WO2013182871A1

WO2013182871A1 - Shape index and polynomial coefficient based pattern analysis and comparison method for cartridge cases and bullets in forensic science

Info

Publication number: WO2013182871A1
Application number: PCT/IB2012/052832
Authority: WO
Inventors: Ufuk Sakarya; Ugur Murat Leloglu; Osman TOPCU; Hasan Saygın ARKAN; Mehmet Ziya KADIOGLU; Sinan CANGA; Sukru Alphan ES
Original assignee: Tubitak
Priority date: 2012-06-06
Filing date: 2012-06-06
Publication date: 2013-12-12

Abstract

Ballistic marks are the striation and impression marks of a firearm imprinted on metallic surfaces of cartridge cases and bullets and their fragments when they are fired from the firearm. Firearm identification is the process of determination of bullets, cartridge cases and their fragments that are fired from the same firearm by examining similarity of ballistic marks. These marks are affected from metal hardness, amount of gun powder, etc. These reasons make automated firearm identification a difficult problem. It is proposed in the present invention that feature comparison based on shape index and polynomial coefficients to determine bullets, cartridge cases and their fragments that are fired from the same firearm. These features can be extracted from surface height map or surface normal vectors using estimation methods.

Description

DESCRIPTION

TITLE

Shape Index and Polynomial Coefficient Based Pattern Analysis and Comparison

Method for Cartridge Cases and Bullets in Forensic Science

TECHNICAL FIELD

This invention is about the approach and feature selection for finding similarities of marks of a gun (MG). It is suited to forensic science practices such as analysis and comparison of patterns on surfaces of cartridge cases, bullets and their pieces. In this method, MG analysis is done by using 3D surface topography and feature images extracted from this topography.

BACKGROUND ART

Firearms leave intrinsic marks onto the bullets and cartridge cases during their discharge. The firearm involved in a crime is identified from the marks onto the evidences (bullets and/or cartridge cases) found in the crime scene. Bullets and cartridge cases fired from the same firearm are called 'sister bullets' and 'sister cartridge cases' respectively.

Generally, MG analysis involves comparison of MGs on evidences of the same type (bullets are compared to bullets). A ballistic expert does this job. The expert tries to decide on the sisterhood of 2 MGs by looking at the similarity between them. This inspection is a time taking process for the expert. By the accumulation of evidences in databases, the number of same diameter, caliber and type MG increases, therefore the workload of the expert increases. Thus, the necessity of automated or semi-automated MG comparison emerges. Automated MG comparison relies on computation of MG similarities or distances. When these similarities or distances are sorted, sister evidences are expected to be in the top positions. It is required that systems for automated MG comparison should place the sisters to a reasonable rank and complete the comparisons in an affordable duration. With semi-automated systems, workloads of experts are reduced by allowing them to compare MG's in top positions.

Within the background art, in order to carry out MG analysis with information processing devices; platforms containing MG is inputted to the data acquisition unit where discrete representation of the MG is generated or alternatively, stored discrete representations can be inputted to the system. Generally, this discrete raw data or features extracted from this data using signal processing techniques are used for the analysis. Similarities or distances of two MG's are determined by comparing them. The larger/smaller the similarities/distances of two MGs are, the more probable that they are sisters. In order to quantify MG similarity with a value, generally, the new MG is compared with all MGs in the database and corresponding similarity/distance values are obtained. According to these similarity/distance values, sisterhood decision can be made automatically or emerging MGs are presented to the expert and the expert decides for their sisterhood in a semi-supervised manner.

TECHNICAL PROBLEMS THAT THE INVENTION AIMS TO SOLVE

With the increase in number of MGs, long time span of MG comparison becomes an important problem. It is required that large quantity of MG data is compared quickly and accurately enough in a supervised or semi-supervised fashion. This invention proposes an MG comparison method which defines methods and features to be used in comparison of MG surface height data so that the sisters are sorted at top positions. With this invention, it is aimed to reduce workload of experts by narrowing the number of items to be examined down by a considerable amount in a semi- supervised manner.

DESCRIPTION OF DRAWINGS

The system designed to conduct the work by using the subject method of this invention is illustrated in the annex, and;

Figure 1 - Block diagram of the system.

The components in this figure are numbered one-by-one, the correspondences of these numbers are given below.

11. Data acquisition unit

12. Database

13. User interface

14. Process management unit 15. Processing

Figure 2 - Illustration of cartridge case and its regions.

21. Headstamp

22. Ejector mark

23. Primer

24. Firing pin mark

Figure 3 - Illustration of a sample comparison algorithm.

31. Reference image

32. Test image

Figure 4- Feature image illustration.

