ES2643138A1 - Method of estimating the number of particles in a given place from a perspective image (Machine-translation by Google Translate, not legally binding) - Google Patents

Abstract

Method for estimating the number of particles of a population in a given place from a perspective image, comprising the steps of: - obtain an auxiliary image without perspective; - identify and limit in the auxiliary image the place where the particles are in the perspective image; - randomly overlaying a grid of sampling windows in the bounded auxiliary image; - take a plurality of contour points in each sampling window, and with projection equations transfer each of said points to the perspective image; - join in the perspective image the points transferred from the contour of each window and form a new grid of sampling windows; - manually count in the perspective image the total number q of particles captured by the projected grid of sampling windows. (Machine-translation by Google Translate, not legally binding)

Description

Density analysis is usually done "in situ" and "by eye", relying on user experience.

For example, in the specific case of demonstrations [Jacobs, 1967, Walson R, Yip P How many were Ihere when il maltered? Significance 2011; 8 (3): 104-107. do ;: 10.111l (j.1740-9713.2011 .00502.x.], the user estimates the area of manifestation in square meters and the average density of people per square meter, covering the manifestation or part of it. number of people is obtained by multiplying the area by density.

However, the definition of area is ambiguous since it is difficult to decide where a mobile demonstration begins and ends, and in the streets there are spaces where there can be no particles, whose areas would have to be subtracted. In addition to this, it should be noted that the density is constant throughout the area considered, and this is not realistic. Finally, density estimates are often made "by eye." All these inconveniences present the following problems:

•

Big mistakes

•

Bias, not predictable error

•

Non-replicable count

•

Result impossible to verify

Sometimes, to estimate the area and / or density, the user is accompanied by still images. In this case, as long as all the particles are distinguishable for counting, the following methods can also be applied: automatic detection in still images with computer vision, manual counting in still images, and method described in Cruz et al. 201 5. However, since aerial images tend to have insufficient resolution, these images are usually terrestrial and therefore have large perspective effects, especially in the case of large agglomerations.

For example, in the case of person counting in a demonstration, the automatic detection programs [V Lel11pilsky and A. Zisserman, "Learning lo counl objecls in images", in Advances in Neural Informalion Processing Syslems, 2010], [M Rodriguez l. Laplev, .J. Sivic, and.J. And Audiberl, "Densily-aware person deleclion and Iracking in crolVds", in 20fl 1nternationa / Conference on Computer Vision, 2011, pp. 2-123-2-130], [H. Idrees, 1. Sa / eemi, e. Seibert, and M Shah, "Mulli-source mu / ti-sca / e counting in extreme / and dense crolVd images", in 2013 IEEE Conference on Computer Vision and Pallern Recognition (CVPR) 1EEE, 2013, pp. 254 7-2554], re. Zhang, j-J. Li, X Wang, and X Yang, "Cross-scene crowd counting via deep convolutiona / new'a / neMorks", in 2015 IEEE Conference on Computer Vision and Pallern Recognition (C VPR), 2015, pp. 833-8-1J work reasonably well for a few dozen well-focused people with homogeneous lighting, but they fail loudly in images of large crowds with thousands of people [S Zhang, R. Benenson, M. Olllran, .J. Hosang, and B. Schiele, "HOlV far are lVe from solving pedestrian delection?" in Proceedings of the IEEE Conference on Computer Vision and Pallern Recognition, 2016. Pp. / 259-1267]. In addition, they have a bias dependent on the image considered and therefore unpredictable.

Manual image counting is slow, tedious and difficult to verify, especially for large agglomerations of more than 10,000 particles, so it is only an alternative when the number of particles is low.

An alternative method to the density method has been developed by the inventors of the present invention, [Cruz M, Gómez D, Cruz-Orive LM (2015) Efficient and Unbiased Estimation ofPopulation Size. PLoS ONE 10 (11): eO / 41 68. doi: / 0./37//journal.pone. 014/868], and allows you to estimate the number of particles in an image. The method is based on systematic sampling using a random window grid. superimposed on the image. The only requirement is that all particles are distinguishable for counting. Manually counting between 50 and 100 particles captured by the sampling grid, an estimate is obtained with a relative error between 5% and 10%.

