CA2044506A1 - Compressed image production, storage, transmission and processing - Google Patents

Abstract

ABSTRACT

Images of objects are produced by:
(1 ) Approximating the object by a model comprising at least one differentiable component.
(2) Establishing the maximum allowable error .epsilon. and the degree k of the polynomials by which the differentiable component(s) of the model are to be approximated.
(3) Constructing a grid of a suitable pitch h.
(4) Computing the coefficients of the Taylor polynomials of the aforesaid differentiable component(s) at selected points of said grid.

Description

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COMPRESSED IMAGE PRODU(: TION~ STORAGE. TRANSMISSION
AND PROCESSING

BACKGROUND OF THE INVENTION

This invention relates to a method for producing an image of an object,storing, transmitting and processing the same.

In this application, "object" means any entity that can be defined, in principle, by geometrical and/or mathematical data and/or geometrical or mathematical or empirical relationships, such as functions, correlations, regressions, lines and surfaces, etc. It is irrelevant whether the object is so complex that the number of data andlor relationships required to define it is so great that complete or exact de~mition is practically impossible. It is also irrelevant how many dimensions the object has. The object may be a physical one, such as a picture, a line, a sur~ace, a solid, a tri-dimensional object or a landscape, etc., or an abstract one, such as a tensor, a form defined in a continuurn having more than three dimensions, etc.; or it may be constituted by an array of data which have only a conceptual relationship with one another.

"Image" means any entity that represents an object exactly, or more or less approximately. The image may have the same nature as the object it rcprescnts, as when, e.g., it is the reproduction of a picture or an array of data representing another array of data; it may be an image in the common 20~06 meaning of the word, as when, e.g., it is a picture of a person or a landscape; or it may be quite dif~erent in nature from the object, as when, e.g., it consists of a plurality of numerical data representing a physical entity. "Intermediate image" means an image that is produced for the purpose of transforming it later into a different image of the same object, as when, e.g., a set of numbers temporarily represent a geometrical form and a geometrical image is to be developed from them. When such a transformation occurs, the image finally produced will be called hereinafter "the ~lnal image". An image which is to be processed in any way elaborated to produce another image of the same nature - e.g. a first set of numbers from which another set of numbers is to be obtained, by any appropriate procedure, said other set of numbers being an intermediate or a final image, will be called a "temporary image", which, if the processing is a correction or adjustment, is an "unadjusted image".

In a great many technical processes, an image of an object must be produced, and quite often must be stored, transmitted or processed. For instance, it is a common occurrence that two-dimensional figures or pictures be represented by digital data which are stored, processed and transmitted, according to needs. This occurs in word processing by computers, message transmission by telefax, etc. Three-dimensional objects, including landscapes, may be represented by a process that is essentially the same. The representation of objects which have more than three dimensions involves in principle no conceptual departure from the said methods. Another common occurrence is the representation, storage and processing of data representing physical relationships, statistical 2 ~ 3 ~ ~

regressions or ways of experimental data. The use of mathematical models is also an instance of object representation by an image, which may be constituted by an array of digital data.

It is obviously desirable to reduce as much as possible the amount of data defining the image which represents a given object, without disorting the image to the extent that it might cease to represent the corresponding object with an acceptable degree of accuracy. Such a reduction of the required data, or "data compression" or "image compression", as it is sometimes called, serves to simplify, reduce and render more economical the equipment required for the storage of an image, its processing and transmission. For instance, it is well known that in modern technology, transmission lines, including frequency bands available for radio transmission, are increasingly overcrowded, and every effort is being made to exploit them as fully as possible, one of the means for so exploiting them being to reduce the amount of data that are sent through a given transmission line in order to convey a given amount of information.

It is a general purpose of this invention to provide a method for producing the image of an object of any kind, storing it, processing and transmitting it, while ltlinimizing the amount of data that are required for carrying out the said operations.

More specific objects of the invention and specific applications thereof, will become apparent as the description proceeds.

BRIEF SUMMARY OF THE INVENTION

The following considerations are preliminary to an understanding of theprocess according to the invention. If the object is defined geometrically or analytically - whether by a graphic representation or a model, depending on the nature of the object, or by an array of numerical data which are assumed to define the object or in any suitable way - it may be broken up into, viz., be considered as defined by, a plurality of components, such as lines or surfaces defined in a space which may have more than three dimensions, arrays of numerical values or functions or operators which can be represented by such lines or surfaces. For the sake of simplicity, the process according to the invention will be described firstly with a reference to an object which may be broken up into a number of plane lines, corresponding to functions of one variable. Description and definition of the process will be then expanded to those objects which must be broken up into surfaces in a three-dimensional space or in hyperspace, having more than three dimensions, corresponding to functions of two or more than two variables. Essentially the process, as described and defined, extends to compressed images of any objects that can be defined by an array of data, by software or hardware for the production and/or elaboration of digital values, such as a special purpose computer or a computer program, or by an analogical circuit or special purpose analogical computer or analogical computer program, or by digital or analogical sensors, or the like~ In what follows, the term "object" will be construed as preferably meaning the physical entities and/or relationships by which the object is defined or into which the objcct has becn translated, and which will have been stored or 2 ~

memorized, as in an electronic memory, e.g. in the form of digital values or instructions relative thereto or analogical representations of functions or relationships.

In one of its simplest forms, the object, an image of which is to be constructed, may be a plane line. The object line, as any other object, may be defined in many different ways, but, for the purposes of illustration only, it will be treated as defined by a graph or by a corresponding function, it being evident that the information conveyed by a graph can be conveyed in other suitable way. In any case, in order to carry out the process according to the invention, the object line must be translated into digital values or intoa computer program or subroutine or an analogical process or into the structure of a special purpose digital or analogical computer, which can be entered and memorized in an elaborator, and which define couples of values x, y for each point of the line. The object line may be considered in itsentirety, or, more frequently, it will be divided into segments, to each of which the process of the invention will be separately applied. Therefore, if the line has been so divided, the expression "object line", when used hereinafter, must be construed as meaning the particular segment under consideration at the moment .

The process, then, comprises, in a restrictive definition, the following steps:
(1 ) Appro~imating a line by a model which includes at least one differentiable component.

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(2) Establishing the maximum allowable error ~ and the degree k of the Taylor polynomials by which the dif~erentiable component(s) of the model are to be approximated.

(3) Establishing at least a pitch grid h and constructing a grid each region of which has one of said pitches h.

(4) Computing the coef~lcients of the Taylor polynomials of the aforesaid dif~erentiable component or components at selected points of said grid.

Two or more of the aforesaid steps may be carried out concurrently, in whole or in part, or divided into successive stages, which may be intercalated to a greater or a lesser extent.

Further operations, hereinafter described, may be carried out and are often desirable to minimize the effect of inaccuracies in the said coef~lcients, for rounding them off, for taking into account different scales which may be prosont in tho data, and for obtainillg, if dcsircd, an image which has the same nature as the object. "Non-dif~erentiable component" means herein a component comprising one or more points at which it is not dif~erentiable, or, a component that is not different;able at all its points.

The process according to the invention can be extended to objects that are more complex than plane lines by simple generalizations, as will be explained hereinafter.

BRIEF DESCRIPTION OF THE DRAWINGS

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The invention will be better understood from the following description of preferred embodiments, with reference to the appended drawings, where:

Figs. la lnd lb illustrate an example of an object line and its image, respectively;

Figs. 2a and 2b illustrate a temporary image line the segments of which do not match at meeting points, and a corresponding adjusted image line, respctively;

Figs. 3a, 3b, 3c, and 3d illustrate respectively an object line and thecorresponding model line, final image and non-differentiable component of the model, with reference to Example l;

Figs. 4a and 4b represent a picture and its image, respectively, with reference to Example 2;

Fig. ~ represents a processed image of the picture of Fig. 4a, with reference to Example 3; and Figs. 6a and 6b replesent the negative of the picture of Fig. 4a and its image, respectively, with reference to Example 4.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

The process steps hereinbefore defined will now be more fully explained.

