ITTO20130683A1 - APPARATUS AND METHOD FOR THE CORRECTION OF PROSPECTIVE DEFORMATIONS OF IMAGES - Google Patents

Description

SVT054

DESCRIPTION of the Industrial Invention entitled: -SVT054-APPARATUS AND METHOD FOR THE CORRECTION OF THE PERSPECTIVE DEFORMATIONS OF THE IMAGES

of SISVEL TECHNOLOGY S.r.l., of Italian nationality, with registered office in Via Castagnole 59, 10060 NONE (TO) and electively domiciled, for the purposes of this assignment, with the Agents of the Agents Eng. Corrado Borsano (Registered No. 446 BM), Eng. Marco Camolese (Registered No. 882 BM), Eng. Matteo Baroni (Registered No. 1064 BM), Dr. Giancarlo Reposio (Registered No. 1168 BM), Eng. Giovanni Zelioli (Registered No. . Register 1536 B) c / o Metroconsult S.r.l., Via Sestriere 100, 10060 None (TO).

Designated Inventor:

- Pietro Porzio Giusto, via Cassia 1110, 00189 - Rome

Filed No.

DESCRIPTION

The present invention relates to an apparatus and a method for correcting images, in order to reduce the deformations that appear, both in the two-dimensional and in the stereoscopic vision, when the images are observed from an observation point that does not correspond to the projection center of the prospect.

As is known, linear perspective, also called simply perspective, is a geometric-mathematical method that allows you to reproduce a three-dimensional scene on a plane. The basic criterion of the perspective construction, shown in fig. 1, consists in projecting on a plane 101, called "projection plane" or "projection frame" or simply "frame", the points of the three-dimensional space as they are seen from a "projection center" C. The straight line that branches off from the center of projection in the direction towards which the observer's gaze is oriented, or the lens of the camera, is called the "optical axis." Generally it is used to define a Cartesian reference such as that of fig. 1, in which the axes originate in the center of projection C, the z axis coincides with the optical axis, the y axis is vertical, oriented from bottom to top, and the x axis is horizontal, oriented from left to right for the In the terminology used in this technical field, the z axis is also called the "depth" axis, being the "depth" of a point in three-dimensional space defined as the distance of the given point from the xy plane. projection plane 101 has distance f from the projection center C and is perpendicular to the optical axis, which intersects the plane of SVT054

projection (101) in point Ic. The projection Q of a point in space A results from the intersection between the projection plane 101 and the "projecting line", that is, the line that passes through the point to be projected A and the center of projection C.

Camera and film cameras produce images theoretically compliant with linear perspective, except that the lenses of real lenses often introduce more or less visible distortions, such as, for example, those called "barrel" and "pillow". The present invention does not deal with this type of distortions, which, on the other hand, are the subject of numerous studies and correction techniques (see for example the European patent EP1333498 B1 in the name of Agilent Technologies Inc. and the international patent application WO 98 / 57292 A1 to Apple Computer).

Nor does the present invention deal with image deformations due to lens positioning errors with respect to the desired location and orientation, such as tapering of images of tall buildings taken from below and deformations of images of documents photographed obliquely or from points of view outside the document axis. These deformations are also extensively discussed in the literature (see for example documents US 7,990,412 A1, US 2009/0103808 A1, BR PI0802865 A2, US 2004/0022451 A1, US 2006/0210192 A1, US 2011/0149094 A1).

The correction technique of the present invention, on the other hand, processes the images as if they were perfectly consistent with the linear perspective. Any distortions with respect to the linear perspective, and in particular those mentioned above, are not considered and are reflected in the results of the calculations.

The present invention relates, in fact, to the deformations that appear in the perspective images when these images are observed from an observation point that does not correspond to the center of projection, such as, for example, if one observes the image of the painting 101 (fig. 1) from point V instead of from the center of projection C.

Fig. 2 shows an example of such deformations. It shows the image of a solid 202 drawn strictly by a calculation program according to the rules of linear perspective. The image of solid 202 is located in the lower right corner of a painting, of which the figure shows only the lower right quadrant (also called fourth quadrant) 201, to enlarge the elements of interest (the solid 202 and the center Ic of the projection plane) with respect to the dimensions that they would have reproducing the SVT054

whole picture.

Observing fig. 2 as one normally looks at a photograph, for example from a point that is on the perpendicular to the center of the figure, the solid 202 seems to have the shape of a parallelepiped with a square front face and side edges longer than the front ones, while the solid 202 is a perfect cube, with the front face on the projection plane and with the coordinates of the center of this face equal to x = 20 units and y = -14 units, measured in relation to the length of an edge of the cube. In the same unit of measurement, the point corresponding to the center of projection is located 25 units from the image plane, on the vertical to the plane of the figure passing through point Ic.

If we observe fig. 2 from a point close to the one corresponding to the center of projection, which is on the straight line for Ic orthogonal to the plane of the figure at a distance from the sheet (the plane of the image) equal to 25 times the edge of the cube , solid 202 appears correctly with the shape of a cube instead of the deformed appearance of a parallelepiped.

Fig. 3 qualitatively explains the reason for the deformation. It represents the horizon plane of a perspective, that is, the xz plane of fig. 1, on which an ABDE square is drawn with the front side adjacent to the projection plane. In fig. 3 element 301 represents the projection plane 101 of fig. 1, seen from the direction of the y axis.

Here and in the following it is assumed, for simplicity, that the projection plane and the image plane coincide, although in reality the image is normally reproduced on a support distinct from the projection plane. In the event that the two floors are distinct, they can be brought back to each other, with all their respective geometric elements, by means of a homothety, as the art technician knows how to do.

In fig. 3 the projections of the vertices B and D from the projection center C coincide with the vertices B and D themselves, while the projection of the vertex A is represented by the point Q. Assuming now to observe the projection plane from the point V, we note that, by attributing to the point Q the representation of a vertex which is on the straight line passing through AB, the observer gives to the point Q the representation of the point A ', so that the square ABDE from the point V appears as if it were the rectangle A'BDE'.

This type of deformation is generally more evident in the viewing of stereoscopic images, as is known to art experts and as will be seen later.

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The linear perspective, therefore, reproduces the vision of reality well if you look at the images from the point corresponding to the center of projection, while if you move away from it, the images appear deformed.

The aforementioned deformations are greater the greater the deviation of the observation point of the image from the position corresponding to the center of projection. Since the images are normally observed from a point not far from the line corresponding to the optical axis of the perspective, in practice the main component that determines the deformation derives from the distance of the observer's point of observation from the point corresponding to the center of projection. Instead of this distance, in practice it is often used to consider the angle of view (the angle that subtends the diagonal of the projection plane from the projection center), considering that between the angle of view and the distance of the projection center from the projection holds the following relationship:

d

ϑ = 2 ⋅ arctan

2 ⋅ f (1)

where the symbols have the following meanings:

ϑ angle of view;

diagonal of the projection plane (diagonal of the photosensitive element, in the case of a camera);

fdistance between projection center and projection plane (focal length, in the case of a camera).

