), a polychromatic palette is used, to comply with the two basic observations R1 and R2:


• increasing luminance (varying in three peaks) is used to show the progression of the numerical values (R2),
• green (a cold color) is linked to negative values (R1)
• red ( a warm color) is linked to positive values (R1)
• gray (a neutral "color") restricts the visual conflicts between red and green (as defined below in paragraph 4.3.3.3),
• finally, a yellow line marks out the zero value level.

As predicted above, shapes stand out from the pictures spontaneously, although algorithmically their automatic extraction from the field would have to be anticipated a priori and this would certainly be a fastidious (or, in most cases, impossible) operation, difficult to communicate and, most importantly, far from complete, while the mind of the observer, which is always in a state of expectation, will react to the unexpected and any surprise.




4.2.5.2-Various representation modes:

Coloring, though, is only the simplest of the representation modes, associating colors with numerical values. As shown in figure "Four different tridimensional visualizations of the same bidimensional scalar (fractal) field", one or more spatial dimensions can code one or more numerical values. On the four corresponding pictures, by playing on lighting and the angle of observation, the values of the field presented in figure "The simple case of a bidimensional scalar (fractal) field presented in four widely differing color palettes" is made to correspond with the "orthogonal height" of the field, which then defines a 3D surface. The highest points have the highest positive value, while the lowest points have the lowest negative value.

This mode may appear superfluous, but while it is difficult to compare the numerical values of the field on two distant pixels using only color and luminance (there is no absolute reference, as shown in paragraph 4.3.3.2), it is easy to make the comparison by means of altitude. Moreover, the surface defined in this way could be colored using a second field and the conventional "mapping" technique (see figure "Wavelet transform of a bidimensional fractal field"), which makes it easier to obtain correspondances and determine correlations. Finally, "spatial" defects that are imperceptible in the original field could then be amplified.

This simple 2D case, then, shows that on the one hand there is no single "natural" solution to the problem of display and that, on the other, precautions are required in order to avoid errors and incoherences (see observations R1 and R2). When traveling in higher dimensions, it is obvious the problem is worse (see figure "Different modes of representation of a tridimensional cross-section of a Quaternionic Julia set").




4.2.5.3-Dynamic display:

finally, the dynamics of a model have to be followed through, for two reasons:


• Tp is an intrinsic part of the model,
• certain (temporal and, in some cases, spatial) correlations and, for example, "spatial" pairings between neighbouring structures (for example, positive and negative value subsets) only appear during motion.

4.3.1-A PICTURE IS ONLY A SINGLE VIEWPOINT:

yet in general several points of view are required (see figures "The simple case of a bidimensional scalar (fractal) field presented in four widely differing color palettes" and "Four different tridimensional visualizations of the same bidimensional scalar (fractal) field"), along with various representation modes and, consequently, various color palettes, if one is to apprehend the complementary aspects of the system being modelled (this has overtones of the wave/particle duality found in quantum mechanics). Unfortunately, the useful modes are not usually known before experimentation begins and, because of this, it is necessary to collect results from a large number of experiments (on the picture itself this time). In this respect, the provisions of a specialist expert system will be appreciated.




4.3.2-TRANSFORMATIONS:

whether they concern primary or secondary results, these transformations are not neutral. In particular, they may weaken essential elements and, inversely, strengthen insignificant details (see figure "Detail filtering and enhancing with colors"). Yet again, we are reminded that a critical eye and a degree of circumspection is absolutely vital for users. And in this case, as in others, the simultaneous use of several complementary modes of representation will often resolve doubts or, on the other hand, call apparent certainties into question.




4.3.3-ARTEFACTS:

they are introduced by, and particularly linked to:


• the digital character (mosaic) of the synthetic picture (common problem of aliasing or a spatial and/or temporal sampling defect caused by the presence of excessively high frequencies),
• the absence of absolute references for colors and luminance,
• optical illusions (Mach bands, simultaneous contrast, etc...),
• and, last but not least, the capacities of the machine itself.