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Step (1) - The object line, the data defining which have been physically stored e.g. in an electronic memory, is approximated by a model, preferably defined in the same way as the object line, which model preferably consists of at least a first component embodying the characteristics of the object, if any, which render it non-differentiable at some points or regions - it being of course possible to omit said first component if there are no significant characteristics of non-differentiability of the object - and at least a second component which embodies all the differentiable content of the object.
Typical cases of models are the following:
Case a) The first component is a base line, which is a simple - desirably, the simplest - line having qualitatively the same discontinuities as the object line, and the second component is a curve which represents the deviations therefrom of the object line, and which will be differentiable and can be called interpolating function. The base line may be constructed in each individual instance, or, more conveniently, may be chosen, according to the actual discontinuities of the object line, from a number of normal forms, which are the simplest functions having the required discontinuities. The following standard l~orm of model can be used in this case:
(1) CP(X) = HxO,a,b,c,d(x) + ~(x) wherein H is a normal form defined by H(x) = a(x-xO) + b, if x > xO or H(x) =
c(x-xO) + d, if x is less than xO ~ The values of the parameters xO,a,b,c,d are determined, in a preferred embodiment of the invention, by minimizing a quantity representing an error, e.g. the quadratical error, as hereinafter set forth. The base line can be predetermined, or chosen, in general 2 ~I L~ L~

according to predermined criteria, from a list prepared in advance, or it can be chosen ir~ each case by the operator. This case is illustrated at Fig 1 a, lb showing respectively an object line and its model.
(~ase b) The model is a differentiable function of another function which embodies the non-differentiable characteristics, viz the discontinuities, of the object line. It can be epressed as:
(2) cP(x) = ~'[~(x)], wherein ~ is the first component, which will be called the base curve, and (P' is the second component. ~(x) can be looked at as defining a change of coordinates: in the differentiable component ~P', the ordinates are referred to abscissae which are not x, but ~(x ).
Case c) This case will be mentioned here, though it is not applicable to a line, but only to surfaces in a space having three or more dimensions. In the case of three dimensions, a coordinate (say, the elevation) z of a sur~ace, is a function Z1 in a certain regrion of the plane x-y of the two remaining coordinates and is another function z2 in another regJiOn thereof, the two regions being separated by a border line defined e.g. by a relationship y=~(x). Then the model (P(z,y) consists of the function Z1 if y is greater than ~(x), and Z2 if y is smaller than ~(x), one or the other of the and z2 applying when y=~(x).
Case d) The object line is differentiable at all points, and the model consists only of a differentiable component.

In a form of the invention, all the parameters of the model the values of which have to be chosen, are determined by minimizing a quantity representing an error - e.g. the quadratical error, viz. ~.[f~xi)-~(xi)]2 - the 2 ~ Q ~

minimization being carried out by means of a predetermined subroutine with respect to all the parameters of the model ~, for the function f~x) representing the object~ the values of f~x) for each x being determined by known subroutines. Programs for this purpose are available, e.g. from the ILSM library.

Step (2) - a) - The maximum allowable error , which is to be tolerated in approximating the object line, viz. which expresses the desired precision of the image, is established.
b) - The degree k of the Taylor polynomials, which will be used to approximate the differentiable component or interpolating curve, is established.

Step (3) - The grid need not be cartesian and its coordinate lines may be curved, although for simplicity's sake a cartesian grid will always be illustrated herein. The grid may be divided into different regions having different grid pitches or even different types of coordinate lines. The grid pitch h (viz., the distance between adjacent coordinate lines which define the grid cells) is selected according to the precision desired of the image, and may be different in different parts of the region, although a regular grid will often be preferred.

In an embodiment of the invention, h is calculated, by a suitable subroutine, f`rom the formula (3) C~hk+l < e 2 ~ o ~

wherein C = 1/(k+1)~ and M is the maximum, at each grid point, of the absolute value of the (partial, in the case of an object which is a ffinction ofmore than one variable) derivatives of degree k+1 of the differentiable component or components, in the segment or zone of the object under consideration, M being determined by using a known subroutine which computes the derivatives of order k+l, produced e.g. by a package such as MAXIMA OR MATHEMATICA.

Step (4) - The nodes of the grid are taken as base points, and a (known, e.g.
a MAXIMA) subroutine is applied at each base point to compute the Taylor polynomials of degree k of the interpolating curve.

~t this stage, the following data have been obtained:
A) The coefficients of the Taylor polynomials of the differentiable component or components of the model;
B) The number or other idcnt;ification or analytical definition of the non-differentiable component(s), if any, of the model, such as the base line or tllo base curvo;
C) The values of the parameters of the said non-differentiable component(s), if any;
and these define an image, which will usually be an intermediate image, but could be a final one, according to cases. Hereinafter it will be assumed that it is an intermediate image, from which the final image, in the same form as the original object, is to be constructed; however this is done merely for the sake of simplicity and involves no limitation.

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In many cases, as will be explained below, the image thus obtained may require further elaboration without changing its nature, viz. while remaining a set of data of the same kind, and it will be only a temporary, in particular an unadjusted image. Then some or all of the steps from (5) on will be carried out.

Step (5) - In the case of the presence of so-called noise or inaccuracies in said temporary image line, or if the numerical noise, viz. the inaccuracies of the computations, which are large in comparison with the accuracy required, the Taylor polynomials which make up the temporary image line or its differentiable component may disagree at their meeting points by more than allowed by the required accuracy, as represented, by way of example, in Fig. 2a.
In this case, an adjusted image line is constructed by applying to each differentiable component a subroutine, hereinafter "Whitney subroutine", which computes W, wherein W is a quantity representing the discrepancies of the Taylor polynomials. In particular, W can be given by a formula:
(3) W = iJ ll Pi - (Pj)i 11 2 E~ere the sum is taken over all the adjacent grid points i, j (possibly belonging to different segments of the image). Pi, pj denote the Taylor polynomials, obtained in steps (1) - (4) at the grid points i, j, and (Pj)i denotes the polynomial pj, e~cpressed in coordinates, centered at the i-th grid point. Il p - q 11~ denotes, for any two polynomials p and q of the same degree and number of variables, the sum of squares of the differences of corresponding coefficients.

2 ~ 3 For any values of the coefficients of Pi, W is computed by using known subroutines, produced e.g. by a package such as MATHEMATICA.
W is then minimized {e .g. by standard gradient methods), using, as starting values of the coefficient of the Taylor polynomials, those obtained by the previous steps, and under such constraints that the result of the minimization do not deviate from the initial data by more than the allowed error, e.g. under the condition that the zero degree coef~lcients of said polynomials remain unchanged. An adjusted image line, corresponding to the unadjusted image l;ne of Fig. 2a, is illustrated by way of example in Fig. 2b.

Step (6) - If the accuracy of the adjusted coefficients of the Taylor polynomials obtained from step (5) is excessive with respect to that desired in the final image, they are rounded off to a maximum allowable error ' by any suitable method (not necessarily the same for coefficients of differcnt degrees). Thc data thus obtained represent the adjusted image line.

Step (7) - Sometimes the data of the object to be represented may require the use of different grid resolutions, or such use may be desirable. An example which clarifies this case is the following.
Let us assume that the object represents a periodic phenomenon, e.g. an oscillatory phenomenon such as an oscillating eletcrical impulse or an electromagentic wave. Such a phenomenon can be analyzed and is usually represented by the combination of two or more superimposed components, speci~lcally, a relatively low frequency carrier wave and a higher frequency modulating wave. The modulation can be sometimes considered as 2 ~

resulting from a first, intermediate frequency modulation, and one or more high frequency modulation or modulations, and in this case the object will have three or more components. The image can be conveniently constructed from images of ther various components, e.g. of the carrier wave and of the modulating wave or waves, and obviously the lower frequencies will require lower resolutions and larger grid pitches will be suitable for them. Likewise, the frequency of an oscillatory phenomenon may vary at dif~erent times or in different spatial regions and its components will not be superimposed, but separated in space. Similar situations may occur in various cases. Generally, many kinds of object may comprise superimposed or separated components which have details of different fineness, which require different degrees of resolution. Since oscillatory phenomena are a typical case of objects requiring different glid resolutions, the word "frequency" will be used to indicate the fineness of the required grid, but this is not to be tal{en as a limitation, since the same proccdurc can bc appliccl to non-oscillatory phcnomcna.

In such cases, the following procedure is pref`erably followed:
a) Steps 1 to 6 (or such among them which are necessary in the specific case) are carried out and a first temporary image is obtained.
b) A new maximum error 2, bigger than ~ (or ', as the case may be) is chosen.
c) A grid which is sparser than the one used for carrying out the steps under a), and the pitch of which is determined by the resolution required by the lowest frequency of the components existing in the object (e.g. that of a carrier wave) is established.

2 ~ r~ ~ 6 d) Steps 1 to 6 are repeated using E2 and the sparser grid and a second temporary image is obtained.
e) The second temporary image thus obtained is substracted from the first and a first residual image is obtained, which contains only data relating to higher frequency components of the object.
f) The same procedure - steps b) to e) - is repeated for successively higher frequencies of components, correspondingly obtaining successive residual images increasingly restricted to higher frequency components.
As a result, coef~lcients of Taylor polynomials are obtained on several grids having increasingly higher resolutions, viz. smaller pitches, separately corresponding to the object components requiring increasingly higher resolutions.

The data obtained after steps (1) to (4) and those among (~), (6), (7), which ithas been found necessary to perform, constitute an intermediate image or someyimes a final one. Usually these are the compressed data which can be stored, transmitted and processed.
If a further compression is desirable, one of the standard methods of encoding coefficients (e.g. ~Ioffman coding) can be applied. If necessary, the resulting string of data can be further compressed by one of the standard methods of unstructured data compression (e.g. entropy compression). However, this last step reduces the possibility of a compressed data processing.