By applying formula (1), it is found that the angle of view corresponding to the point of the image in fig. 2 furthest from Ic corresponds to a field angle of 90 °. The deformations illustrated by means of FIG. 2 and fig. 3 were already noticed by Leonardo da Vinci. In architecture and painting, perspective was, in fact, introduced in the fifteenth century by Brunelleschi with contributions from Donatello and Masaccio. Leonardo, to prevent such deformations from appearing, recommended painting from a distance at least twenty times greater than the size of the object to be portrayed, corresponding to field angles of about 3 ° (see "Leonardo da Vinci on Painting - A SVT054

lost book (Book A) - by Carlo Pedretti - Foreword by Sir Kenneth Clark, University of California Press, Berkely and Los Angeles, California, © 1964 by the Recent of the University of California, Library of Congress Catalog Card Number: 64-17171 ).

In the eighteenth century Giovanni Paolo Pannini, to reduce the deformations of the linear perspective already noted by Leonardo, devised a method of projection consisting in the succession of two projections. With the first projection, the scene is projected from a first projection center onto a vertical cylindrical surface with the axis passing through the projection center. With the second, from a second projection center different from the first, the image obtained on the cylindrical surface is projected onto a flat surface. This technique gives good results in the representation of scenes in which there are vertical lines that must remain vertical and in which there is a central vanishing point towards which lines of many elements converge. On the other hand, it does not give good results in representing generic images. Using this technique, Pannini painted suggestive views of architectural complexes, with wide angles of view without visible perspective deformations (see Thomas K. Sharpless1, Bruno Postle, and Daniel M. German, "Pannini: A New Projection for Rendering Wide Angle Perspective Images" , Computational Aesthetics in Graphics, Visualization, and Imaging (2010), The Eurographics Association 2010, http://vedutismo.net/Pannini/panini.pdf).

In 1995 Denis Zorin, Alan H. Barr (see Zorin D., Barr A. H., "Correction of geometric perceptual distortions in pictures", SIGGRAPH '95: Proceedings of the 22nd annual conference on Computer graphics and interactive techniques (1995), pp . 257-264) proposed to circumvent the occurrence of the aforementioned deformations by first projecting the images onto a spherical surface, centered on the projection center, and then returning this spherical surface to a plane. But by reproducing a spherical surface in a plane distortions inevitably arise, so the Zorin-Barr method can give some improvement only for narrow field angles.

More recently Robert Carroll et al. (Carroll R., Agrawal M., Agarwala A., "Optimizing content-preserving projections for wide-angle images", SIGGRAPH '09: ACM SIGGRAPH 2009 papers (New York, NY, USA, 2009) , ACM, pp. 1–9) proposed to minimize the deformations by adapting the projection to the content. For this purpose, a man-machine interface is provided through which the user can characterize the areas and elements of the images to be corrected in particular ways, SVT054

such as straight lines that must remain so, people's faces, etc. However, this method is laborious and inconvenient, and requires specific calibrations for various types of elements.

Finally, we cite the American patent application US 2011/0090303 A1 in the name of Apple Inc., which, with reference to videoconferencing systems, describes a method for correcting the deformations of images shot by cameras. This method is based on the identification of particular elements contained in the images taken (for example, the face of participants in a teleconference), and on the application of specific corrections to the deformations of these elements. The identification of these elements can take place through recognition algorithms and through indications provided by the user through appropriate man-machine interfaces. The user can, for example, indicate the central point of a face. The correction of deformations and distortions is then implemented by comparing the image taken with reference images, such as, for example, photographs of the participants taken before the start of the videoconference. Image recognition and corrections are facilitated by determining the orientation of the camera using sensors (accelerometers, gyroscopes). As we can imagine, this method is very complex and is applicable only in particular circumstances. It also does not solve the problem of deformations that are generally evident when viewing a perspective image from an observation point that does not correspond to the center of projection.

The present invention provides an appropriate solution to the above problem by describing a method, and the relative apparatus, for the correction of the deformations that appear in the images, when these are observed from a point not corresponding to the projection center of the perspective. The apparatus and its method are applied both for the two-dimensional reproduction of a single image and for the reproduction of a pair of stereoscopic images.

Such apparatus comprises appropriate means for acquiring a two-dimensional image, or a pair of stereoscopic images, with sufficient data to determine the coordinates of the point corresponding to the projection center of the perspective (for example, center of the image and focal length) and with the depth map kit. This is defined as the set of depths of the points of the scene represented in the image, that is, with reference to fig. 1, as the set of z coordinates of the points of the three-dimensional scene. As an alternative to the map of SVT054

depth data can be provided to allow it to be derived.

Furthermore, this apparatus comprises memory means and processing means (for example a processor which executes an appropriate software code) configured to correct the position of the points of the acquired images, according to a technique called "Partial Perspective Gradient" (PPG). The corrections that this technique makes tend to represent, on the plane of the images, the position of each point as if it had been taken with the lens pointed at it. This technique, of which numerous variants can be defined, is based on the calculation of a gradient of the positioning of the points on the image plane. By integrating the components of this gradient there are the functions according to which to place the points in order to achieve the aforementioned correction.

It should be noted that the representation of the images, on which one operates to apply the correction techniques, is often made up of a limited number of discrete elements of finite size, called "pixels". Strictly speaking, each pixel represents, in an approximate way, a small area of the image, but for simplicity of exposure, in some passages of this description, a geometric point will be identified with a pixel, accepting the approximation of assuming that the discrete coordinates of the pixels correspond to those of the geometric point that it is intended to represent. Considering the reference system shown in fig. 4, this gradient is calculated considering a generic point A of the three-dimensional space, to which corresponds, according to the linear perspective, the point Q of the image plane (401), and the point P resulting from the intersection between the projecting line of A and plane 402, which we will call π (or auxiliary plane π) and which, in the preferred embodiment of the invention, is orthogonal to the projecting line of A.

At an incremental shift to <�>

of A, corresponds, on the π (402) plane, one <�>

incremental shift pdi P. From the incremental shift components p <�>

the components of the gradient, i.e. the partial derivatives, of the functions Fx (xQ, yQ, zA) and Fy (xQ, yQ, zA) are calculated with which to represent, on the plane of the images 401, the correct coordinates of the image of point A .

By integrating the aforementioned partial derivatives, the coordinates are determined according to which to place the representation of point A on the plane of the images.

SVT054

These characteristics and further advantages of the present invention will become clearer from the description of an embodiment thereof shown in the attached drawings, provided purely by way of non-limiting example, in which:

fig. 1 illustrates in a geometric way the operation of the linear perspective;

fig. 2 illustrates the perspective representation of a cube in an offset position with respect to the optical axis;

fig. 3 qualitatively illustrates the deformations inherent in the linear perspective in observing images from a point that does not correspond to the center of projection;

fig. 4 geometrically illustrates an embodiment of the invention;

fig. 5 illustrates in a geometric way the functioning of the linear perspective in the stereoscopic case;

fig. 6 shows a plan view of part of fig.4;

fig. 7 illustrates a representation of the Cartesian references;

fig. 8 illustrates the trend of the raised cosine function and its complement used in the method and apparatus according to the invention;

fig. 9 shows stereoscopic images of the cube of fig. 2;

fig. 10 illustrates a block diagram of an apparatus according to the invention;

fig. 11 illustrates a flow diagram of a process in which the perspective correction of the present invention is applied.

The present invention relates to the correction of single two-dimensional images, or pairs of stereoscopic images, intended to reduce the deformations that appear in the perspective images, when these are observed from a point not corresponding to the projection center of the perspective. In the following, an apparatus is described, and the relative method that it applies, in a preferred embodiment and in some exemplary, but not limiting, variants.