DETAILED DESCRIPTION OF THE INVENTION

In this invention, MG comparison operations are carried out in the following ways:

For the comparison of MG data, platforms containing MG is inputted to the data acquisition unit (11) where discrete representation of the MG is generated or alternatively, stored discrete representations can be inputted to the system. This discrete representation is either height map which represents surface of MG or normal vectors that are generated from local features of the surface. Database (12) is a storage environment such as RAM, disk, hard disk, CD, DVD, cartridge where MG data (including similarity/distance values) are stored for re-accessing and is the software managing the data. Discrete data is transmitted by the data acquisition unit (11) and stored in the database (12). If necessary, transformations and/or filters can be applied to come up with new representations before storing this data. Meanwhile, user controls these processes via the user interface (13).

The process management unit (14) gets the MG data required for comparison from the database (12). The process management unit (14) transmits MG comparison data and the necessary commands to the processing unit (15). The process management unit (14) saves results from the processing unit (15) to the database (12). The database (12) presents these results to the user via the user interface (13). The processing unit is composed of a single or a group of electronic computation hardware (CPU, GPU, etc.) that is capable of the required comparison operations.

Within the proposed method, 2 separate MGs are compared according to the following steps: a) Surface height maps or normal vectors containing 3D surface information of both MGs are generated by the data acquisition unit (11). b) Shape index (SI) images for each of the MGs are extracted from surface height maps (J. J. Koenderink, A. J. van Doom, "Surface Shape and Curvature Scales", Image and Vision Computing, pp. 557-564, 1992 and Bober, M. "MPEG-7 Visual Shape Descriptor", IEEE Transactions on Circuits and Systems for Video Technology, pp. 716-719, 2001). For this purpose, a shape index value is calculated for each pixel of the surface height map (Figure 4). SI images of different MGs are compared and corresponding similarity values are generated. Comparison involves placing feature images one over another by aligning their centers. The feature image chosen as reference is rotated and translated. Similarity values are computed for each of these rotation and translation pairs (Figure 3). Maximum similarity value and the corresponding translation and rotation values are determined by rotating and translating reference image within a predetermined interval. These values for each of the compared image pairs are stored. c) A feature vector is generated for each pixel of the surface. Feature vectors are generated using surface fitting methods. For each pixel of the surface height map, a quadratic (or higher degree) bi-variate polynomial is fitted to pixels within a neighborhood of the pixel. The coefficients of this polynomial is represented in vector form. Feature images/matrices whose pixels contain feature vectors are subjected to rotation, translation and similarity computations. The rotation and translation at maximum similarity along with this similarity value is determined. These values are stored for each compared feature image/matrix pair.

The method explained below can be used for the extraction of shape index feature from evidence surface.

Shape index allows quantitative comparison of shapes by mapping them into a numerical scale (J. J. Koenderink, A. J. van Doom, Surface Shape and Curvature Scales, Image and Vision Computing, pp. 557-564, 1992), (M. Bober, "MPEG-7 Visual Shape Descriptor", IEEE Transactions on Circuits and Systems for Video Technology, pp. 716-719, 2001). Shape index is a local feature of a 3D surface. Quadratic bi-variate polynomial is fitted to a surface patch. Then, Hessian matrix is formed from the second order derivatives of the polynomial. The eigenvalues and eigenvectors of this matrix indicate the principal axes of the surface and the rate of changes in these axes. Shape index is invariant to Euclidean transformations such as rotation, translation and scaling. For example, shape index features take values within the [0,1] interval in this invention (M. Bober, "MPEG-7 Visual Shape Descriptor", IEEE Transactions on Circuits and Systems for Video Technology, pp. 716-719, 2001). Shape index equation is given in Equation (1).

In Equation

ve K₂are eigenvalues of the Hessian matrix. This feature is within the class of dense features. In other words, shape index value is computed for each pixel of bullet, cartridge case and/or regions of cartridge case. Comparisons are done using all the pixels. Extraction steps of shape index are detailed in the following lines.

1) For a pixel, a window (rectangular, circular or any other shape) that is centered on the pixel is defined.

2) A quadratic bi-variate polynomial is fitted using all those pixels within the window.

3) The Hessian matrix of the polynomial is computed.

4) The eigenvalues of this matrix is calculated. These eigenvalues are κ_χ ve κ₂ in Equation (1).

5) The shape index is calculated as in Equation (1). Shape index feature can be extracted from the whole evidence as well as its regions. For example, it can be extracted from cartridge case regions such as primer (23), firing pin mark (24) and ejector mark (22). The features extracted from a region can be contained in a rectangular frame like images are contained in. Those pixels within the frame that do not contain feature values are highlighted or masked so that they do not affect the comparison.