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If a final image, which has the same nature as the object, is to be constructed, the following procedure is followed:
Step (8) - a) The Taylor polynomial coefficients obtained after completion of steps (1) to (4) and of those among steps (~) and (6) which it has been found necessary to perform, are treated as if they represented an unadjusted temporary image, which is affected by noise, and are subjected once more to step (~), using them as starting data.
b) The domain in which the temporary image has been defined is divided into regions by means of a grid, each region being a portion of the grid around a grid node or base point. These regions may overlap.
c) A curve or curves representing the Taylor polynomials of degree k in the above regions are constructed from the coefficients defining the temporary image - e.g. obtained as in step (8) a) - at each node of the grid or of that grid having the highest resolution (smallest pitch), if there are more than one grid (particularly if step (7) has been carried out), using a known subroutine.
Said curve or curves constitute the final image of the object line.
The aforementioned curves may diverge at the meeting points of the regions mentioned above under b) (or on their overlapping parts). If this disagreement does not exceed the allowable error ~, any of the overlapping curves curves can be used at the meeting points on the overlapping parts of the above regions.
If as the result of the noise of the data or the computational noise, the above discrepancies are large in comparison with the accuracy required, average values can be used on the overlapping parts. This is done by 2~ ~

averaging the values of the overlapping curves with the appropriate weights.

Actually, other polynomials or functions could be used for approximation purposes, such as Tchebicheff polynomials, trigonometric exponential functions, etc., without departing from the invention, but Taylor polynomials are preferred.

The above described process applies, with obvious generalization, to a wide range of objects. Some examples follow.

I - A surface in a three-dimensional space corresponds to a function of two variables. If the surface is defined in a space that has more than three, say, n+1 dimensions, the independent variables will be more than two, say, n (x1,x2,...xn), but the operations to be carried out will be essentially the same, and the necessary generalizations will be obvious to skilled persons.
In any case, any surface can be translated, as well as a line, to digital values, which can be entered and stored. The model will be constructed in l;he same way as for a line. Case c) of model construction, already described, applies to surfaces in any space. Analogously to case a), a model may consists of a simple base surface, which presents the discontinuities of the object surface, and by an differentiable or interpolating surface, which represents the deviations of the object from the base surface. One can also operate analogously to case b), by using functions of more than one variable. The minimization of the quadratical error is effected in the same way as in lhe casc Or an objcct line, using valucs of ~'i; n and f~xi,xj,....,xn) 20~ 0G

which depend on n variables. The remaining steps are likewise adapted to the existence of n variables. All derivatives, of course, will be partial derivatives. The construction of the final image from the temporary image -step (8) - can likewise be carried out with obvious generalizations in the case of images defined in a space having any number of dimensions.

II - A surface can be considered as a family of lines, which are obtained by the intersection of the surface with a family of planes, e.g. vertical planes the orientation of which is taken as that of the x-axis, identified by a parameter, e.g. their y coordinate. A family of curves in a plane, depending on one parameter, as may result from the representation of any number of phenomena, is obviously equivalent to that of a surface and may be treated as such, or vice versa.

III - A particular case of an object which is a surface is, e.g., a terrain, whcrein the surracc is dcfincd by the clevation as a function of two plane (cartesian or polar) coordinates.

lV - A building can be represented in the same way, if it is very simple. If its shape is comple~, however, it must be broken up into a number of component parts. However, if it is desired to represent it as it is seen from the outside, say by an observer which can place himself at any vantage point v~ithin a certain distance from the building, the oberver's position can be identified by three coodinates, x, y and z (or polar coordinates), or by two,if it assumed that the observer's eye is at a given level. From each position of the observer point it is possible, if the configuration of the terrain is 20~4~06 known, to determine the distance D on each line of sight from the observer's eye to the building surface, and this will determine how the building is seen. Each line of sight can be identified by two coordinates: e.g.
its inclination (the angle thereof with the vertical in a vertical plane which contains it), and its azimuth (the angle of said vertical plane with a reference vertical plane, e.g. one that contains the geographic or magnetic north). The way in which the building appears to the observer, is therefore defined by a function D of five variables, viz. by a surface in a six-dimensional space.

V - A family of curves in a plane, depending on more than one parameter, is obviously equivalent to a surface in a space having more than three dimensions.

VI- If in example IV above the coordinates of the observer are known as a function of a single variable, say, when he approaches the building along a given line, in which case the variable is the distance covered from a start;ng po;nt;, or in motion, as ;n a vchicle, along a given line, in which case the vari~ble is time. In this case the variables of the surface become three (e.g. distance or t;me and inclination and azimuth) and the space is only four~dimensional, but the four-dimensional surface is subject to the constraint represented by the definition of the observer's motion. In general, in many cases, the degree of the space in which the surface is defined may be reduced by the introduction of suitable constraints.

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VII - The final image of a colour picture is another colour picture, that is not identical, but sufficiently similar to the object picture. The object picture can be scanned by known apparatus (scanners), by means of white light, and for each point the intensity of the three basic colours (magenta, cyan and yellow) may be measured and registered. The object is thus reduced to three partial or component objects, each consisting of the distribution of one basic colour over the picture and having a physical reality, as it is equivalent to the colour picture that would be contained by exposing the original through three filters, having colours complementary to the three basic colours, or, in practice, to an array of digital data representing such one-coloured picture. Each of said partial objects can be subjected to the process of the invention, to produce a reduced or compressed array of data, constituting a partial image, and the partial images can be transformed into a combined final irrage approximating the original object, by processes known to those skilled in the art. If the partial images must be stored and/or transmitted, the process of the invention will facilitate doing this and render it more economical. In the same way a dynamic coloured picture, such as a movie or a TV broadcast, can be reduced to a final dynamic image.

A particular advantage and a preferred aspect of the invention consists in the possibility of processing the compressed intermediate image obtained as set forth hereinbefore and producing from it a processed final image, which does not represent the object but represents what would have been the result of processing the object. The processed intermediate image can be stored and transmitted with the already mentioned savings and 2 ~

advantages inherent in the reduction of the number of data, but said reduction is even more advantageous in the processing, for it is obviously more convenient to process a reduced instead of a larger amount of data.
Said processing in a compressed form, as it may be called, is made possible by the following property:
Let F be an operator which is analytic in nature, viz. can be defined by mathematical relationships. Let O be an object of any nature, but which can be represented by Taylor polynomials Pi Then by applying operator F to the pi's, one obtains polynomials which represent the object that would be obtained by applying the operator F to the object 0. If one uses the symbol ~
to indicate that an array of polynomials represents an object, one can write:
if pi~O? then F(pi)-FtO).
Elementary examples of analytic operators are algebraic operations, rotations of geometrical figures, changes of coordinates in general, etc.
These operators are represented by mathematical operations. If F(O) is to be constructed, such opcrations must bc carricd out on all the data, e.g.
digital data, which define the object. But if a compressed image has been obtained as set forth above, and an array of Taylor polynomial coefficients has been obtained, which are in a much smaller number than the said digital data, said mathematical operations can be carried out on said coefficients, and a processed intermediate image will be obtained, which represents F(O) and from which F(O) can be constructed as set forth in step (9) above.

The following examples illustrate a number of embodiments of the invention.

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Example_l An object line f in thé plane (x, y) is given by an array A = (yo, Y1, -, Y100), where y; = f(xi), x; = i/100, i = 0,1, ... ,100. In this specific example the array (array 1) is the following:

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0.1152 0.1155 0.1191 0.1131 0.1174 0.1133 0.1108 0.1149 0.1105 0.1182 0.1167 0.1206 0.1238 0.1196 0.1264 0.1282 0.1313 0.1315 0.1299 0.1330 0.1366 0.1409 0.1402 0.1462 0.1569 0.1608 0.1631 0.1604 0.1693 0.1779 0.1797 0.1826 0.1826 0.1888 0.1963 0.2011 0.2034 0.2084 0.2170 0.2244 0.2265 0.2327 0.2~29 0.2468 0.2472 0.2523 0.2661 0.2673 0.2702 0.2796 0.2811 0.2845 0.2949 0.3022 0.3078 0.3049 0.3121 0.3157 0.3256 0.3270 0.3346 0.3413 0.3405 0.3428 0.3503 0.3515 0.3530 0.3571 0.3675 0.3616 0.4648 0.4665 0.4659 0.4607 0.4600 0.4536 0.4473 0.4441 0.4427 0.4330 0.4329 0.4268 0.4243 0.4185 0.4135 0.4107 0.3961 0.3925 0.3877 0.3774 0.3698 0.3671 0.3583 0.3449 0.3397 0.3338 0.3271 0.3091 0.3031 0.2929 2~5~ ~

An object line itself is shown in Fig. 3a. The required accuracy of representing this line is 0.035. The compressed image of this line is produced as follows.
Firstly it is subdivided into three segments lying over the segments [0.0, 0.6], [0.6, 0.8], [0.8, 1.0] in the x-axis. The following model is chosen on the segments [0.0, 0.6] and [0.8,1.0]:
y = Q(x) = cl sin (cl)l x + ~1) + c2 cos (0)2 x + ~2) + c3 x2 + c4 x + cs with cl, c2, , 0)2, ~ 2, C3, C4, C5 - the parameters.
On the segment [0.6, 0.8] the following model is chosen:
y = Q(x) ~ Hxo~ a, b, c, d (xj, where Q(x) is as above, and the normal form H
is defined by H(x) = a(x - xO) + b, if x 2 xO or H(x) = c(x - xO) + d, if x is less than xO. Said normal form is illustrated in Fig. 3d. Approximation on each segment is carried out by minimization, with respect to the corresponding parameters, of the quadratic error:
~ (Yi - Q(xi) )2 (~ (Yi - Q(xi) - H(xi) )2 on [0.6, 0.8] ).
The values of the parametors found are given in the following array 2.