In this respect, the work carried out by artists (especially those of Itten) should be consulted without any false sense of shame for it often contains useful advice with regard to chromatic balance and proportions (for example, the use of the golden rectangle for the format of "windows"). As we shall see, a picture is not a neutral tool. It is evident that there is no such thing as a neutral tool, yet unfortunately, in this context, picture synthesis is considered to be just that, and the use to which it is put often confuses the aesthetic and the informative. Let us remember that the picture produced in this way is, in this context at least, usually arbitrary, and that in fact, using the same set of data, a large number of very different representations could be imagined and built up. Moreover, the eye (or to be more precise "the entire visual system") is subject to optical illusions that are well-known but rather too often forgotten. There are four categories of illusion, each of them responsible for a failure to appreciate correctly what is being seen:



4.3.3.1-Geometric illusions:

They cause geometric deformities in the perceived shapes. For example, the well known Zšllner illusion introduces waves on lines that are objectively straight and parallel.




4.3.3.2-Luminance illusions:

this type of illusions shows quite "clearly" that the eye has no absolute reference for luminance (luminance represents the intensity of gray at a given point, its minimum representing the intensity of black and the maximum the intensity of white). This being so, two zones with identical luminance separated by several other zones with a very different luminance will appear to be dissimilar (see figure "2 identical grey squares moving over a grey scale"). It is, therefore, a total "illusion" to attempt to interpret numerical results quantitively and generally if the results are represented by sets of luminances. Major relative errors might well be committed but, more seriously, inversions in the orders of size might be introduced. Moreover, when the luminance gradient changes sign, "over-luminous" bands (the so-called Mach bands) appear. Lack of knowledge of this phenomenon might result in erroneous conclusions about a numerical field.




4.3.3.3-Chrominance illusions:

luminance illusions have already shown that the visual system functions globally rather than locally. This being so, the neighbourhood of a given pixel (or even the entire picture) affects the perception of the pixel itself. The same is true of chrominance: the "subjective" color of a zone is strongly influenced by the color of adjacent zones. As to the black-and-white broadcasting of a color picture, it can occasionally give rise to rather unpleasant surprises. For example, a TV color scale seems to have non-monotonous luminance whereas its black and white version shows that this is not the case.




4.3.3.4-"Computing" illusions:

the previous three categories of illusion were known to physiologists and those interested in plastic arts long before the arrival of computers. The fourth and final category, however, is much more recent and arises from the notion of false color. Most display systems in use at the present time provide a means of coloring and re-coloring a picture. A digital picture in false colors can be defined as a matrix of pixels with integer coordinates X and Y. Each of these pixels contains a single memorized numerical value L, which often takes the form of a byte. When video signals are generated, the matrix is reread. The L bytes are retrieved in succession and "colored" before being displayed. The operation involves the use of a Look Up Table, which has as many entries as there are possible values L in each pixel (28=256 in pixel coding on one byte). Each of its entries is programmed and contains a triplet (R,G,B) that give the intensity of Red, Green, and Blue respectively to be applied to every pixel with the same value L. This means that an identical picture can be screened differently depending on the definitions of color loaded in the "LUT". This is one of the major dangers, but it is also the least well-known and the most underestimated. To take a stark example, which some might find arguable or even naive but which has the advantage of being simple and devoid of any physical consideration (there is, in fact, nothing to see in this field), we use four different color palettes, each of which can be used to draw a number of conclusions (all of them unfortunately incompatible) in respect of the subjacent structure. Figure "The same bidimensional scalar field displayed with 4 different color palettes" -bottom left- apparently shows two characteristics of the field. Horizontally, it is uniform; vertically, it increases and decreases. Note in passing that no quantitative data can be deduced from this observation. In particular, it is impossible to define whether the increase (resp. decrease) is linear, sinusoidal or gaussian. As for figure "The same bidimensional scalar field displayed with 4 different color palettes" -bottom right-, it also indicates horizontal uniformity but this time there are vertical undulations. Finally, on figure "The same bidimensional scalar field displayed with 4 different color palettes" -top left and top right-, there is a notable variation in the horizontal plane. In these conditions, which would be the only possible display for this arbitrary field? The response is obvious: a single display does not exist, and none of the displays described above is complete. Again, it will be possible to criticize this example, perhaps for its simplistic aspect or its lack of physical significance. Yet let us remember that, for a given field, there is not usually any a priori display mode and, moreover, everyday experience shows that this type of usage is made of color (examples drawn from well-known reputable scientific publications are available...). From the above considerations, it is possible to draw a number of conclusions. First and foremost, there is no single a priori mode of representation; in fact, several modes, all of them complementary, will be required in order to avoid interpretation errors or to apprehend various aspects of the "objects" represented. The second lesson is that, contrary to a widely-held belief, display, in particular, the display of results from numerical simulations, is an "art form". Coloring is not a puerile child's game; it is a very difficult operation demanding a "certain instinctive grasp of plastic arts" and considerable knowledge of the subjacent phenomena. As we have seen, when carried out unthinkingly, it can lead to conclusions that are completely false. It is important to remember that an attractive picture is seldom a good picture (as far as its informative aspect is concerned).