Comparisons of ballistic evidences are one-to-many comparisons. We name the evidence of interest reference image (31) evidence and name the others test image (32) evidence.

Similarity of shape index features can be computed using one of these methods:

1- ) Normalized cross-correlation (D.P. Bertsekas, J. N. Tsitsiklis, "Introduction to probability", Second Edition, Athena Scientific, ISBN 978-1-886529-38-0, 2008.). The equation of this method is given as;

In Equation (2), / is the mean of f [x, y] and t is the mean of t [x, y] . The numerator in Equation (2) contains the inner product of the functions. The denominator contains the standard deviation of the functions. Similarity scores of this method takes values within [- 1,1] .

Comparison can be done by using cross-correlation, that is without making normalization:

2- ) Cross-correlation r(f, 0

y] - 1) (3)

3-) Mean square distance method

Y(f, t) = ^∑x,yeD [f(x> y) - t( , y)]² (4) 4-) Mean absolute difference

5-) Average cosine similarity

Y(f, *) = Z∑x,ye_D cos(( (x, y) * pi) - (t(x, y) * pi)) (6)

6-) Variance of the difference

Y(f, t) = var( (x, y) - t(x, y)) (7)

The method explained below can be used for computation of images/matrices of polynomial coefficients.

Bivariate quadratic polynomial is fitted to a surface patch within a local window. The result of this fitting is the estimated polynomial coefficients. Bivariate quadratic polynomial equation is given in Equation (8). The vector-form representation of this polynomial is as in Equation (9). f(x, y) = a x + a₂y² + a₃xy + a₄x + a₅y + a₆ (8)

f(x, y) = [x² y² xy x y 1] (9)

Writing Equation (9) for all pixels within the local window (assuming that all pixels satisfy the same relation), the system of linear equations in Equation (10) is obtained.

The number of equations has to be larger than or equal to the number of unknowns. The number of equations is 9 for a window size of 3x3 and it is 25 for windows of size 5x5. Equation (10) is written in matrix-vector form representation in Equation (11).

Aa = f (11)

The estimation of the polynomial coefficients is done using the pseudo-inverse of the matrix A as in Equation (12). a = (A^TA)^~1A^Tf (12)

These coefficients are computed for all pixels representing the surface in the same way. Thus, these features (shape index and polynomial coefficients) are in the class of dense features.

The first 3 polynomial coefficients contain the second order surface variation. Feature images containing feature vectors of the first 3 coefficients are given here as an example. Feature vectors of any configuration of these coefficients can be handled similarly as in the following lines.

Similarity of polynomial coefficients is computed with a method based on vector inner product. This method is formulated as:

In Equation (13), a(x, y) and ¾ (x, y)are the vectors that contain the first 3 polynomial coefficients. The expression (a, b) denotes inner product of two vectors and || a|| denotes vector £₂ norm. The similarity values takes values within [-1,1] interval.

When comparing reference and test evidences, reference (or test) evidence region is rotated. Polynomial coefficients have to be modified according to the rotation. For this purpose, the relation between rotated and original coefficients has to be determined.

Writing Equation (8) in matrix-vector representation,

Equation (14) is obtained. In Equation (14), /represents the height value at (x, y) coordinate. When the surface patch centered on a pixel is rotated by Θ £ [— π, π] radians around the region (evidence) center, pixel coordinates change accordingly and the surface patch rotates around the center pixel by Θ £ [— π, π] radians. Polynomial coefficients are calculated in terms of the surface patch's coordinate system; therefore, accounting for the rotation of the surface patch is sufficient. Let' s denote the coordinate of a pixel with respect to the coordinate system of the surface patch before the rotation as The

relation between these two coordinates is,

or

Equation (15) is written as Equation (16) and is replaced in Equation (14).Thus, Equation (17) is obtained.

When Equation (17) is written in terms of the coefficients corresponding to the rotated coordinates,

is obtained. Primed coefficients, such as <¾', represents the coefficients of the rotated surface patch. The relation between coefficients before and after rotating the surface patch is given in Equations (19-21).

. (21)

Shape index is related to the first 3 polynomial coefficients. It can be obtained from these three coefficients according to Equation (22).

Rotation invariance of shape index can be seen by replacing relations in Equation (19) in Equation (22).

Shape index and the first 3 polynomial coefficients represent the second-order changes of the surface. However, shape index is invariant to rotation and scale while polynomial coefficients are not. Although this makes shape index more appealing in terms of computational efficiency, polynomial coefficients can distinguish shapes in different scales. Shape index can be thought as a value derived from the first 3 polynomial coefficients.