-25- 20~ 0~3 Q (x) = 2 . 0 + 0 . l*x - 0 . 2*x*x - 0 . 15*cos ( -0 . 4+4*x) 0.2*sin( -0.3 + 0.5*x ) H(x) = 1.0/7.0 * (x-0.7) + 0.1 , x < 0.7 H(x) = -1.0/3.0 * (x-0.7) -~ 0.2 , x >= 0.7 20~0~

The corresponding model curve is shown in Fig. 3b.
The error of the approximation of the object line by the model found turns out to be 0.006. Respectively, on the step 2, iS chosen to be 0.03. k is chosen to be 2 on each segment.
M, equal to the maximal absolute value of the third derivative of the smooth component in the above model, as computed by the standard subroutine, is 8. The maximal possible pitch h of the grid to be constructed, is defined by (1/6) M (h/2)3 = ~, or h ~ 0.24. In order to provide a uniform grid, a smaller value h = 0.2 is chosen on each segment. The corresponding grid points are the follov~ing: 0.1, 0.3, 0.5 on [0.0, 0.6], the only grid point 0.7 on [0.6, 0.8] and the only grid point 0.9 on [0.8,1.0]. Taylor polynomials at these points, as computed by the standard "MATHEMATICA" subroutine, are given in the follov~ing array 3.

-27- 2 0 ~ 6 zi aO al a2 0.1 0.1215820312500.105957031250 0.993652343750 O .3 0.1801757812500.454345703125 0.632080078125 0.5 0.2856445312500.542724609375 -0.236083989375 0.9 0.381103515625-0.727050781250 -1.394042968750 0.7 0.2724609375000.125244140625 -1.083496093750 a= 0.142822265625 b= 0.099853515625 c= -0.333251953125d= 0.199951171875 2 ~ 3 Now the coefficients of order O are rounded off up to 3 digits, the coefficientsof order 1 are rounded off up to 2 digits and the coefficients of order 2 up to 1 digit. The parameters of the normal form H are rounded off up to three digits. These data, listed in the following array 4 represent the intermediate compressed image.

- - 2~
Zi aO al a2 0.1 0.121 0.10 0.9 0.3 0.180 0.45 0.6 0.5 0.285 0.54 -0.2 O .9 0.381 -0.72 -1.3 0.7 0.272 0.12 -1.0 a= 0.142 b= 0.100 c= -0.333 d= 0.200 2 ~

The compression ratio is 4*100 digits/ 37 digits ~ 10.8.
The final image is obtained by computing the values of the Taylor polynomials (and the normal form H on [0.6, 0.8]) at the initial points x, i =
0, ...,100. Each polynomiai is used for x, belonging to the corresponding cell of the grid Zi. The result is shown in the following array 5.

0. ].196 0.1190 0.1185 0.1183 0.1182 0.1183 0.1186 o2lq90 o 1197 0.1205 0.1215 0.1227 0.1240 0.1256 0.1273 0.1292 0.1313 0.1335 0. ~ 360 0.1386 0.1426 0.1460 0.1496 0.1532 0.1570 0.1609 0.1649 0.1691 0.1733 0.1777 0.1822 0.1868 0.1916 0.1964 0.2014 0.2065 0 ~ 2117 0.2171 0.2225 0.2281 0.2318 0.2376 0.2433 0.2490 0.2546 0.2602 0.2658 0.2713 0.2768 0.2822 0.2876 0.2930 0.2983 0.3036 0.3088 0.3140 0.3192 0.3243 0.3294 0.3344 0.3380 0.3424 0.3466 0.3506 0.3545 0.3581 0.3615 0.3648 0.3678 0.3706 0.4709 0.4685 0.4660 0.4633 0.4603 0.4572 0.4539 0.4503 0.4466 0.4427 0.4376 0.4328 0.4276 0.4223 0.4166 0.4107 0.4046 0.3981 0.3915 0.3845 0.3773 0.3699 0.3621 0.3542 0.3459 0.3374 0.3287 0.3196 0.3104 0.3008 2 ~

The corresponding fnal curve is shown in Fig. 3c. The maximal error in representing the object curve by the final one is 0.033.
~ mple 2 The object (black and white, continuous tone) picture is the standard test picture, called "Lena" (see Fig. 4a). It is represented by a 512 x 512 array, each pixel containing 8 bits, representing one of the gray levels between O
and 255. The file representing this picture is available in test collections in the field of imaging . A part of this array, representing the piece S, marked on Fig. 4a, is the following.

-33~ 2 ~ 0 97 97 97 97 97 97 100 98 96 94 92 90 89 86 8q 84 86 97 97 97 97 ~7 97 96 94 92 90 88 86 86 82 81 81 82 84 89 92 97 103 112 122 132 1~1 154 171 191 215 220 224 217 198 167 125 J.28 127 125 124 123 122 121 118 115 111 108 105 169 127 124 123 122 122 121 120 121 118 115 ll:L 108 105 -34- 2Ot~L~3~

65 72 79 85 92 g9 104 104 104 104 104 104 107 105 103 99 95 95 83 83 85 89 97 107 123 138 153 167 182 197 207.214 210 193 79 83 ~8 93 97 102 94 69 48 30 15 4 40 51 59 65 68 58 66 67 69 72 75 80 81 97 114 132 151 171 179 ~.66 155 lq4 134 127 119 117 116 114 113 lll 110 109 107 104 101 96 88 87 86 86 85 84 89 80 74 72 7~ 79 83 85 84 81 75 -35- 2~ 3~

68 62 71 82 96 113 132 158 162 166 1~8 170 171 154 148 141 134 144 119 128 139 153 170 189 187 186 184 182 178 1?4 152 145 138 132 124 118 121 ~.18 116 113 110 108 102 100 99 98 96 g5 118 118 118 118 11~ 118 114 113 114 117 121 127 136 161 178 186 187 121 115 112 114 11~4 112 108 102 103 106 100 ~7 65 35 118 118 118 11.8 118 118 122 121 122 125 129 135 152 175 190 198 197 -36- 2~ ~

112 113 112 lll llO 109 108 107 121 90 64 44 29 19 110 lll lll 110 109 109 108 107 111 82 58 39 26 18 13g 186 178 174 174 180 189 193 189 186 184 183 182 160 139 123 113 lll 114 117 119 122 125 130 130 130 132 134 138 158 173 183 189 l90 166 189 178 172 170 172 179 156 156 156 158 160 163 156 134 ].17 106 114 117 119 122 12~ 127 132 130 128 128 129 130 112 126 136 140 141 93 lO0 91 110 120 121 114 97 29 27 25 24 22 20 82 91 100 114 120 116 104 83 25 24 23 21 20 l9 91 103 ~.02 98 93 86 78 61 66 71 76 81 86 78 72 69 72 78 89 107 117 118 110 ~4 68 23 22 2:1 20 19 18 83 93 91 87 82 74 65 50 56 62 68 '73 79 73 68 68 73 66 64 62 62 62 63 60 58 57 57 58 60 65 69. 73 75 77 2 ~ J ~

The compressed image is produced as follows: first, the picture is subdivided into square segments, c ontaining 6 x 6 = 36 pixels each one. See Fig. 4a and the array S above, where one of such segments is marked.
The step 1 consists in approximating the picture on each segment by the model, which is chosen to be the quadratic polynomial z=ao+alx+a2y+allx2+2al2xy+a22y2 where z represents the gray level, and x and y are the coordinates on the picture plane centered at the center of the corresponding segment.
The values of the coef~lcient "a" are found by the standard subroutine, minimizing the quadratic error of the approximation of the gray level on each segment by the model chosen.
The array of 8 x 8 = 64 polynomials, obtained on the segrnents, covering the piece S of the picture, is given in the following array 7. `