The apparatus of the present invention, with its method, processes the images using a technique, called "Partial Perspective Gradient" (PPG), which corrects SVT054

position of the points (pixels) of the images so as to place them as if each point of the scene represented in the image had been taken by a lens pointed at it. To determine this correction, this technique uses the coordinates of the point, to correct the representation, and the data that define the geometry of the perspective according to which the image was generated.

With reference to fig. 4 and the preferred embodiment, the coordinates of the point A under consideration are obtained from the image, that is, from the x, y coordinates of the point Q, and from the distance that the point A has from the xy plane. In the terminology of art, this distance is called "depth" and the set of distances of the points of three-dimensional space from the xy plane is called "depth map".

The geometry of the perspective according to which the image was generated is essentially defined by the focal length and size of the painting.

With reference also to fig. 5, in the case of stereoscopic images, the inter-optic distance b is added to the focal length and the size of the picture, ie the distance between the projection centers according to which the two stereoscopic images were generated. The data sufficient to determine the depth of the points represented in the image, also called depth data, can include the depth map and can be obtained with various methods known to the person skilled in the art, both for an image intended for two-dimensional vision, and for a pair of stereoscopic images. In the case of drawings or paintings, these data are implicit in the artist's project.

In the case of a pair of stereoscopic images, the depth map can be obtained from the disparity map, which represents the difference between the horizontal coordinates of the homologous points of the two images, as illustrated in fig. 5. In this figure it is assumed that the two projection centers are aligned horizontally and that the horizontal axes of the Cartesian references lie on the plane of the horizon, or on the horizontal plane that contains the projection centers. In fact, fig. 5 clearly shows that the depth axes zLe zR originate on the straight line passing through the projection centers CLe CR, while the x and y axes are represented on the image planes to simplify the graphics; however, it is evident to the skilled in the art that this representation on the image planes is entirely equivalent to the representation of the same axes on the plane passing through the aforementioned projection centers CLe CRed orthogonal to the z axis.

SVT054

Fig. 5 represents an example in which a point A of the three-dimensional space projected by two distinct projection centers, CLe CR, on two distinct projection planes, the plane 501 and the plane 503, on which the coordinates are respectively referred to the Cartesian references IcLxLyLe IcRxRyR . The optical axes that branch off from the projection centers, zLda CL, and zRda CR, are parallel to each other and at a distance b (inter-optic distance) from each other. The planes 501 and 503 are orthogonal to these optical axes and equidistant from their respective projection centers by a distance equal to f, while their horizontal axes, xLe xR respectively, lie on the same line, which is parallel to the joining the two centers of projection CLe CR. and intersect the optical axes respectively at the points IcL and IcR.

In the respective reference systems, the homologous points QLe QR, resulting from the projection of A respectively on the xLyLe xRyR planes, have the same vertical coordinate, not indicated in fig. 5 for simplicity, and have the horizontal coordinates xQL and xQR respectively.

The disparity "disp" of the homologous points QLe QR, is given by disp = xQL- xQR.

The following mathematical relationship holds between "depth" and "disparity":

<f ⋅ b>

zA <= (2)> avail

where the symbols have the following meanings:

z A “depth” of the point considered, ie the z coordinate of point A in fig.4 or the coordinates of point A in fig.5 on the zLe zR axes;

binteroptic distance, i.e. the distance between the two projection centers CLe CR; ffocal length, i.e. the distance between the projection center CLo CRe respective projection plane 501 or 503;

disparity, i.e. the difference between the horizontal coordinate of a point in the left image (QL) and the corresponding coordinate of the homologous point in the right image (QR), referring the coordinates to the centers of the respective images (IcLe IcR) or to other homologous points of reference.

Equation (2) is defined with reference to the projection centers and the plane of SVT054

projection, but, as known to the skilled in the art, the same equation, mutatis mutandis, can also be used to relate the depth of a point represented on the two images of a stereoscopy with the disparity measured on the pair of stereoscopic images and with the value of the equivalent inter-optic distance relative to the geometric configuration according to which the stereoscopy was generated. The projection center of the perspective according to which an image was generated can be located by providing the center of the projection plane and the distance between the projection center and the projection plane (the focal distance, in the case of cameras), or by providing the dimensions of the projection plane and the angle of view, since the angle of view, the diagonal of the image and the distance of the projection center from the projection plane are related by equation (1).

The apparatus claimed by the present invention comprises appropriate means for acquiring, in numerical form, a two-dimensional image or a pair of stereoscopic images, accompanied by depth or disparity data and by data sufficient to determine the coordinates of the corresponding projection center C in the projection center of the perspective, as mentioned above.

Once the geometry of the image has been acquired, the apparatus claimed by the present invention applies the method ("Partial Perspective Gradient") according to the invention, where said method is developed on the basis of the principle of representing the surroundings of each point of an image as if, in the shot or in the drawing, the gaze was turned towards it. This principle is realized by calculating how, in such conditions of gaze orientation, small displacements of the point considered would be perceived (for example point A in the attached figures).

With reference also to fig. 6, this calculation is described, for the sake of simplicity, considering the representation of a point in space lying on the horizon plane (xz plane in the attached figures) and an incremental displacement of this point in a direction parallel to the x axis. The person skilled in the art is able to extend the calculation to the case in which point A is located in any position in space and the incremental displacement of said point occurs along any direction.

The plane 402 is as distant from the projection center C as the projection plane 401. This configuration is to be considered only as an illustrative, non-limiting example of the preferred embodiment. As the art expert understands, and as will be said in SVT054

then, the plane 402 can be fixed at any distance from the projection center and with any orientation.

<�>

As the skilled in the art knows, in general an incremental shift∆ a is defined by a composition of a plurality of components, where these components, preferably three (three-dimensional space), are oriented along various directions. The most common decomposition is that made using the three directions corresponding to the axes of the Cartesian reference, to which reference is made in this description. Since in this illustrative example a <�>

has a direction parallel to the x axis, ∆ a <�>

it is completely characterized by its component along this axis, which we will call x�

A. Similarly, in this example, the incremental displacement ∆q of Q is completely characterized by its component along the x axis, which we will call �

∆xQ. Note that, in any case, a ∆aparallel displacement to the x axis of point A <�>

corresponds to a displacement q, also parallel to the x axis, of the point Q.

To deal with the incremental displacement that an observer would perceive if his <�> gaze were directed towards point A, it is necessary to represent a projection p

<�>

of the incremental movement∆ on plane 402.

<�>

With reference also to fig. 7, this projection∆ can be defined using the Cartesian reference Cαβγ. This reference has its origin in C, coinciding with that of the reference Cxyz, and its γ axis develops along a direction coinciding with that of the straight line passing through C and A. The αβ plane is orthogonal to the γ axis. The α axis is identified by the intersection of the αβ plane with the xz plane (the α axis lies on the xz plane), while the β axis, also passing through C, is orthogonal to both the α axis and the γ axis . In general, a shift a <�> parallel to the x axis of point A results in a shift p <�> on the 402 plane

of P with components both in the direction of the α axis and in the direction of the β axis. In the particular case of fig. 6, where the point A is on the plane of the horizon and, consequently, the β axis is orthogonal to the x axis, the component in the direction of the β axis is zero, therefore only the component in the direction remains to be treated of the α axis, called α P. Since con∆α Psi wants to determine the partial derivative that the component along the x axis of the point Q, also called xQ, must take, ∆α P is related to x Q. The calculation can be done with the common SVT054 rules

of geometry and mathematics, which the art expert knows. They consist

essentially in the change of the reference system that makes the <�> representation of the displacement pass from the Cartesian reference Cxyz to the reference

Cartesian Cαβγ. The formulas for changing the reference system can be found in

textbooks and on various Internet sites, including, for example, the following:

http://www.cns.gatech.edu/~predrag/courses/PHYS-4421-10/Lautrup/space.pdf.