However, everything stated in this article refers to the estimation of the number of people in a single image of conventional size (few Mb) and with a small perspective effect. To cover a large agglomeration of more than 10,000 people it is usually impossible to capture all people in a normal size image, preferably images with sizes of at least I Gb (gigapixel) are preferably necessary. These images usually have large perspective effects that can cause an increase in relative error from 5% -10% present in normal images to 30% -40%.

SUMMARY OF THE INVENTION

The present invention tries to solve the aforementioned drawbacks by means of a method for estimating the particle number of a population in a place defined from a perspective image, comprising the steps of:

-

obtain an auxiliary image without perspective or at least with a less perceptive effect than that of the image with perspective, which contains at least said determined place, and such that in said auxiliary image the projection equations can be found or known, so that any point P of the auxiliary image can be transformed to a point P 'in the perspective image; -identify and delimit in the auxiliary image the place where the particles are in the image with perspective, thus obtaining a bounded auxiliary image; - randomly overlap a grid of sampling windows in the bounded image that covers it completely, thus being able to estimate the size of the entire population; - take a plurality of contour points in each sampling window superimposed on the bounded auxiliary image, and with projection equations, transfer each of these points to the perspective image, transferring any point P of the bounded auxiliary image to a point P 'in the perspective image, such that the more points selected, the greater the accuracy achieved; - join in the image with perspective the points transferred from the contour of each window and form a new grid of sampling windows (projected grid) with new sampling windows (projected windows), so that the greater the perspective effect in each window, the greater the deformation suffered by each projected window; - manually count in the perspective image, using an unbiased counting rule, the total number Q of particles captured by the projected grid of sampling windows, such that by counting only in a sampling fraction of the image, the manual counting is feasible and fast, being the estimate (N) of the total number of particles:

~ a

N = - · Q

to'

where:

a: area of the sampling window a ': area of the fundamental tile, the fundamental tile being a minimal or "tile" motif with which the entire infinite plane containing the auxiliary image can be tessellated, such that the entire infinite plane remains completely covered by said fundamental tiles placed in parallel and non-overlapping series and such that each fundamental tile contains at least one sampling window;

In a possible embodiment, the perspective image is a photograph.

In a possible embodiment, the auxiliary image is accompanied by elevation data above sea level.

In a possible embodiment, the sampling window grid has a rotation angle such that said grid is prevented from having the same orientation as the particles, the number of non-empty windows of the sample window grid superimposed on the auxiliary image bounded is between 20 and 50, the sampling windows and tiles are square, and the side of each sampling window is such that the number of particles per window is between I and

5.

In a possible embodiment, neither the type of projection used in perspective images nor the projection parameters are known, so it is necessary to estimate them by identifying as many reference points as necessary to achieve the desired accuracy, such that said reference points they must be identifiable both in the perspective image and in the bounded auxiliary image, in order to propose an overdetermined system and obtain an approximate solution using the least squares method.

In one possible embodiment, the unbiased counting rule used is that of the forbidden line, such that only particles that have a portion within the sampling window and that do not touch a prohibited line that theoretically extends to infinity are counted.

Preferably, the method further comprises the final stage of error estimation. In a possible embodiment, the formula for calculating the error estimate, considering square sampling window and square tile is:

(1 -t) 2. (3 (C _ v) _ 4C + C J + vn

6 r (2 _ r) I '[4

n

or 2

n-k

LQ¡Oj + k 'k = 0, 1, 2.

;_ one

Vn = (; =; ~ 2 .t (Qm -Q.i

Ll

being:

r = li T E (O, 1], linear sampling traction.

n: number of rows of sampling windows covering the entire population, (n>

2). ni: number of sampling windows within row number ¡, i = 1, 2, ..., n. qi / number of particles captured by the sampling window} of row i,} = 1, 2, ... , neither·

Qoi> Qei: total number of particles captured by odd sampling windows and pairs respectively within row i. Q¡ = I:; ~ l q¡¡, total number of particles sampled in row i. Note that Qi =

Qoi + Qú Q = I: 7-1 Qi, total number of particles sampled.