4.3.4-UNKNOWN OBJECTS:

this problem has already been mentioned (see figure "Black and white display of a tridimensional function") and is becoming increasingly common, especially during the study of fractal objects and chaotic systems. One possible solution would be a high level of interactivity at simulation and display level. A structure could then be explored in real time, whatever its complexity, just as a pilot training on a flight simulator can explore an unknown landscape; clues must be added to help the understanding of the third dimension (see figures "Reconstruction of a 3D structure -a cubic lattice-", "Cauliflowers, seaweeds, shells,... with fog" and "Autostereogram of a quaternionic Julia set"). Moreover, a large number of tools must be accessible to the user instantly so that he can follow a new path without delay. Finally, these same tools must be suitable for sequencing and combining as required. This point will be discussed at length in Chapter 5.




4.3.5-THE QUALITY OF THE DISPLAY SYSTEM:

all these considerations show that a display system adapted to the analysis and comprehension of the results of numerical simulations must have a certain number of qualities, viz:


• a capacity to manipulate n-dimensional data easily (in general, n<=4 -which includes the time-);
• good spatial definition: the 512x512 format is a minimum and the problem of compatibility with standard "video" equipment as regards recording should also be looked at. In this context, the use of High-Definition TeleVision will probably be very useful;
• good chromatic definition: 256 redefinable colors (in a much larger palette) are, again, a minimum for use in "false colors", and operation in "true colors" is vital in many cases (e.g. transparency management and "mapping" function). The palette-definition tools available must be simple, powerful and interactive;
• high level of interactivity: this is the only factor that enables a scientist to test, explore, and experiment without letting his mind wander while he waits for the computer to respond (beyond a one-second delay, a user, whether artist, engineer, or scientist, runs the risk of "switching off" intellectually). This is vital in both simulation and display systems so that the feedback loop is not "broken";
• high communication potential: this provides for effective dialogue between simulation and display, as well as assisting user/system exchanges.

This results in the architecture of a specialist system for virtual experimentation which will be described in detail in Chapter 5.

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• can the reliability of software be enhanced using a straigthforward programming environment?
• can this programming environment be machine and system independent?
• how to optimize the use of available resources?
• how to guarantee the reliability of the results produced?
• how to analyse these results?

• the "objects" being simulated could be processed as objects. This would facilitate the transition from model to computer creation, the development of the object, the use of the parallelism in the architecture and, most importantly of all, interaction with the modelized system.
• a new scientific language could be defined. Its principle would be the same as the one found in expert systems i.e. a distinction between knowledge (in this case, numerical resolution methods) and "concrete" problems to be solved; then, several methods of resolution could be tested and compared on a single identical problem.
• dialogue with the user could be provided by an expert system. It would provide a natural (i.e. user-friendly) dialogue, implicit implementation functions for the system as a whole (e.g. automatic selection of representation modes depending on the current problem) and assistance in result analysis.

• simplicity,
• generality,
• harmony,

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