The result of the processing is the expression of the formula (3αx), shown below together

with the expressions of the other components of the displacements, which are derived in mode

similar.

<∆α> 2

P f

=

∆ x <Q> (x <2 2> (3αx)

Q + f <2>) ⋅ (x <2>

Q + y Q f <2>)

∆ <α> P = 0

∆y (3αy)

Q

<∆α> −f 2

P ⋅ x

= Q

∆ z (3αz) A zA⋅ (x <2 2>

Q + f) ⋅ (x <2>

Q + y <2>

Q f <2>)

<∆β> P f⋅x

= Q ⋅ y Q

∆ x <2> (3βx) Q (x <2>

Q + y <2>

Q + f <2>) ⋅ x <2>

Q f

<∆> 2

<β> P f⋅ x2

Q f

=

∆ y 2 2 2 (3βy) Q xQ + yQ + f

<∆β> P −f 3 ⋅ y

= Q

∆ z A zA⋅ (x <2>

Q + y <2>

Q + f <2>) ⋅ x <2>

Q f <2> (3βz)

In formulas (3 ..) the symbols have the following meanings:

incremental displacement of point P on plane 402 in the direction

∆αP

of the α axis;

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incremental displacement of point P on plane 402 in the direction

∆βP

of the β axis;

incremental displacement of the Q point on the projection plane in the direction ∆xQ

of the x axis;

incremental displacement of the Q point on the projection plane in the ∆yQ direction

of the y axis;

<∆z> Incremental shift of point A in the direction of the z axis;

fdistance between projection center and projection plane;

<x> Q abscissa of the point Q;

<y> Q ordinate of point Q;

The expression (3αy) indicates that the incremental displacement α P of the point P in the direction of the α axis does not depend on the displacement component <∆y> Q of the point Q in the direction of the y axis, since the y axis is orthogonal to the plane on which the α axis lies, as mentioned above in the comment of fig.7 (the α axis lies on the xz plane).

By calculating the integrals of the formulas (3 ..) we obtain the coordinates Fx (xQ, yQ, zA) and Fy (xQ, yQ, zA) with which to represent, on the plane of the images, the correct position of point A. These integrals are given by the following formulas:

x ∆ <α>

Fx <(> x <Q,> y <Q,> z <A) => ∫QP z <α>

0A∆P

∆x (4xx) Q ∫ f =

∆z A

x 2 ⋅ dt

<=> ∫Qf

0 (4αx) (t2 + f 2) ⋅ (t2 + y2

Q + f 2)

- f 2 ⋅ x

<Q> � zA� (4αz) �

(x2 2

Q f 2) ⋅ (x2

Q y Q f 2 ln

) ��

� f <�>

� (*) SVT054

xQyQ∆ <β>

FyPz <β> P <(> x <Q,> y <Q,> z <A) => ∆ <β>

∫ P

0

∆x 0 f = (4yy) Q ∫ ∆x Q ∫A∆

∆z A

� �

f � arctg� x2

Q + f 2� � ��

= ⋅ � −arctg� <f> ��

� � y�� � (4βx) <(*)> � � Q � � y Q��

�

� y

f ⋅ arctg� <Q> �

�

� x <2> + f <2> + � (4βy)

� Q �

−f 3 ⋅ y

<Q> � z (4βz)

⋅ �A�

<�> �

(x2

Q + y2

Q + f2) ⋅ x2

Q f 2 ln�

� f � (*) <(*)> For yQtending to zero the expression (4βx) tends to zero, while the arguments of the logarithms of the expressions (4αz) and (4βz) are always positive.

The symbols of the formulas (4 ..) have the same meanings as those of the formulas (3 ..). The expression (4αx) can be approximated by the following formula:

x <Q f> 2 ⋅ <dt> 2 ⋅ f 2 � 2 ⋅

∫ ≈ ⋅ arctg� x Q �

�

0 (t2 + f2) ⋅ (t2 + y2 2) ⋅2 + 2� <�> (4αxa)

Qf 2 f y Q � 2⋅f2 + y2

Q�

having approximated the integrand function (3αx) using the following expression:

<∆α> f 2 2 ⋅ f 2

P =

∆ x <2>

Q (x2 ≈

Q + f2) ⋅ (x2

Q + y2

Q + f 2) 2⋅xQ + y <2>

Q + 2 ⋅ f <2> (4αxb)

The formula (4αxa) gives results that differ little from the integral calculus (4αx). For example, in the case of application to the cube shown in fig. 2, where there are points corresponding to field angles of 90 °, the calculations performed with the formula (4αxa) differ less than 1% from those performed with the formula (4αx). As you can see, SVT054

the use of the formula (4αxa) does not require, unlike that of the formula (4αx), the execution of numerical integration steps, so as to advantageously reduce processing times and load.

Formulas (4 ..) constitute the first embodiment of the "Partial Perspective Gradient" technique of the present invention, which consists in representing, in approximate terms on the plane of the images, what, around each point of the scene to be reproduced , it would appear from the center of the perspective if the optical axis passed through that point. In accordance with this criterion, a certain number of variants of the above technique can be considered.

For example, instead of the components (3αz) and (4βz), which represent the dependence on the "depth" of the image points, we can use formulas that represent the dependence on the "disparity" between the homologous points of stereoscopic images, consistent with formula (2).

A second variant is the use of formulas determined by assuming that the point P (fig. 4, fig. 6, fig. 7), instead of being at a fixed distance equal to f from the projection center C, has a distance CP from C dependent on some parameter. In particular, we can impose CP = CQ, so that P coincides with Q.

A third variant consists in determining the formulas by imposing that the point P is on a segment VA, instead of on the segment CA, with V distinct from C (fig. 1, fig. 3, fig.

4). In this case, the V point can preferably be found on the optical axis that passes through the C and IC points, or outside this optical axis. Furthermore, the distance of P from VPV can be prefixed to a constant value or be variable depending on some parameter. For example, PV = f ⋅VA / CA can be imposed.

The aforementioned variants of the "Partial Perspective Gradient" technique are shown as non-exhaustive or limiting examples, being the art expert able to imagine many others without departing from the teachings of the present invention.

A second embodiment of the idea consists in applying in part the formulas (4 ..) and in part the formulas of the linear perspective.

The known art has, in fact, highlighted that the linear perspective reproduces images well within certain limits, therefore within these limits it may be convenient to maintain the reproduction provided by the linear perspective, combining the formulas (4 ..) with those SVT054

of linear perspective. Typically, the combination can take place in such a way as to gradually pass from the exclusive application of the formulas (4 ..) to the exclusive application of the linear perspective formulas, but it can also be done in other ways that the art expert can imagine.

One way to realize the above combination is to multiply the results of the formulas (4 ..) by a first factor, preferably between the unit value and the null value, and multiply the results of the linear perspective formulas by a second factor, preferably complementary to the first factor; the results of the products thus obtained are then added together.