BRIEF DESCRIPTION OF THE FIGURES

In order to help a better understanding of the characteristics of the invention, in accordance with a preferred example of practical realization thereof, and to complement this description, a set of drawings is attached as an integral part thereof, whose character is Illustrative and not limiting. In these drawings:

Figure 1 shows the cylindrical perspective image of the example.

Figure 2 shows the auxiliary image without perspective of the example.

Figure 3 shows the sample window grid of the example on the selected bounded auxiliary image.

Figure 4 shows the grid of sampling windows projected on the image

with cylindrical perspective of the example.

Figure 5 shows three projected and enlarged sample windows of the example, containing 2, 2 and 3 points respectively.

DETAILED DESCRIPTION OF THE CONVENTION

In this text, the term "comprises" and its variants should not be understood in an exclusive sense, that is, these terms are not intended to exclude other technical characteristics, additives, components or steps.

In addition, the terms "approximately", "substantially", "around", "ones", etc. should be understood as indicating values close to which these terms accompany, since due to calculation or measurement errors, it is impossible to achieve Go with total accuracy.

In addition, particle means any person, animal or object that you wish to count in the image. If it is not completely observed, the fragment observed in the image is considered as a particle (for example: heads, whole bodies, eyes ...).

In addition, population is understood as the set of particles whose number you want to estimate.

In addition, an image with perspective is understood as that image in which at least

two palms have a relative size in the image different from the real one. In the

actuality, perspective effects on especially important

30 mostly in images with sizes of at least I Gb (gigapixel images), presenting a high resolution of a given population and taken from a

oblique position

In addition, it is understood by fundamental tile, a minimum motif or "tile" with which the entire infinite plane containing the auxiliary image can be tessellated, such that the entire infinite plane is completely covered by said fundamental tiles placed in parallel and not superimposed series (for example, it could be assimilated to the prismatic cells of a honeycomb) and such that each fundamental tile contains at least one sampling window.

In addition, a sampling window is understood as a window contained in the fundamental tile in which the manual counting is performed. If the result of the counting in a window is zero, the window is considered to be "empty".

In addition, a grid of sampling windows is understood as the set of sampling windows of all the fundamental tiles that cover the study image.

The following preferred embodiments are provided by way of illustration, and are not intended to be limiting of the present invention. In addition, the present invention covers all possible combinations of particular and preferred embodiments indicated herein. For those skilled in the art, other objects, advantages and features of the invention will be derived partly from the description and partly from the practice of the invention.

The method of estimating the number of particles (ie people, animals or objects) of a population in a specific place from a perspective image is described below. The fundamental requirement of the method of the invention is that all particles are distinguishable for counting the acquired images.

The method of the invention requires an image with perspective, for example a photograph, showing the determined place where it is desired to estimate the number of particles, for example an avenue through which a manifestation of people circulates.

Next, an auxiliary image (for example Google image or map) must be obtained without perspective or at least with less perspective effect than the image with perspective, which at least contains said determined place. Optionally, said auxiliary image is accompanied by elevation data above sea level. In case of not saying the same, it can be assumed that the elevation above sea level is the same for all points of the determined place. The auxiliary image has to be chosen so that the projection equations can be found, that is, so that any point P of the auxiliary image can be transformed to a point p. In the image with perspective.