-38- 2~
0.37500000 -0.0058594 -0.0055246 0.001883 -0.007146 0.010986 0.35937500 -0.0104353 -0.0276228 0.007010 0.013632 0.009208 0.31640625 -0.0081473 0.0078125 0.010149 -0.001263 0.033064 0.39843750 -0.0127790 0.0998326 0.003348 0.014263 0.062360 0.74218750 -0.0247767 0.1643973 -0.015172 0.000459 -0.049700 0.70703125 0.0304130 -0.2262835 -0.009312 -0.027809 -0.038191 0.50781250 -0.0092076 -0.0348772 -0.015067 -0.003071 0.007847 0.43359375 0.0021763 -0.0233817 0.007010 0.004908 -0.004918 0.33593750 -0.0344308 ~0.0031808 0.003348 0.027637 0.026263 0.37109375 0.0164063 -0.0194196 -0.005022 0.008954 -0.022600 0.32421875 0.0092634 -0.0077009 0.016532 -0.015383 -0.003871 0.38671875 -0.0045201 0.0807478 0.006069 -0.026834 0.035575 0.63671875 -0.0199777 0.1973214 0.009626 -0.011422 0.066127 0.81250000 0.0224331 -0.2329241 0.003558 0.016875 -0.205811 0.48828125 -0.0261161 -0.0174107 -0.025321 0.024337 0.006069 0.43750000 0.0035156 -0.0440290 -0.002511 -0.010188 -0.007533 0.32031250 0.0101563 0.0811384 0.006278 0.006256 0.002511 0.40234375 -0.0292969 -0.0090960 -0.058594 -0.026260 0.015172 0.40234375 -0.0013g51 -0.0376674 -0.050642 0.022643 -0.022391 0.33984375 -0.0376116 0.0617188 0.004290 -0.008265 0.046980 0.63281250 -0.0425223 0.1786830 -0.036516 0.004305 0.006173 0.80468750 -0.0712053 -0.2079241 -0.040388 0.029043 -0.234375 0.45703125 -0.0006696 0.0003348 -0.000314 -0.009184 -0.011300 0.41015625 0.0016741 -0.0319196 0.020299 0.019056 -0.000732 0.33203125 -0.0089286 0.0193080 0.020194 -0.052519 0.014544 0.21093750 0.1237165 -0.0967076 0.060059 0.156036 0.077009 0.25781250 0.0914063 0.0062500 0.087995 -0.093673 -0.055455 0.26562500 -0.0221540 0.0343192 0.074916 -0.002899 0.018101 0.47265625 -0.0229911 0.2156250 0.078055 0.008839 0.015904 0.57812500 -0.0305245 -0.1201451 -0.003557 0.072006 0.021240 0.49531250 0.0000558 -0.0220424 0.014230 -0.028096 0.001674 0. ~1015625 -0.0073103 -0.0318638 -0.019198 0.008409 -0.016950 0. q3750000 0.0410714 -0.0239955 -0.033378 0.020663 -0.000732 0.36718750 0.0566406 0.0056362 0.019962 -0.005309 0.016218 0.48046875 0.1619397 0.0278460 0.038295 0.096687 -0.109236 0.53906250 0.1313058 0.1659040 -0.128697 0.012025 0.059618 0.71093750 0.0371094 -0.0090960 -0.059326 -0.061617 -0.031076 0.53125000 -0.0079799 -0.0807478 0.012347 -0.010418 0.015486 0.44921875 -0.0116630 -0.0317522 -0.008789 0.0079.50 0.000000 0.38671875 -0.0111607 -0.016071~ -0.011300 0.000402 0.000000 0.95703125 0.0071929 -0.006138~ 0.010568 0.007691 -0.002302 0.96093750 0.03~2639 0.0351562 0.00198~ 0.005568 0.035261 0.73828125 0.0527394 0.0965960 -0.012660-0.018109 -0.15077q 0.59765625 0.0304687 0.0779553 -0.012556-0.091212 0.1~6589 0.71875000 0.0060268 -0.0338170 -0.019880 -0.024952 -0.011905 0.51171875 -0.0079799 -0.0932~78 -0.001569 -0.002612 0.015695 0. ~3750000 -0.0099085 -0.024~978 0.00931~ -0.00~334 -0.03306~
0.37500000 -0.0677456 -O . lqlO714 0.003557 -0.121397 -0.143032 0.45703125 -0.0029559 0.0328125 0.0299890.014522 -0.005650 0.51171875 -0.0028460 0.0307478 -0.024170 -0.036993 0.026681 0.75390625 -0.125167q 0.0778960 -0.168298 -0.019142 -0.088518 0.68359375 0.0162997 0.0053571 -0.122001-0.059694 0.078265 0.70703125 -0.1199218 -0.0175781 -0.155797 0.070226 0.024379 0.44921875 -0.0365513 -0.1284040 -0.030866 -0.027522 0.101597 0.41796875 -0.0125000 0.0001116 0.0006280.019687 -0.008161 0.1523q375 -0.0695313 -0.1944197 0.006906 0.062679 0.108608 0.29296875 -0.1353237 -0.0140067 0.060582 0.000373 0.015695 0.27734375 -0.1326451 -0.0152344 0.019357 0.018683 0.025635 0.32031250 -0.0872768 0.0389509 0.044155-0.000804 -0.024902 0.34765625 -0.1336496 -0.0653460 0.070103 -0.026145 -0.042585 0.26171875 -0.1049665 0.0693639 0.1640620.038772 -0.000419 0:26562500 0.0065848 0.0437500 0.130371 0.086556 0.090193 0.43750000 -0.0600446 -0.1228755 -0.031076 -0.157902 -0.156948 0.0742].875 -0.0148995 -0.0127790 0.015904 0.030048 0.013079 2 0 C~ v (The coeff~lcients are given after rescaling the x and y variables to the square [-1,1] [-1, 1], and the gray level z to [0,1] ).
Step 2 The required accuracy ~ is chosen to be ~ gray levels, k is fixed to be 2, and the grid on each segment is chosen to contain the only point - the center of this segment. Thus the Taylor polynomials computed on this step are identical to the approximating polynomials found on the step 1.
The 6 digits accuracy with which the coefficients of these polynomials are given in the array P above is excessive, and the coefficients are rounded off up to 8 bits in degree 0, up to 7 bits in degree 1 and up to 6 bits in degree 2 The corresponding binary array is the intermediate compressed image. It is approximately represented by the following digital array P' (corresponding to the same piece S of the picture, as the above array P).