With reference also to fig. 8, an example of a function that lends itself to realizing multiplicative factors is the raised cosine function represented by curve 801, together with its complement 802. As can be seen from the figure, curve 801 is maintained at the unitary value for values of the abscissas between zero and a limit ts; subsequently, in the interval from tsa to tf, it gradually decreases to zero, and then remains at zero value. Its complementary function 802, on the other hand, has the opposite trend.

The abscissa of fig. 8, and therefore the limits tsa tf, can be related to the offset angle with which the point to be represented is seen from the projection center, or with the distance of the point from the projection center, or with other parameters or combinations of parameters which allow the person skilled in the art to satisfy the requirements of the specific application of the method according to the invention.

The simplest way to combine the results of formulas 4 (..) with the results of linear perspective is to:

a) multiply the function expressed by the formula (4xx) by a first function 801;

b) multiplying the formula which provides the abscissas of the points of the images according to the linear perspective by a second function 802, which is preferably complementary to said first function 801;

c) derive the function that expresses the abscissa of the points in the image as the sum of the results of operations a) and b);

SVT054

d) derive the function that expresses the ordinate of the image points in a similar way to what was done for the function that expresses the abscissa according to operations a) -c).

However, the art expert is able to achieve the combination in other ways as well. For example, you can:

e) multiplying each of the terms (4α.) and (4β.) by a distinct function;

f) obtain the function Fx (xQ, yQ, zA) as the sum of the terms obtained from the above multiplications;

g) calculate a weighted average of the aforesaid terms with which the function Fx (xQ, yQ, zA) in the previous point f) was calculated;

h) multiply the formula that provides the abscissas of the points of the images according to the linear perspective by the complement of the aforementioned weighted average obtained from the calculation g);

i) derive the function that expresses the abscissa of the points in the image as the sum of the results of operations f) and h);

j) derive the function that expresses the ordinate of the points of the image in a similar way to what was done for the function that expresses the abscissa with operations e) -i).

These two examples of combinations are neither exhaustive nor limiting. The art expert is, in fact, able to propose others without departing from the teaching of the present invention. The functions represented in fig. 8 are to be considered only as non-exhaustive or limiting examples. The art expert knows, in fact, many other types of functions that can be used to combine the terms of the formulas (4 ..) with those of the linear perspective.

With the aforementioned second embodiment of the invention there are then all the variants that can be considered for the first embodiment.

With reference also to fig. 9,

aF is the enlarged image of cube 202 (cube projected from the center of SVT054

projection Ic);

aE is the image of the same cube 202 projected from a projection center with abscissa equal to twice the side of the cube;

bF is the image obtained by processing the AF image with the "Partial Perspective Gradient" technique;

bE is the image obtained by processing the aE image with the "Partial Perspective Gradient" technique.

By applying the method according to the invention (PPG) described above to the image of the cube 202 of fig. 2, which is shown enlarged in the image "aF" of fig. 9, you get the "bF" image of fig. 9. Comparing the aforementioned “aF” and “bF” images, the improvement brought about by the present invention on two-dimensional images is evident. It should also be noted that the image "bF" shows a cube that has no face perpendicular to the optical axis, as it would be in reality if the gaze were directed towards it, since this cube is seen with a horizontal offset of 21 ° and a vertical offset of -15 °.

The improvement is even more evident in the stereoscopic vision, which can be obtained by observing fig. 9 with the measures indicated below, assuming that the images “aF” and “bF” in fig. 9 are the images for the left eye, while their counterparts "aE" and "bE" are the images for the right eye. The latter were generated by a projection center distant from the projection center relative to "aF" and "bF" by an interoptic distance ("b" in fig. 5) equal to twice the side of the cube.

To obtain the stereoscopic vision of the images in fig. 9, the figure must be reproduced so that the distance between the vertical lines hanging from the upper horizontal line is approximately equal to, or slightly less than, one's own interpupillary distance. An adequate size is normally obtained by reproducing the sheet on which fig. 9 in A4 format (21 cm wide). Ensuring that the line joining the centers of one's pupils is parallel to the lines that delimit the figure above and below, it is then necessary to fix the gaze on the figure, in order to obtain the fusion of the images on the right with those on the left. The fusion could be facilitated by initially fixing the arrows that go from the lower to the upper images or, better, by placing an SVT054 near one's forehead.

cardboard in an orthogonal position with respect to the aforementioned horizontal lines of delimitation, so that the right eye does not see, at least in large part, the left image, and vice versa.

While in the stereoscopic view of fig. 9, the image "b" (obtained from the fusion of the images bF and bE) represents the shapes of a cube well, the image "a" does not even look like that of a parallelepiped, because the dimensions of the rear face appear larger than those of the front face. Furthermore, in the linear perspective the horizontal and vertical lines maintain their directions, in particular the segments that lie on the projection plane, such as the edges of the front face of the cube reproduced in fig. 9, maintain their length and orientation, so that the front face of the cube appears as a perfect square. The result of processing using the PPG method instead shows a cube seen obliquely, consistent with the fact that it is offset from the optical axis by 21 ° horizontally and -15 ° vertically.

The PPG technique can therefore be applied to stereoscopic images even more advantageously than to monoscopic images.

Even in the case of stereoscopic vision, however, the images of linear perspective correctly reproduce the shape of a cube if they are observed from the observation point corresponding to the center of projection. In the case of the aF and aE images (with the sheet on which fig. 9 is reproduced in A4 format), the point corresponding to the center of projection is located approximately 48 cm to the right and 34 cm above the center of the image. “AF”, and 60 cm away from the plane of the figure.

Typically, the PPG method can be applied to each of the images of the stereoscopic copy, with the possible variants mentioned above, but in the case of stereoscopy there are additional variants and tricks.

Referring to fig. 5, it is assumed to apply the same embodiment, with the same possible variants, to both images of the couple, projecting the scene from the projection centers CLe CR. For simplicity of exposition, it is assumed that the two optical axes zLe zR are parallel, but the art expert knows how to treat stereoscopic images produced with non-parallel optical axes, and in particular convergent, as are those of the eyes in real vision.

The main alternatives dealing with stereoscopic images, and which are added to the SVT054

embodiments described above are the following:

1. keep the projection centers CLe CR (fig. 5) in their real position; in this case also the inter-optic distance remains that of the real position, but the distances of each point of the scene from the two distinct projection centers are different and can lead to an annoying difference in vertical correction between the two images, as indicated by the formulas (4βx) and (4βz); to avoid this difference, in the above two formulas instead of the abscissas xQL and xQR (fig. 5), which should take the place of xQ for the left image and the right image respectively, a common value can be used, for example their average value xQm, given by:

xQL x

x QR

Qm = (5)

2

By applying the formula (4yy), i.e. the formulas (4βx) and (4βz), with xQmal instead of xQ, a negligible error is made on the value of the vertical coordinate, because in practice the square of the difference between xQLo xQR and their mean value xQm is much less than f <2>, to which this square is always added in the formulas (4β.);

2. rotate the segment CLCR, together with the same projection centers CLe CR, around its midpoint C, keeping it in the plane to which the axes zLe zR belong, so that it assumes a direction orthogonal to the projecting line joining C with A; in this case the distances of each point of the scene are equalized from the two projection centers, but the projecting lines no longer correspond to the original ones and the inter-optic distance increases;

3. rotate the CLCR segment as in the previous point, moving the projection centers CL and CR towards the center of the CLCR segment in order to keep the projecting lines unchanged; this alternative better approximates the criterion of representing the surroundings of each point of the scene as if, in the shot or in the drawing, the gaze was turned towards it.