Next, the place where the particles are located in the perspective image, must be identified and bounded in the auxiliary image to subsequently overlap a grid of sampling windows in said bounded auxiliary image that completely covers it, thus being able to estimate the full population size. That is, the grid of sampling windows is superimposed randomly on the bounded auxiliary image, there being a series of parameters that determine the grid and that the user of the method must select at his convenience:

o Rotation angle 6: Rotation angle of the grid with respect to the horizontal axis. The angle of rotation is fixed and arbitrary. One skilled in the art will understand that in order to reduce the variance, it is advisable to avoid that the grid has the same orientation as the particles in case they are aligned, such as rows of spectators.

o Distances between the centers of two consecutive sampling windows: To maximize efficiency these distances must be chosen so that the number of non-empty windows superimposed on the bounded auxiliary image is preferably between 20 and 50, such that an error is expected relative between 5% and 10%. These recommendations have been analyzed by the inventors of the present invention and embodied in Cruz et al. 2015 described in the state of the art. Choosing an empty sample window number less than 20 increases the error

Relative, while a number of windows greater than 50 implies that the

Manual counting is very tedious, although it is feasible to do it and it is required

higher accuracy

Preferably, the sampling windows are square, therefore being

It is necessary to select a single distance T between the centers of two consecutive sampling windows. For example, in the case of sampling windows and square tiles, the distance T is calculated:

T = fF [1]

10 where Lx is the width and Ll is the height in pixels of the bounded auxiliary image, and k the number of windows to be obtained, of which there should be no more than 20 gaps to comply with the recommendation mentioned above.

o Lengths of the sides of the window: Length of each side (in pixels) of

15 the window of death (t in the case of square windows). To maximize efficiency, the number of particles per window must be chosen

or approximately between I and 5, according to the recommendations of the inventors in the document Cruz et al. 201 5. In this way, by multiplying by the number of non-empty windows, between 50 and 150 particles will be counted in

20 total, obtaining an expected error between 5% and 10%. An expert in the field will understand that if the number of particles per sampling window is greater, the count would be long and tedious, but the relative error could be reduced. The opposite occurs if the number of particles is less than recommended.

Next, the method of the invention takes a plurality of contour points in each sampling window superimposed on the bounded auxiliary image, and with projection equations each of these points is transferred to the perspective image. Then, in the perspective image, it joins the points transferred from the contour of each window and forms a new grid of sampling windows

(projected grid) with new sampling windows (projected windows), so that the greater the perspective effect in each window, the greater the deformation suffered by each projected window. Therefore, one skilled in the art will understand that in the perspective image, the projected windows in the background will appear smaller than the projected windows closest to the camera.

Imaging with perspective can use different types of projections such as linear, spherical or cylindrical. The type of projection must be known, and ideally the projection parameters as well, so that any point P of the auxiliary image bounded to a point P 'can be transformed in the perspective image. In the case of not knowing the projection parameters, it is necessary to estimate them, for example, by identifying as many reference points as necessary to achieve the desired precision. These reference points must be identifiable both in the perspective image and in the bounded auxiliary image, in order to propose an overdetermined system and obtain an approximate solution using the least squares method.

As mentioned above, once the equations and parameters necessary to transform P into p. Are known, as many points per sampling window are selected as desired to achieve the desired accuracy, and projected to the image with perspective to form the grid projected joining them by rectilinear segments. It must be taken into account that some projections, such as the spherical one, do not preserve straight lines, so make sure that you take a number of points high enough to join them by segments. Alternatively, fewer points can be used and the projected windows can be drawn with the analytical equations that are obtained from projecting rectilinear segments.

The projection of the grid of sampling windows on the perspective image, using the corresponding projection equations, allows to realize a particle count in an ordinary way but having taken into account the perspective effect, thanks to the transformation undergone by each window of sampling. Is

In other words, the method of the invention then proceeds to manual counting in the perspective image of the total number Q of particles captured by the projected grid of sampling windows. By counting only in a sampling fraction of the image, manual counting is feasible and fast. The estimate (N) of the total number of particles is:

~ a

N = - · Q

a '[2]

being:

a: sampling window area a ': fundamental tile area

A counted rule is used for counting. There are several unbiased counting rules that avoid double counting. For example, the most common is that of the forbidden line: only particles that have any portion within the sampling window and that do not touch a prohibited line that theoretically extends to infinity [[Gundersen HJG "Notes on the estimation of the numerical density of arbitrary profiles: the edge effect ". J Microsc. 1977; 111 (2): 2019-223. Doi: 10.1111 / j.1365-2818.1977.tb00062.x]].