_40_ 2~ 0~
0.37500000 0.0000000 0.0000000 0.000000 0.000000 0.000000 0.35937500 -0.0078125-0.02343750.000000 0.000000 0.000000 0.31640625 -0.00781250.00000000.000000 0.000000 0.031250 0.39843750 -0.00781250.09375000.000000 0.000000 0.046875 0.79218750 -0.02343750.16406250.000000 0.000000 -0.046875 0.70703125 0.0234375-0.21875000.000000 -0.015625 -0.031250 0.50781250 -0.0078125-0.03125000.000000 0.000000 0.000000 0.43359375 0.0000000-0.01562500.000000 0.000000 0.000000 0.33593750 -0.03125000.00000000.000000 0.015625 0.015625 0.37109375 0.0156250-0.01562500.000000 0.000000 -0.015625 0.32421875 0.00781250.00000000.015625 0.000000 0.000000 0.38671875 0.00000000.07812500.000000 -0.015625 0.031250 0.63671875 -0.01562500.19531250.000000 o . oooooo 0.062500 0.81250000 0.0156250-0.22656250.000000 0.015625 -0.203125 0.49218750 -0.0234375-0.0156250 -0.015625 0.015625 0.000000 0 - 43750000 0.0000000 -0.0390625o . oooooo o . oooooo o . oooooo 0.32031250 0.00781250.0781250 0.000000 0.000000 0.000000 0.40234375 -0.0234375-0.0078125-0.046875 -0.015625 0.000000 0.40234375 0.0000000-0.0312500-0.046875 0.015625 -0.015625 0.33984375 -0.03125000.05468750.000000 0.000000 0.046875 0.63281250 -0.03906250.1718750-0.031250 0.000000 0.000000 0.80968750 -0.0703125-0.2031250-0.031250 0.015625 -0.218750 0.45703125 0.00000000.00000000.000000 - -0.41015625 0.0000000-0.03125000.015625 0.015625 0.000000 0.33203125 -0.00781250.01562500.015625 -0.046875 0.000000 0.21093750 0.1171875-0.09375000.096875 0.140625 0.062500 0.25781250 0.08593750.00000000.078125 -0.078125 -0.046875 0.26562500 -0.01562500.03125000.062500 0.000000 0.015625 0.47656250 -0.01562500.21093750.062500 0.000000 0.015625 0.57812500 -0.0234375-0.11718750.000000 0.062500 0.015625 0.44531250 0.0000000-0.01562500.000000 -0.015625 0.000000 0.41015625 0.0000000-0.0312500-0.015625 0.000000 -0.015625 0.93750000 0.0390625-0.0234375-0.031250 0.015625 0.000000 0.36718750 0.05468750.00000000.000000 0.000000 0.015625 0.48437500 0.15625000.02343750.031250 0.093750 -0.093750 0.53906250 0.12500000.1640625-0.125000 0.000000 0.046875 0.71093750 0.0312500-0.0078125-0.0~6875 -0.046875 -0.015625 0.53125000 -0.0078125-0.07812500.000000 0.000000 0.000000 0.49921875 -0.0078125-0.03125000.000000 0.000000 0.000000 0.386-71875 -0.0078125-0.01562500.000000 0.000000 0.000000 0.45703125 o . ooooooo o . ooO0000o . ooooooo . ooO000 o . oooooo 0.96093750 0.03125000.03125000.000000 0.000000 0.031250 0.73828125 0.04687500.09375000.000000 -0.015625 -0.140625 0.59765625 0.02343750.07031250.000000 -0.031250 0.190625 0.71875000 0.0000000-0.0312500-0.015625 -0.015625 0.000000 0.51171875 -0.0078125-0.08593750.000000 0.000000 0.015625 0,43750000 0.0000000-0.023~3750.000000 0.000000 -0.031250 0.37500000 -0.0625000-0.14062500.000000 -0.109375 -0.190625 0.45703125 0.00000000.03125000.015625 0.000000 0.000000 0.51171875 0.00000000.0234375-0.015625 -0.0312500.015625 0.75390625 -0.12500000.0703125-0.156250 -0.015625-0.078125 0.68359375 0.01562500.0000000-0.109375 -0.0468750.078125 0.70703125 -0.1171875-0.0156250-0.140625 0.062500 0.015625 0.44921875 -0.0312500-0.1250000-0.015625 -0.015625 0.093750 0.41796875 -0.00781250.00000000.000000 0.0156250.000000 0.15234375 -0.0625000-0.18750000.000000 0.0625000.093750 0.29296875 -0.1328125-0.00781250.046875 0.0000000.015625 0.27734375 -0.1250000-0.0~781250.015625 0.0156250.015625 0.32031250 -0.08593750.03125000.031250 0.000000-0.015625 0.34765625 -0.1328125-0.06250000.062500 -0.015625-0.031250 0.26171875 -0.10156250.06250000.156250 0.0312500.000000 0.26562500 0.00000000.03906250.125000 0. b78125 0.078125 0.43750000 -0.0546875-0.1171875 -0.015625 -0.156250 -0.156250 0.07421875 -0.0078125-0.00781250.015625 0.015625 0.000000 2 ~ i3 The compression ratio is 512*512*8 bits/86*86*(8 + 2*7 + 3*6) bits ~ 6.7.

The final image is obtained by computing the values of the Taylor polynomials, representing the intermediate image, at each pixel of the corresponding segment. The part S' of the obtained array, representing the final image tand corresponding to the piece S of the initial picture), is the ~ollowing array 9.

-42- 2~

65 63 61 55 67 72 66 73 69 73 74 73 ~3 70 78 80 76 83 88 90 102 106 111 110 116 118 125 128 126 127 136.

129 13~ 129 136 135 ~.34 129 130 131 136 136 136 132 132 132 127 131 209 197 183 156 131 96 77 77 71 79 76 ~3 73 85 87 89 91 88 90 90 94 93 lOl 93 88 88 92 96 92 ~7 94 lO0 92 97 96 95 lOl 95 97 93 89 lOl 95 97 97 88 94 82 - 43 - ~ ?3 96 104 98 96 98 100 95 106 108 102 103 lOO 100 96 99 104 91 95 102 97 105 102 102 103 103 99 103 96 106 102 99 102 lOl 106 97 107 105 104 106 101 100 lO0 lOl 99 102 97 98 99 103 105 103 97 96 95 101 95 97 93 89 101 95 97 97 8~ 9q 137 137 136 138 129 138 134 140 136 135 134 130 139 135 129 13~ 131 149 143 154 145 ~.42 135 133 118 120 127 110 98 82 86 74 63 65 89 92 95 92 90 92 91 lOl 97 92 89 94 104 100 ~ 3 106 100 104 103 101 102 100 100 100 104 108 103 104 99 104` 99 105 99 110 104 101 100 102 102 98 100 1~3 103 99 98 100 84 88 89 84 91 93 107 139 144 lq7 1~9 148 130 99 137 137 137 136 138 l~9 138 139 140 136 135 139 130 139 135 129 13~
131 138 139 127 132 129 13~ 131 133 130 125 130 129 127 131 127 134 109 98 96 98 lO0 95 106 108 102 103 100 lO0 96 99 104 91 98 95 105 101 98 lOl 97 102 97 97 105 104 105 103 95 102 97 105 102 102 103 103 99 103 96 106 102 99 102 101 106 lO0 L ~
105 106 101 105 lOS 105 101 109 106 104 102 105 104 105 102 104 112 134 128 135 133 144 135 142 143 142 143 150 148 154 153 14~ 149 149 143 154 1~5 142 135 133 118 120 127 110 98 82 86 79 78 78 73 75 73 75 '79 79 ~9 77 81 86 90 86 78 89 89 92 95 92 90 92 91 101 97 92 89 94 104 lO0 lOl 96 104 98 96 98 100 95 106 108 102 103 lO0 lO0 96 99 104 91 lO0 104 103 101 102 100 100 100 104 108 103 104 99 104 99 105 106 The picture representing the final image is shown in Fig. 4b.
Example 3 (~ota~on of a ~icture) The object picture is the same as in the Example 2. The required operation is the rotation by 90 in the counterclockwise direction (Fig. 5a represents the result of a rotation of the object picture).
The array of the gray levels of the rotated piece S' of the object picture is the following array 10.

_47- ~9 ~ 6 56 62 71 82 96 113 ll9 118 117 117 118 119 118 118 118 118 118 56 62 71 82 96 113 125 123 ' 22 122 123 125 118 118 118 118 118 111 107 102 97 93 88 83 88 74 6q 56 50 48 92 99 103 105 114 107 102 97 93 88 83 90 72 58 46 36 30 90 97 102 lOg 50 58 67 77 88 lO0 129 132 134 136 136 136 132 129 126 124 121 118 108 103 98 94 89 84 95 .74 55 39 26 15 88 96 101 104 -48- 20~ 6 83 86 93 ~.03 117 134 181 189 190 186 175 158 174 169 165 161 157 61 66 74 86 102 121 159 172 179 180 17~ 162 173 172 171 171 170 - 49 - 2 ~ .3 ~ 33 158 157 152 153 154 156 157 158 i78 180 182 184 186 188 187 187 189 ~.90 191 193 194 195 203 205 207 209 211 213 96 81 73 '72 78 91 101 106 110 113 115 116 126 126 127 128 128 202 205 20~. 199 198 196 195 193 176 174 172 171 169 167 ` --`` 2 ~ 6 112 113 114 114 115 116 116 114 114 114 114 115 115 1.18 118 118 118 118 11~ 118 121 123 125 125 125 124 125 126 126 127 128 10~ 100 101 102 102 103 104 106 108 109 109 108 106 111 110 110 110 112 114 118 118 118 118 118 118 114 114 114 114 114 llg 21 20 ~0 21 23 25 33 39 46 52 58 64 93 96 100 104 108 107 98 98 99 100 100 101 102 104 104 104 lOq 102 108 106 105 105 22 20 19 19 20 22 lS 18 21 23 26 29 45 55 65 75 85 The above rotation acts on the Taylor polynomials, representing the intermediate image, obtained in the Example 2, as follows: let the 6 x 6 pixel square segments, into which the original picture has been subdivided, be indexed by two indices i and j, in such a way that the middle segment has indices 0, O. Denote the Taylor polynomial corresponding to the segment i, j by pjj. Then:
a. The indices i, j of each p;j are replaced by j, i b. x is replaced by y, and y by -x.
Using the notations already used in discussing processing, F(pjj(x,y) )=
P j, i ty, -x).
The result of the application of the corresponding subroutine to the Taylor polynomials in the intermediate range, obtained in the Example 2, is the intermediate range of the rotated picture. Its part P' corresponding to the rotated piece S', is the following array 11.