These three alternatives are to be considered as non-exhaustive or limiting examples of the ways in which the reference geometry for the method of SVT054 can be set in practice

calculation of the present invention. The art expert is able to propose other equally applicable ones. In the example shown in fig. 9 the first of the three described above was applied.

Fig. 10 shows a simplified block diagram of an apparatus which applies the correction method according to the present invention. An image processing apparatus 1 according to the invention, such as for example a camera or a video camera or other, comprises image acquisition means 1001, input / output means 1002, a central processing unit (CPU) 1003 , a read-only memory 1004, a random access memory (RAM) 1005 and means for producing processed images 1006. All these elements of the apparatus 1 are in signal communication, so as to allow the mutual exchange of data between any pair of such elements. The image acquisition means 1001 are capable of acquiring both two-dimensional images and pairs of stereoscopic images. If the images are accompanied by the respective depth map and by data that allow to trace the geometry with which they were generated (for example the focal length, the angle of view, the inclination of the sensor or other), the Image Acquisition 1001 also acquire this data. Otherwise some data can be set through a user interface (not shown in the figures) which is in signal communication with the input / output means 1002, while, in the case of stereoscopic images, the depth map can also be produced by the apparatus 1 itself, as will be discussed later. The user interface also allows you to set the options and variants, with the related parameters, that the user prefers to use in the contingent circumstance. For example, you can choose the following settings:

- the type of combination of the PPG method with linear perspective, assigning the parameters that characterize it (tse tf in the case of the function illustrated in fig.

8);

- the alternative between the use of the formula (4αxa) and the use of the formula (4αx);

- if the formula (4αx) is used, define the precision to be achieved in numerical integration;

- the position of the point of view with respect to which the calculations of the PPG method are applied;

SVT054

- in the case in which the point of view with respect to which the processing of the PPG method is applied has a variable position, the parameters that characterize its variability;

- in the case of stereoscopy, the alternative to be taken on the positioning of the two projection centers.

The point of view with respect to which the elaborations of the method of the invention are applied may be different from the projection center C (see fig. 1) with respect to which the image to be processed was generated. Given the coordinates of Q and the depth of A and calculated with them the position of A in three-dimensional space, it is possible to determine the projection of A on any plane and from any point of view. Processing the image from a point of view other than the projection center C allows you to make special corrections to the image and to produce useful artificial images, such as those that can be used in stereoscopy to fill gaps, which will be discussed later.

The central processing unit (CPU - Central Processing Unit) 1003 is the part of the apparatus that executes the calculation algorithms, including complementary operations which, after the application of the PPG technique, are used to complete the correction of the images to be returned. . We will discuss these operations in commenting on fig. 11.

As is known to the person skilled in the art, the central processing unit 1003 could actually comprise specially developed integrated circuits, one or more microprocessors, programmable logic circuits (eg CPLD, FPGA) and the like. These and other realization possibilities do not anticipate or make obvious the teachings of the present invention.

The read-only memory 1004 is preferably used to permanently store some instructions for managing the apparatus and the instructions that implement the calculation algorithms, while the random access memory (RAM) 1005 typically serves to temporarily store the images and results. intermediates of the elaborations.

Finally, the means for the production of processed images 1006 provide for the return of the processed images, for example by transferring them from the RAM memory 1005 to the input / output means 1002, so that said means 1002 can provide for the saving of said processed images in a permanent memory (for example an SVT054

hard-disk or a Flash-type memory such as a Secure Digital, MMC or other), to their display on one or more screens or other display means (not illustrated in the attached figures), to their printing, and more.

With reference to fig. 11, the process of correcting a two-dimensional image or a pair of stereoscopic images will now be described. For simplicity, in FIG. 11 no distinction is made between monoscopic images and stereoscopic images, since, having seen the previous description, it will be clear to those skilled in the art how the method according to the invention is applied to both cases. In the case of stereoscopic images, processing can take place separately for each image, in subsequent processing, or in parallel on the two images. There are no substantial differences between these two alternatives, considering that the method of the present invention is applied in each case separately to the individual images.

The process of carrying out the invention can include the following steps:

- start-up phase 1101, during which the apparatus 1 is configured to process at least one image;

- setting phase 1102, during which the settings and data that the user intends to provide manually for the processing of at least said image are acquired;

- image acquisition phase 1103, during which the image to be processed is acquired by means of the image acquisition means 1001 and preferably transferred into the RAM memory 1005 (for simplicity it is assumed that, downstream of phase 1103, the images are in numerical form and, if they are available in another form, the person skilled in the art knows how to convert them into numerical form); it is intended that the image acquired in step 1103 is accompanied by the data that define the geometry according to which the image was generated and that the depth map is possibly also provided with it, or data is provided that allow it to be determined; in the subsequent steps 1104 and 1105 the case is considered in which the depth map is not provided in this step 1103;

- phase of checking the presence of the depth map 1104, during which it is determined whether or not the depth map is available;

- depth map calculation phase 1105, during which the apparatus 1 generates the SVT054

depth map according to one of the methods known to those skilled in the art;

- processing step 1106, during which the apparatus 1, using the depth map, acquired in step 1103 or calculated during step 1105, determines, in three-dimensional space, the position of the points corresponding to the pixels of the image acquired during step 1103 and applies to them the position correction algorithm, according to the invention; the result of step 1106 is a matrix that indicates the position that the pixels of the image acquired in step 1103 must assume following the aforementioned processing; in this phase there are no pixel displacements, to avoid having to repeat this type of operation after the further processing envisaged in the subsequent phases;

- resizing phase 1107, during which the apparatus 1 can enlarge or reduce the image processed in phase 1106; resizing consists in recalculating, starting from the result of step 1106, the position that the pixels of the processed image must assume following the resizing that is applied in this step 1107;

- phase of elimination of overlaps, during which, as will be explained later, the conflicts between pixels that are overlapped are resolved;

- phase of displacement of the pixels and filling of gaps 1109, during which the displacement of the pixels is carried out and, as will be explained later, the gaps are filled; - phase of restitution of the processed image 1110, during which the image is preferably stored in an area of the RAM memory 1005, or other memory, and made available for viewing, printing, sending to other devices, and other operations;

- final phase 1111, which denotes the end of the process.

When the apparatus 1 is in an operating condition, said apparatus 1 enters, after the start-up phase 1101, in the setting phase 1102 and then enters, simultaneously or subsequently, in the image acquisition phase 1103. At the end of the phase 1103, the apparatus 1 checks the availability of a depth map (step 1104) for what has been acquired during the step 1103; if the depth map is available, the apparatus 1 enters the processing phase 1106, if the map is not present, the apparatus enters the depth map calculation phase 1105 before SVT054

proceed with the processing step 1106. Subsequently to step 1106, the apparatus can optionally perform the resizing step 1107. After that, the apparatus 1 proceeds with the step of eliminating the overlaps 1108, followed by the phase of moving the pixels and filling lacune 1109 and then from the phase of restitution of the elaborated image 1110. At the end of the process we arrive at the final phase 1111.

In step 1106 the PPG technique of the present invention is applied, in one of the embodiments described above.