Obviously at a higher sampling fraction, greater accuracy is obtained but efficiency is reduced, since it requires more time for manual counting.

A person skilled in the art will understand that since the superimposition of the grid on the image is random, every time the method of the invention is used, a different estimate will be obtained, although being unbiased, the average of infinite measurements is the correct value.

Finally, and preferably, the error estimate is calculated. For example, a formula for calculating the error estimate, considering square sampling window and square tile is:

1 (1 -'d V

VdrQv (&) = -. . [3 (C -V) -4C + C J + .... !!.

r 4 [3] 6 r (2 -r) or "1 2

n-k

LQjQj + k 'k = 0, 1, 2.

[4]

j-I

(1-r) 2 ~ 2 Vn = 3 _ 2r. L (QOi -Q.J [5]

i = l

being:

r = LIT E (O, 1], linear sampling fraction.

n: number of rows of sampling windows covering the entire population, (n> 2). n¡: number of sampling windows within row number i, i = 1, 2, ..., n.

qi / number of particles captured by sampling window j of row i, j = 1,2,

..., neither' Qoi, Qu total number of particles captured by odd sampling windows and

pairs respectively within row i. Q¡ = 2 ::; '1 qij, total number of particles sampled in row i. Note that Qi =

Qoi + Qú Q = 2 :: 7-1 Qi, total number of particles sampled.

Example

A concrete example of embodiment of the invention and the results obtained are shown below.

1) It is based on an image with cylindrical perspective (Figure 1) in which it is desired

Estimate the number of particles. This image is a simulation of the image giga pi xe l http://Iab.elespanol.com/eslaticos/gigapan_sol/ taken in the Plaza del Puerta del Sol in Madrid. The cylindrical perspective is the most used in gigapixel photographs since it allows a viewing angle greater than 1800 and preserves the vertical straight lines (the others would be curved). The position of the camera is noted in Table l. Each black dot (A) of the perspective image simulates the position of a person in the photograph. A total of 20,000 points have been simulated (note that in a real image the total number of people is known and is precisely what you want to estimate). It is also represented with a circle with white center 6 reference points (B) that are easily identifiable on a map, since they are corners of buildings. These reference points are used to estimate projection parameters. The number of reference points varies according to the image and the perspective. In this example 6 they are enough.

2) Next, an auxiliary image without perspective is obtained, which in this case is a map (Figure 2) that contains the place of the Puerta de Sol square that appears in the perspective image, and identifies the place where the particles are found (marked in figure 2 with a white polygon). It is not necessary to define the polygon precisely, but it is enough to choose a rectangle large enough to ensure that all the particles observed in Figure I are contained (bounded auxiliary image). The white triangle in Figure 2 marks the position of the camera. The 6 reference points identified in Figure 1 are represented again in white circles. Elevation data above sea level can be obtained by knowing the coordinates of longitude and latitude of a point, using for example the Google Earth Pro tool.

3) Next, the grid of sampling windows is superimposed on the bounded auxiliary image, this auxiliary image corresponding to the polygon corresponding in this example. The grid of sampling windows must completely cover the bounded auxiliary image, thus being able to estimate the size of the entire population. That is, the grid of sampling windows is superimposed randomly on the bounded auxiliary image, and are read as grid parameters:

-

side of the sampling window t = 1.5 meters, so that the sampling windows contain at most about 9 people in high density areas where there are 4 people per square meter; -To choose the eparation between windows T, it is considered that the dimensions of the square are approximately 150 by 500 m and therefore, applying equation 1, and obtains approximately T = 27 m. For greater accuracy, it is rounded off by default and T = 20 m is set.

-

to prevent the grid from being aligned with the buildings that mark the edge of the square, a 60 ° rotation of the grid is read. Figure 3 shows the grid superimposed randomly on the polygon marked in Figure 2 (bounded auxiliary image). The only thing that needs to be preserved for the next step is the positions of the vertices of the sampling windows.