-52- 20~
0.2929687s -0.00781250.0078125 0.015625-0.0000000.015625 0.45703125 0.0312500-0.0312500 0.000000-0.0000000.000000 0.45703125 0. ooooooo -0.0000000 o . oooooo -0.000000 0.0000000.43750000 -0.02343750.0234375 0.000000-0.0156250.000000 0.33203125 0.0156250-0.0156250 0.0000000.0468750.000000 0.32031250 0.0781250-0.0781250 0.000000-0.0000000.000000 0.33593750 0.0000000-0.0000000 0.015625-0.0156250.015625 0.37500000 0.0000000-0.0000000 0.000000-0.0000000.000000 0.27734375 -0.00781250.0078125 0.015625-0.0156250.015625 0.51171875 0.0234375-0.0234375 0.0156250.0312500.015625 0.46093750 0.0312500-0.0312500 0.031250-0.0000000.031250 0.36718750 0.0000000-0.0000000 0.015625-0.0000000.015625 0.21093750 -0.09375000.0937500 0.062500-0.1406250.062500 0.40234375 -0.00781250.0078125 0.0000000.0156250.000000 0.37109375 -0.01562S00.0156250 -0.015625-0.000000-0.015625 0.35937500 -0.02343750.0234375 0.000000-0.0000000. oooooo 0.32031250 0.0312500-0.0312500 -0.015625-0.000000-0.015625 0.75390625 0.0703125-0.0703125 -0.0781250.015625-0.078125 0.73828125 0.0937500-0.0937500 -0.1406250.015625-0.140625 0.48437500 0.0234375-0.0234375 -0.093750-0.093750-0.093750 0.25781250 0.0000000-0.0000000 -0.0468750.078125-0.046875 0.40234375 -0.03125000.0312500 -0.015625-0.015625-0.015625 0.32421875 0.0000000-0.0000000 - - - -0.31640625 0.0000000-0.0000000 0.031250-0.0000000.031250 0.34765625 -0.06250000.0625000 -0.0312500.015625-0.031250 0.68359375 0.0000000-0.0000000 0.0781250.0468750.078125 0.59765625 0.0703125-0.0703125 0.1406250.0312500.140625 0.53906250 0.1690625-0. ].6~0625 0.096875-0.0000000.096875 0.26562500 0.0312500-0.0312500 0.015625-0.0000000.015625 0.33984375 0.0546875-0.0546875 0.046875-0.0000000.046875 0.38671875 0.0781250-0.0781250 0.0312500.0156250.031250 0.39893750 0.0937500-0.0937500 0.046875-0.0000000.046875 0.26171875 0.0625000-0.0625000 0.000000-0.0312500.000000 0.70703125 -0.01562500.0156250 0.015625-0.0625000.015625 0.71875000 -0.03125000.0312500 0.0000000.0156250.000000 0.71093750 -0.00781250.0078125 -0.0156250.046875-0.015625 0.47656250 0.2109375-0.2109375 0.015625-0.0000000.015625 0.63281250 0.1718750-0.1718750 0.000000-0.0000000.000000 0.63671875 0.1953125-0.1953125 0.062500-0.0000000.062500 0.74218750 0.1640625-0.1640625 -0.046875-0.000000-0.096875 0.26562500 0.0390625-0.0390625 0.078125-0.0781250.078125 0.94921875 -0.12500000.1250000 0.0937500.0156250.093750 0.51171875 -0.08593750.0859375 0.015625-0.0000000.015625 0.53125000 -0.078~.250 0.0781250 0.000000 -0.000000 0.0000000.57812500 -0.11718750.1171875 0.015625-0.0625000.015625 0.80468750 -0.20312500.2031250 -0.218750-0.015625-0.218750 0.81250000 -0.22656250.2265625 -0.203125-0.015625-0.203125 0.70703125 -0.21875000.2187500 -0.0312500.015625-0.031250 0.43750000 -0.11718750.1171875 -0.1562500.156250-0.156250 0.91796875 0.0000000-0.0000000 0.000000-0.0156250.000000 0.43750000 -0.02343750.0234375 -0.031250-0.000000-0.031250 0.44921875 -0.03125000.0312500 0.000000-0.0000000.000000 0.44531250 -0.01562500.0156250 0.0000000.0156250.000000 0.45703125 0.0000000-o . ooooooo o . oooooo _o . ooO000 o . oooooo 0.49218750 -0.01562500.0156250 0.000000-0.015625~ .000000 0.50781250 -0.03125000.0312500 0.000000-0.0000000.000000 0.07421875 -0.00781250.0078125 0.000000-0.0156250.000000 0.15234375 -0.18750000.1875000 0.093750-0.0625000.093750 0.37500000 -0.14062500.1406250 -0.1406250.109375-0.140625 0.38671875 -0.01562500.0156250 0.000000-0.0000000.000000 0.41015625 -0.03125000.0312500 -0.015625-0.000000-0.015625 0: 41015625 -0.0312500 0.0312500 0.000000 -0.015625 0.0000000.43750000 -0.03906250.0390625 0.000000-0.000000o . oooooo 0.43359375 -0.01562500.0156250 0.000000-0.000000O . OOC000 2~4~06 The final image, produced from the data rotated in a compressed form, is shown in Fig. 5b.
E~ample 4 OE'roducing a negat;ive picture) The object picture is the same as in the Example 2. It is required to produce a negative of this picture. Under this operation each gray level value z must be replaced by z' = 255 - z.
The negative of the original picture is shown in Fig. 6a. The array S" of the gray levels, corresponding to the negative of the piece S, is the following.

_54_ 2~L~A;~3 169 165 163 15'7 149 139 125 113 94 77 63 52 43 25 41 59 79 90 133 12~ 126 127 129 130 132 13~ 137 1~0 1~4 147 150 17~. 166 163 158 152 143 133 123 11~ 101 89 64 40 35 31 38 57 88 130 127 128 130 131 132 133 134 137 1~0 144 197 150 169 164 162 158 152 144 134 125 llS 102 85 65 41 34 30 37 56 -55~ 204~06 190 184 177 170 ` 164 157 155 155 155 154 154 153 153 155 158 161 166 188 182 175 168 162 155 150 151 152 153 15~ 155 148 149 150 153 156 166 177 178 176 171 16q 154 144 129 114 lO0 85 70 68 59 63 79 176 172 167 162 158 153 161 186 207 225 240 251 215 204 196 l90 187 194 184 183 181 179 175 171 169 153 136 118 99 79 70 84 98 llO
122 133 138 139 140 140 141 142 146 147 149 lSl 155 159 174 ~.72 170 168 166 164 169 186 199 209 216 219 202 196 192 191 192 197 189 188 186 183 180 175 174 158 l~l 123 104 8~ 76 89 lO0 lll 1.21 130 137 138 139 141 192 1~3 145 1~6 1~8 lSl 154 159 121 128 136 138 139 141 192 144 1~5 146 148 lSl 154 159 -56- 2 ~ 0 6 127 134 131 133 136 139 ~.41 144 150 151 153 154 155 157 138 141 143.145 147 149 160 161 162 162 161 160 156 144 138 138 142 132 139 131 ll9 105 89 69 71 71 71 73 75 78 103 109 116 123 130 137 133 136 139 141 144 147 153 154 155 15? 158 159 75 98 107 108 100 85 62 67 69 71 73 75 77 102 lll 119 127 134 140 143 141 ~41 143 147 153 152 149 155 168 190 220 137 137 137 137 137 137 130 131 130 ].28 123 117 98 75 60 54 55 2~ 3~

147 147 146 146 147 147 147 147 153 181 203 220 232~238 143 140 137 135 ~.32 129 124 125 125 125 123 121 115 101 91 86 85 155 138 139 142 147 153 161 179 170 166 162 158 154 16~ 173 177 177 164 152 153 157 162 169 177 194 189 18q 179 17q 169 177 183 186 183 172 162 164 i68 173 181 190 205 199 193 187 182 176 182 187 187 182 189 191 193 193 193 192 195 197 198 198 197 195 l90 186 182 180 178 2 ~

The above operation on Taylor polynomials is the following:
F(aO+alx~a2y+all x2+~al2 XY+a22Y2)=
1 - aO- a1 x - a2 y - a11 x2 - 2al2 xy - a22 y2 (in the same rescaling as above).
The corresponding subroutine, applied to the Taylor polynomials of the intermediate image obtained in the Example 2, gives the intermediate image of the negative. The part of the polynomials array P", corresponding to the piece S" of the negative, is the following.