In summary, the processing means 1003 and memory 1004 and 1005 of the apparatus 1 are configured to correct the image represented on the plane 401, which has been generated according to the rules of linear perspective with respect to the projection center C and comprises at least a first point Q, where said first point Q is the result of the perspective projection through the projection center C of the second point A of a region of three-dimensional space; to do this the processing means 1003 and memory 1004 and 1005 carry out the method according to the invention which comprises the following steps:

a) calculate the position of the second point A in three-dimensional space;

<�>

b) calculate, on the basis of a first incremental displacement of the second point A in three-dimensional space, a second displacement <�>

incremental π of a third point P, where said third point P is the result of the perspective projection on an auxiliary plane π of said second point A through the point of view V, and where said auxiliary plane π is different from the plane on which the 'image 401; c) obtaining, on the basis of said second incremental shift p <�> calculated during step b), a gradient of a new position in which to represent said second point A on the plane of image 401; d) obtain, by calculating an integral of at least one of the components of said gradient obtained during step c), a new position in which to represent the second point A on the plane of image 401;

e) move, in the plane of image 401, the first point Q from the SVT054 position

resulting from the perspective projection with respect to the projection center C to said new position obtained during step d).

It should be noted that the corrections made by the PPG method cause pixel displacements which typically bring the parts of the image that are significantly offset from the optical axis closer to the center of the picture, resulting in the possibility that gaps and pixel overlaps are created.

In fact, when as a result of perspective corrections or resizing, an area of the image moves, some pixels of this area may overlap the pixels of areas that do not move, or that move less, or that move in different directions. To resolve these conflicts it is possible, for example, to make the pixel having a smaller depth occupy a disputed position (the pixel that is at a shorter distance from the xy plane covers the vision of the farthest one).

The person skilled in the art can resolve the conflicts between pixels using techniques different from the one described above, without however departing from the teachings of the present invention.

Just as a correction of the position of a pixel can create an overlap, the same or another correction, it can generate a gap, if the pixel that leaves one position to occupy another is not replaced by one that moves into the position vacated. Even the filling of gaps is a problem for which the skilled in the art knows solutions (eg interpolations according to one of the "inpainting" methods), but, in the case of stereoscopy, the method of the present invention can be advantageously used for a new solution. With stereoscopy there are, in fact, two images corresponding to the same scene seen from two different points of view. By processing the two images with the PPG method, during phase 1109, the corrective shift of a point in the left image is generally different from the corrective shift of the homologous point in the right image. Furthermore, the contents of the two images are different (think for example that some areas of the scene are covered by objects in front of them and that the areas that are invisible from one of the two points of view may be visible from the other). Therefore, the gaps in each of the two processed images can be filled, at least in part, with an appropriate processing of the other. For example, one way to fill in the gaps in the left (right) image is to superimpose this on an "artificial" image, obtained by processing SVT054

the right (left) image assuming the same projection center as the left (right) image as the projection center. After treating the gaps with this technique, if there are still empty pixel positions, known art methods (eg. "Inpainting" methods) can be applied to complete the fills starting from known pixels of the image.

It is advantageous that the pixel displacement and the filling of the gaps, which take place during the phase 1109, are carried out on the images already resized and after the resolution of the conflicts, i.e. downstream of the phase 1108, since the resizing can themselves generate overlaps and gaps .

In the common circumstances in which the images reproduced on screens or printed matter are observed from points not corresponding to the projection center with respect to which they were generated, the application of the PPG method, described in the present invention, improves the vision of the images of the objects that they are offset from the direction the shooting lens is pointing. These improvements are evident in monoscopic vision and even more so in stereoscopic vision.

In the absence of the PPG method of the present invention, in order to contain the characteristic deformations of linear perspective within acceptable limits, in painting, photography and cinematography an attempt is made to confine the field angles of the shots within stringent values. However, this leads to undesirable conditions, since there is not always the possibility of shooting scenes with telescopic lenses that have a reduced angle of view. The PPG technique, applied in real time during shooting or at a later stage on the acquired images, allows you to shoot scenes with wider angles of view than the currently recommended limits, with considerable advantages both for stereoscopic and monoscopic shooting.

The approximation of integrals whose primitive does not exist with expressions in closed form, such as the approximation of the integral 4αx with the formula 4αxa, substantially reduces the processing load necessary to apply the PPG method, producing considerable advantages from the point of view of 'applicability of the method, so as to make it particularly interesting for applications where it is necessary to process images in real time (for example live TV or other).

Thanks to numerous variants that can be applied in full consistency with the novelty principles inherent in the inventive idea of the present invention, the PPG SVT054 technique

it is very versatile and can be optimized for different types of applications and devices, characterized by very different processing capacities.

For example, the PPG technique can be used to correct the deformations present in video streams, applying it to each image that composes them. This applies to both 2D and 3D video streams, in which case the technique must be applied to each image of the stereoscopic pairs forming the 3D video stream.

Some of the possible variants have been described above, but it is clear to those skilled in the art that, in the practical implementation, there are also other embodiments, with different elements that can be replaced by other technically equivalent ones. The present invention is therefore not limited to the illustrative examples described, but is susceptible to various modifications, improvements, replacements of parts and equivalent elements without involving deviations from the basic inventive idea, as specified in the following claims.

Claims (28)