4) The grid is then projected from the auxiliary image bounded to the perspective image. In the event that the parameters and transformation equations are not known, first you have to obtain transformation equations of a point P with real coordinates (x, y, z) in meters of the auxiliary image (for example the vertex from a sampling window) to another point p. with XI and X2 coordinates in pixels in the bounded auxiliary image. The x and y axes are selected on the horizontal plane that forms the ground. a vertical z axis and the origin O, in the position of the camera. The XI and X2 axes follow the image analysis convention with the XI axis increasing from left to right, the X2 axis increasing downwards and the origin in the upper left vertex of the image. It is now considered a cylinder of radius r> O Ycentro with axis of immetry the z axis. The cylindrical projection is equivalent to projecting the point P with cylindrical coordinates P = (p, ep, z) by means of a ray that emanates from O to a point in the cylinder P '= (T, C!>,:'). It is easy to see that:

p): 1., 2 + y2 </> atan2 (± y, ± x)

r

z '± -z.

p

In the last two equations the negative sign is selected so that the image has the desired orientation.

Unwinding the cylinder,

transform at the point Peyl =

r Zcyl = --z. p

it goes from three to two dimensions, and the point P 'is (Xeyl, Zcyl) so that:

10 Since the origin is arbitrary, a displacement t :. = (t :. I, t: .2) to obtain the coordinates of the point in the image 'Pl'hoto = Pcy1 tJ. so that the final coordinates of PphOlO are worth:

Xl r4> + he

r X2 --z + him2

p

15 Therefore it is necessary to estimate 3 parameters that can be grouped into a vector

To estimate the projection parameters, several reference points are needed

whose coordinates can be identified both in the auxiliary image without perspective P

= (x, y, z) as in the image with perspective p. = (XI, X2).

20 In this example, the 6 reference points marked in the respective images and whose coordinates are listed in the following table are sufficient:

Number

Length x (degrees 0) Latitude and (degrees O) Z elevation (m) XI coordinate (pixels) X2 coordinate (pixels)

Camera

-3.702702 40.4 17105 670.96 - -

one

-3.702243 40.4 16656 649.47 16280 6250

2

-3.703388 40.4 16607 648.32 37420 5740

3

-3.704986 40.4 16462 646.5 1 43220 3360

4

-3. 704697 40.41678 1 645.75 45290 3870

5

-3 .7041 2 1 40.41709 1 647.28 48 180 4440

6

-3.70 1658 40,417 159 651.70 2490 4430

. .

Table 1: Reference points used to adjust projection parameters.

Longitude and latitude coordinates are easily obtainable from a map and can be transformed to coordinates (x, y, z) in meters, taking an arbitrary origin and 5 using tools such as Python Geocoding Toolbox or Google Earth elevation data Pro.

With the above equations, an overdetermined system can be obtained:

-z

which can be written in matrix equation as AS = b, being A the matrix of coefficients and b the vector of independent terms. Substituting the values of X, y, z, xl, x2 from Table I in the matrix equation, the vector can be estimated

15 A = (1 \ 1, 1 \ 2.; r), by least squares by solving m in ll A9 b ll using the normal equations S = CA / ArlA / b, provided there is CA / A) -l. Solving the following values are obtained:

T = 1428.6m III = 2584.ópixeLó III = 158.9pixels.

Once the projection parameters and equations are known, the

Sampling windows of the auxiliary image bounded to the perspective image. In this specific case, it is verified by simulations, that sufficient precision is obtained by approximating each projected sampling window with the quadrilateral resulting from joining the 4 projected vertices. Figure 4 shows the grid of 5 sampling windows projected on the image with cylindrical perspective. Since the cylindrical projection only preserves vertical straight lines, in general it is not enough to take the four vertices of each window and project them. For total accuracy, we should find the equations of the curves resulting from projecting the straight segments that make up each sampling window, or project a number

10 large enough points in each segment to then join them forming the projected window.