2~o~Qo6 o ~ ~ ~ooooo -o . ooooooo -o ~ ooooooo -o . oooooo -o . oooooo -o . o o o 0.64062500 0.0078125 0.0234375 -0.000000 -0.000000 -0.000000 0.68359375 0.0078125 -0.0000000 -0.000000 -0.000000 -0.031250 0.60156250 0.007~'`125 -0.0937500 -0.000000 -0.000000 -0.046875 0.~57~1~50 0.0234375 -0.]690625 -0.000000 -0.000000 0.046875 0.29296~ 15 -0.023~375 0.21~'7500 -0.000000 0.015625 0.031250 0.49218750 0.0078125 0.0312500 -0.000000 -0~ 000000 -0.000000 0.56640625 -0.0000000 0.0156250 -0.000000 -0.000000 -0.000000 0.66406250 0.0312500 0.0000000 -0.000000 -0.015625 -0.015625 0.62890625 -0.0156250 0.0156250 -0.000000 -0.000000 0.015625 0.67578125 -0.0078125 -0.0000000 -0.015625 -0.000000 -0.000000 0.61328125 -0.0000000 -0.0781250 -0.000000 0.015625 -0.031250 0.36328125 0.0156250 -0.1953125 -0.000000 -0.000000 -0.062500 0.18750000 -0.0156250 0.2265625 -0.000000 -0.015625 0.203125 0.50781250 0.02343750.0156250 0.015625 -0.015625 -0.000000 0.56250000 -0.00000000.0390625-0.000000 -0.000000 -0.000000 0.67968750 -0.0078125-0.0781250 -0.000000 -0.000000 -0.000000 0.59765625 0.02343750.00781250.046875 0.015625 -o.ooooao 0.59765625 -0.00000000.03125000.046875 -0.015625 0.015625 0.66015625 0.0312500-0.0546875-0.000000 -0.000000 -0.046875 0.36718750 0.0390625-0.17187500.031250 -0.000000 -0.000000 0.19531250 0.07031250.20312500.031250 -0.015625 0.218750 0.54296875 -0.0000000- - - - - - -0.58984375 -0.00000000.0312500-0.015625 -0.015625 -0.000000 0.66796875 0.0078125-0.0156250-0.015625 0.046875 -0.000000 0.78906250 -0.11718750.0937500-0.096875 -0.140625 -0.062500 0.74218750 -0.0859375-0.0000000-0.078125 0.078125 0.046875 0.73437500 0.0156250-0.0312500-0.062500 -0.000000 -0.015625 0.52343750 0.0156250-0.2109375-0.062500 -0.000000 -0.015625 0.42187500 0.02343750.117'1875-0.000000 -0.062500 -0.01562S
0.55468750 -0.00000000.0156250-0.000000 0.015625 -0.000000 0.58984375 -0.00000000.03125000.015625 -0.000000 0.015625 0.56250000 -0.03906250.02343750.031250 -0.015625 -0.000000 0.63281250 -0.0546875-0.0000000-0.000000 -0.000000 -0.015625 0.51562500 -0.1562500-0.0234375-0.031250 -0.093750 0.093750 0.46093750 -0.1250000-0.16406250.125000 -0.000000 -0.046875 0.28906250 -0.03125000.00781250.046875 0.046875 0.015625 0.46875000 0.00781250.0781250-0.000000 -0.000000 -0.000000 0.55078125 0.00781250.0312500-0.000000 -0.000000 -0.000000 0.61328125 0.00781250.0156250-0.000000 -0.000000 -0.000000 0.54296~75 -0.0000000-0.0000000-0.000000 -0.000000 -0.000000 0.53906250 -0.0312500-0.0312500-0.000000 -0.000000 -0.031250 0.26171875 -0.0468750-0.0937500-0.000000 0.015625 0.1~0625 0.40234375 -0.0234375-0.0703125-0.000000 0.031250 -0.140625 0.28125000 -0.00000000.03125000.015625 0.015625 -0.000000 0.48828125 0.00781250.0859375-0.000000 -0.000000 -0.015625 0.56250000 -0.00000000.023~375-0.000000 -0.000000 0.031250 0.62500000 0.06250000.1406250-0.000000 0.109375 0.190625 0.54296875 -0.0000000-0.0312500-0.0156~5 -0.000000 -0.000000 0.48828125 -0.0000000-0.02343750.015625 0 031250 -0 015625 0.24609375 0.1250000-0.07031250.156250 0.0156250.0i8125 0.31640625 -0.0156250-0.00000000.109375 0.046875 -0.078125 0.29296875 0.11718750.01562500.140625 -0.062500-0.015625 0.55078125 0.03125000.12500000.015625 0.015625-0.093750 0.58203125 0.0078125-0.0000000-0.000000 -0.015625 -0.000000 0.84765625 0.06250000.1875000-0.000000 -0.062500-0.093750 0.70703125 0.13281250.0078125-0.046875 -0.000000-0.015625 0.72265625 0.12500000.0078125-0.015625 -0.015625-0.015625 0.67968750 0.0859375-0.0312500-0.031250 -0.000000 0.015625 0.65234375 0.13281250.0625000-0.062500 0.0156250.031250 0.73828125 0.1015625-0.0625000-0.156250 -0.031250 -0.000000 0 :73437500 -0.0000000 -0.0390625 -0.125000 -0.078125 -0.0781250.56250000 0.05468750.1171875 0.015625 0.156250 0.156250 0.92578125 0.00781250.0078125-0.015625 -0.01562,5 -0.000000 2 ~

The final image produced from the intermediate negat*e image, obtained as above, is shown in Fig. 6b.

While a number of embodiments of the invention have been discussed and illustrated, it will be understood that the invention may be carried out in a number of ways and with many modifications, adaptatlons, and variations, by persons skilled in the art, without dcparting from its Spilit and from the scope of the appended claims.

Claims (22)

1 - Process for the production of images of objects, as hereinbefore defined, comprosing the steps of:
(1) Approximating the object by a model comprising at least one differentiable component.
(2) Establishing the maximum allowable error .epsilon. and the degree k of the polynomials by which the differentiable component(s) of the model are to be approximated.
(3) Constructing a grid of a suitable pitch h.
(4) Computing the coefficients of the Taylor polynomials of the aforesaid differentiable component(s) at selected points of said grid.

2 - Process according to claim 1, wherein the object is defined in a space having more than three dimensions.

3 - Process according to claim 1, wherein the object is a line.

4 - Process according to claim 1, wherein the object is a surface.

5 - Process according to claim 1, wherein the object is a solid.

6 - Process according to claim 1, wherein the model further comprises at least one non-differentiable component.

7 - Process according to claim 1, comprising carrying out the said steps at least in part concurrently.

8 - Process according to claim 1, wherein the object is defined by data which are values and/or relationships embodied in physical entities.

9 - Process according to claim 8, comprising the preliminary step of storing the data defining the object in an electronic memory.

10 - Process according to claim 1, comprising determining the parameters of the components of the model by minimizing a quantity representing an error

11 - Process according to claim 10, wherein the quantity representing an error is the quadratical error.

12 - Process according to claim 1, wherein the non-differentiable component(s) of the model embody the same discontinuities as the object, and the differentiable component(s) represent the deviations of the object lfrom the non-differentiable component.

13 - Process according to claim 12, wherein the model has the form:

(1) .PHI.(x) = Hx0,a,b,c,d(x) + .PHI.(x) wherein H is defined by H(x) = a(x-x0) + b, if x ? x0 or H(x) = c(x-x0) + d, if x is less than x0.

14 - Process according to claim 1, wherein the model is a differentiable function of another function which embodies the non-differentiable characteristics of the object.

15 - Process according to claim 1, wherein each grid pitch is calculated from the formula (3) CMhk+1 ? .epsilon.
wherein C = 1/(k+1)! and M is the maximum, at each grid point, of the absolute value of the derivatives of degree k+1 of the differentiable component or components of the model.

16- Process according to claim 1, further comprising constructing an adjusted image line by applying to each differentiable component the Whitney subroutine, and minimizing the quntity W thus computed, under such constraints that the results of the minimization do not deviate from the initial data by more than the allowed error.

17 - Process according to claim 1, further comprising rounding off the coefficients of the Taylor polynomials to a maximum allowable error greater than the original one.

18 - Process according to claim 1, further comprising separating a temporary image into components of increasing fineness, constructing a grid which is sparser than the one used for obtaining said image and the pitch of which is determined by the resolution required by the lowest fineness of said components, obtaining thereforom a second temporary image, subtracting said second temporary image from the original one to obtain a first residual image, and repeating the same steps for successively finer components, correspondingly obtaining successive residual images, whereby to compute coefficients of Taylor polynomials on several grids having increasingly higher resolutions.

19 - Process according to claim 1, further comprising applying to the coefficients of the Taylor polynomials any desired known encoding method.

20 - Process according to claim 1, further comprising applying to any data obtained in carrying out the process any desired known encoding method.

21 - Process according to claim 1, further comprising constructing a final image by a procedure comprising the steps of dividing the domain, in which the temporary image has been defined, into possibly overlapping regions by means of a grid, each region being a portion of the grid around a grid node, and constructing curves representing the Taylor polynomials of degree k from the coefficients defining the temporary image at each grid node.

22 - Process according to claim 1, further comprising processing the obtained data, representing an intermediate image, by applying thereto an operator, whereby to obtain an image representing an object which is the result of applying to the original object the said operator.