SVT054 CLAIMS 1. Apparatus (1) comprising - image acquisition means (1001) suitable for acquiring an image (401) residing on a given projection plane, where said image (401) has been generated according to the rules of linear perspective by projecting a region of the three-dimensional space on said plane of projection through a projection center (C), and where said image (401) comprises at least a first point (Q) which represents the projection, according to said linear perspective, of a second point (A) included in said three-dimensional region, - means for acquiring a set of data sufficient to determine the position of said projection center (C) with respect to said projection plane and the depth of said second point (A) in three-dimensional space, - memory means (1004,1005) , - processing means (1003), characterized by the fact that said processing (1003) and memory (1004,1005) means are configured to correct the image (401), so as to reduce the deformations of the representation of said three-dimensional region, which appear when said image (401) is observed by an observation point not corresponding to said projection center (C), by carrying out the steps of a) calculate the position in three-dimensional space of said second point (A) based on the position of the first point (Q), the position of the projection center (C) and the depth of said second point (A), <�> b) calculate, on the basis of a first incremental displacement (∆ a) of said second point (A) in three-dimensional space, a second incremental displacement (∆ p <�> ) of a third point (P), where said third point (P) is the result of the perspective projection on an auxiliary plane (402) of said second point (A) through a point of view (V), and where said auxiliary plane (402) is different from the plane on which the image (401) lies, c) derive, on the basis of said second incremental shift (∆ p <�> ) calculated during phase b), a gradient of a new position in which to represent said second point (A) on said projection plane, SVT054 d) obtain, by calculating an integral of at least one of the components of said gradient obtained during step c), a new position in which to represent the second point (A) on said projection plane, e) move, in the image (401), the first point (Q) to the new position obtained in the course of step d). Apparatus (1) according to claim 1, wherein, in the calculation of at least one integral made during step d), the integral function is approximated by a function of the integration variable of said integral function. Apparatus (1) according to claims 1 or 2, wherein said auxiliary plane (402) is orthogonal to the straight line passing through the second point (A) and the point of view (V). Apparatus (1) according to any one of the preceding claims, in which the position of said point of view (V) is constant and coincident with the position of said center of projection (C). 5. Apparatus (1) according to any one of claims 1 to 3, wherein the position of said point of view (V) is variable as a function of the position of the second point (A), or as a function of the position of the first point ( Q), or as a function of the distance of the first point (Q) from the projection center (C), or as a function of the angle formed by a straight line, passing through the projection center (C) and the second point (A), and an optical axis which is perpendicular to said projection plane and passes through the projection center (C). Apparatus (1) according to any one of the preceding claims, in which the distance of said viewpoint (V) from said auxiliary plane (402) is preset to a constant value. 7. Apparatus (1) according to any one of claims 1 to 5, wherein the distance of said viewpoint (V) from said auxiliary plane (402) is variable as a function of the distance of said viewpoint (V) from first point (Q) either as a function of the distance of said point of view (V) from said second point (A), or as a function of the angle formed by a straight line, passing through the center of projection (C) and the second point (A), and an optical axis which is perpendicular to said projection plane and passes through the projection center (C). Apparatus (1) according to any one of the preceding claims, wherein SVT054 at least one of the components of the new position in which to represent the second point (A) calculated during phase d) is multiplied by a variable that is a function of one or more parameters that may include the distance of the second point (A), or of said first point (Q), from the center of projection (C), the angle formed by a straight line, passing through the center of projection (C) and the second point (A), and an optical axis which is perpendicular to said plane of projection and passes through the projection center (C), other parameters forming part of the context of the space represented in said image (401) and / or of the context in which said image (401) is generated and / or of the context of reproduction to which said image (401) is intended. 9. Apparatus (1) according to any one of the preceding claims, in which at least one of the components of the new position in which to represent the second point (A), calculated during step d), is combined with at least one component of the previous position of said second point (A). Apparatus (1) according to any one of the preceding claims, wherein the image (401) comprises a pair of stereoscopic images, and where steps a) -e) are applied to the two images of said pair of stereoscopic images. Apparatus (1) according to claim 10, wherein at least one of the components of the gradients calculated during step b) is calculated assuming a common horizontal coordinate. Apparatus (1) according to claim 11, wherein said common horizontal coordinate corresponds to the average of the horizontal coordinates of at least two points, which are respectively included in the two images of the pair of stereoscopic images, and where said at least two points represent the same part of the three-dimensional region. Apparatus (1) according to any one of claims 10 to 12, wherein said processing (1003) and memory (1004,1005) means are configured to correct the image (401) also performing a gap filling step , where, in the case in which the execution of step e) on a first image of said pair of stereoscopic images generates a first processed image which includes at least one gap, this gap filling step attempts to SVT054 fill at least said one gap by superimposing said first processed image on a second processed image, where the second processed image is obtained, at least in the area of said gap, by applying steps a) -e) to a second image of said pair of stereoscopic images , assuming as the center of projection the same as the first image of the pair of stereoscopic images. 14. Method for correcting an image (401) which resides on a projection plane and which has been generated according to the rules of linear perspective, by projecting on said projection plane a region of three-dimensional space through a projection center (C), so as to reduce the deformations of the representation of said three-dimensional region which appear when said image (401) is observed from an observation point not corresponding to said projection center (C), the image (401) comprising at least a first point ( Q) and the three-dimensional region comprising at least a second point (A), in which said first point (Q) is the result of the perspective projection on said projection plane of the second point (A) through the projection center (C), characterized by understanding the phases of a) calculate the position in three-dimensional space of said second point (A) based on the position of the first point (Q), the position of the projection center (C) and the depth of said second point (A), <�> b) calculate, on the basis of a first incremental displacement (∆ a) of said second point (A) in three-dimensional space, a second incremental displacement (∆ p <�> ) of a third point (P), where said third point (P) is the result of the perspective projection on an auxiliary plane (402) of said second point (A) through a point of view (V), and where said auxiliary plane (402) is different from the plane on which the image (401) lies, <�> c) derive, on the basis of said second incremental shift (∆ p) calculated during phase b), a gradient of a new position in to represent said second point (A) on said projection plane, d) obtain, by calculating an integral of at least one of the components of said gradient obtained during step c), a new SVT054 position in which to represent the second point (A) on said projection plane, e) move, in the image (401), the first point (Q) to the new position obtained in the course of step d). Method according to claim 14, wherein, in the calculation of at least one integral made during step d), the integral function is approximated by a function of the integration variable of said integral function. Method according to claims 14 or 15, wherein said auxiliary plane (402) is orthogonal to the straight line passing through the second point (A) and the point of view (V). Method according to any one of claims 14 to 16, wherein the position of said point of view (V) is constant and coincident with the position of said center of projection (C). Method according to any one of claims 14 to 16, wherein the position of said point of view (V) is variable as a function of the position of the second point (A), or as a function of the position of the first point (Q), o as a function of the distance of the first point (Q) from the projection center (C), or as a function of the angle formed by a straight line, passing through the projection center (C) and the second point (A), and an axis optical which is perpendicular to said projection plane and passes through the projection center (C). Method according to any one of claims 14 to 18, wherein the distance of said viewpoint (V) from said auxiliary plane (402) is preset to a constant value. Method according to any one of claims 14 to 18, wherein the distance of said viewpoint (V) from said auxiliary plane (402) is variable as a function of the distance of said viewpoint (V) from the first point ( Q) or as a function of the distance of said point of view (V) from said second point (A), or as a function of the angle formed by a straight line, passing through the center of projection (C) and the second point (A) , and an optical axis which is perpendicular to said projection plane and passes through the projection center (C). 21. Method according to any one of claims 14 to 20, in which at least one of the components of the new position in which to represent the second point (A) calculated during step d) is multiplied by a variable which is SVT054 function of one or more parameters which may include the distance of the second point (A), or of said first point (Q), from the center of projection (C), the angle formed by a straight line, passing through the center of projection ( C) and the second point (A), and an optical axis which is perpendicular to said projection plane and passes through the center of projection (C), other parameters forming part of the context of the space represented in said image (401) and / either of the context in which said image (401) is generated and / or of the reproduction context to which said image (401) is intended. Method according to any one of claims 14 to 21, wherein at least one of the components of the new position in which to represent the second point (A) calculated during step d), is combined with at least one component of the previous position of said second point (A). Method according to any one of claims 14 to 22, wherein the image (401) comprises a pair of stereoscopic images, and where steps a) -e) are applied to the two images of said pair of stereoscopic images. Method according to claim 23, wherein at least one of the components of the gradients calculated during step b) is calculated assuming a common horizontal coordinate. Method according to claim 24, wherein said common horizontal coordinate corresponds to the average of the horizontal coordinates of at least two points, which are respectively included in the two images of the pair of stereoscopic images, and where said at least two points represent the same part of the region three-dimensional. Method according to any one of claims 23 to 25, also comprising a gap filling step, where, in the case where the execution of step e) on a first image of said pair of stereoscopic images generates a first processed image which comprises at least one gap, this gap filling step attempts to fill at least said one gap by superimposing said first processed image on a second processed image, where the second processed image is obtained, at least in the area of said gap, by applying steps a) - e) to a second image of said SVT054 pair of stereoscopic images, assuming as the projection center the same as the first image of the pair of stereoscopic images. 27. Computer product that can be loaded into the memory of an electronic computer and comprising a portion of software code for carrying out the steps of the method according to any one of claims 14 to 26. 28. Computer readable means comprising a recorded program, said computer readable means comprising program coding means adapted to carry out the method according to any one of claims 14 to 26, when said program is run on a computer.