5) After projecting the grid, we proceed to the manual counting of points contained in each projected window. In Figure 5, three sampling windows are expanded

15 projected containing 2, 2 and 3 points respectively. In total, adding all the points captured by the grid, Q = 143 is obtained, so applying the formula [2]:

~ a N = - · Q

to'

= 20143

1.52 ~ 25422

In addition, by writing down the individual accounts of each window, formula (3) can be applied, obtaining vaTca ~ (Ñ) = 2526529 which is equivalent to an estimated standard deviation of approximately 1590, or what is the same as an estimated relative standard error of approximately 6%

Claims (8)

1. Method for estimating the number of particles of a population in a given place from an image with perspective, which includes the steps of:

-

obtain an auxiliary image without perspective or at least with a perspective effect less than that of the image with perspective, which contains at least said determined place, and such that in said auxiliary image the projection equations can be found or known, so that any point P of the auxiliary image can be transformed to a point P 'in the perspective image; -identify and delimit in the auxiliary image the place where the particles are in the image with perspective, thus obtaining a bounded auxiliary image; - randomly overlap a grid of sampling windows in the bounded auxiliary image that completely covers it, thus being able to estimate the size of the entire population; - take a plurality of contour points in each sampling window superimposed on the bounded auxiliary image, and with projection equations transfer each of said points to the perspective image, transforming any point P of the bounded auxiliary image to a point P 'in the image with perspective, such that the more points selected, the greater the accuracy achieved; - join in the image with perspective the points transferred from the contour of each window and form a new grid of sampling windows (projected grid) with new sampling windows (projected windows), so that the greater the perspective effect in each window, the greater the deformation suffered by each projected window; - manually count in the perspective image, using an unbiased counting rule, the total number Q of particles captured by the projected grid of sampling windows, such that by counting only in a sampling fraction of the image, the manual counting is feasible and fast, being the estimate (N) of the total number of particles:

~ a N = - · Q to' where: a: area of the sampling window a ': area of the fundamental tile, the fundamental tile being a minimal or "tile" motif with which the entire infinite plane containing the auxiliary image can be tessellated, such that the entire infinite plane remains completely covered by said fundamental tiles placed in parallel and non-overlapping series and such that each fundamental tile contains at least one sampling window;

2.

The method of claim 1, wherein the perspective image is a photograph.

3.

The method of any of the preceding claims, wherein said auxiliary image is accompanied by elevation data above sea level.

Four.

The method of any of the preceding claims, wherein the sampling window grid has a rotation angle such that said grid is prevented from having the same orientation as the particles, where the number of non-empty windows of the sampling window grid superimposed on the bounded auxiliary image is between 20 and 50, where the sampling windows and the tiles are square, and where the side of each sampling window is such that the number of particles per window is between I and 5.

5.

The method of any of the preceding claims, wherein neither the type of projection used in the perspective images nor the projection parameters are known, and both are estimated by identifying as many reference points as necessary to achieve the desired accuracy, such that said reference points must be identifiable both in the image with perspective

as in the bounded auxiliary image, in order to propose an overdetermined system and obtain an approximate solution using the least squares method.

6.

The method of any of the preceding claims, wherein the unbiased counting rule used is that of the prohibited line, such that only particles that have any portion within the sampling window are counted and that do not touch a prohibited line that theoretically is extends to infinity.

7.

The method of any of the preceding claims, further comprising the final stage of error estimation.

8.

The method of the preceding claim, wherein the formula for calculating the error estimate, considering square sampling window and square tile is:

n-k Ck -QQ j + k 'k = 0, 1, 2. j-I (1-r / "2 VII = 3 _ 21:. (Qoi - ~ j) 1-1 being: í = liT E (O, 1 J, linear sampling fraction. n: number of rows of sampling windows covering the entire population, (n> 2). ni: number of sampling windows within row number i, i = 1, 2, ..., n. q¡ / number of particles captured by the sampling window j of row i, j = 1, 2, ..., ni · Qob Qe {total number of particles captured by odd sampling windows and pairs respectively within row i. Q¡ = ¿tl qij, total number of particles sampled in row i. Note that Qi = Qoi + Qei '5 Q = ¿;' = l Qi, total number of particles sampled.