In the methodology explained up until here, feature extraction using surface height map data is explained. Apart from this, when surface height map is not available, these features can be extracted using surface normal vectors. As an example, surface normal vectors are generated using photometric stereo method (U. Sakarya, U.M. Leloglu, E. Tunali, "Three-dimensional surface reconstruction for cartridge cases using photometric stereo", Forensic Science International, vol. 175, no. 2-3, pp. 209-217, 5 March 2008). The polynomial fitting is done in terms of the coefficients of the tangent plane to the surface at the pixel location. For this purpose, first order gradients, p and q in Equations (23-24), are used for fitting. ^A dh

p =— = 2a₁x + a₃ y + a₄ , (23) dx dh „

— = 2a₂y + a₃x + a₅ . (24) dy

For instance, the solution for 3x3 window size is detailed in Equation (25). Using Equation (25), the coefficients, a_{l 5} a₂, a₃, a₄, as are estimated.

The system of equations can be solved using least squares method and/or singular value decomposition.

Surface normals can be obtained by using photometric stereo as follows (U. Sakarya, U.M. Leloglu, E. Tunali, "Three-dimensional surface reconstruction for cartridge cases using photometric stereo", Forensic Science International, vol. 175, no. 2-3, pp. 209-217, 5 March 2008):

If the number of images with different illuminations equals to 3 (for Lambertian surfaces), then the irradiance equation can be given as:

E = p S n (26) where E is the irradiance values at a pixel on the surface, p is the albedo (reflectance factor), S is the light source illumination direction matrix and n is the surface normal at that pixel. The equation can be solved as

P = S-^lE\

(27) and n =—S^~1E

(28)

If the number of images with different illuminations is higher than 3, the irradiance equation can be solved using the least squares estimation:

P S^PE\

(29) n = - S^PE

(30) where S^P is the pseudo-inverse of S and can be calculated as

(31)

INDUSTRIAL APPLICABILITY

As the number of items increases by time, MG comparison operation takes longer time that becomes an important problem. The necessity arises to compare large numbers of MG data with a computerized system fast and accurately enough in an automatic or semi- supervised manner. This invention is about the methods and features to compare MG surface height map (or MG surface normal vectors) to place sister MGs in top positions in the sorted list. With this invention, it is aimed to reduce workload of forensic ballistics experts by narrowing the number of items to be examined in a semi-supervised manner.

Claims

1. A method which, compares and computes similarity of the surface height matrices (SHM) of cartridge cases and/or bullets expelled from firearms in forensic science or marks on the parts thereof (MG), characterized by the use of shape index matrices in similarity computation of two SHM MGs, the similarity of which are to be measured.

2. A method which, compares and computes similarity of the surface normal matrices (SNM) of cartridge cases and/or bullets expelled from firearms in forensic science or marks on the parts thereof (MG), characterized by the use of shape index matrices in similarity computation of two SNM MGs, the similarity of which are to be measured.

3. A method, as described in Claim 1 or Claim 2, characterized by, shape index matrices in similarity computation of two SHM/SNM MGs, when one shape index matrix is fixed, another shape index matrix is rotated and translated; and the rotation angle and the translation values that give the maximum similarity are found.

4. A method as described in Claim 1 or Claim 2, characterized by the use of at least one of the following methods in similarity computation of two SHM/SNM MGs:

a. Normalized cross-correlation,

b. Cross-correlation,

c. Mean square distance method,

d. Mean absolute difference,

e. Average cosine similarity,

f. Variance of the difference.

5. A method which, compares and computes similarity of the surface height matrices (SHM) of cartridge cases and/or bullets expelled from firearms in forensic science or marks on the parts thereof (MG), characterized by, a quadratic (or higher degree) bi- variate polynomial is fitted to each pixel of the SHM using a neighborhood of the pixel and the use of these polynomial coefficients matrices in similarity computation of two SHM MGs, the similarity of which are to be measured.

6. A method which, compares and computes similarity of the surface normal matrices (SNM) of cartridge cases and/or bullets expelled from firearms in forensic science or marks on the parts thereof (MG), characterized by, a quadratic (or higher degree) bi- variate polynomial is fitted to each pixel of the SNM using a neighborhood of the pixel and the use of these polynomial coefficients matrices in similarity computation of two SNM MGs, the similarity of which are to be measured.

7. A method, as described in Claim 5 or Claim 6, characterized by, the use of the first 3 polynomial coefficients (which model the second order surface variations) in similarity computation of two SHM/SNM MGs.

8. A method as described in Claim 7, characterized by the use of following formula in similarity computation of two SHM/SNM

9. A method, as described in Claim 5 or Claim 6, characterized by using the polynomial coefficients matrices as described in Claim 5 or Claim 6 (SHM/SNM MGs), when one of the polynomial coefficients matrix is fixed, the other polynomial coefficients matrix is rotated and translated; and the rotation angle and the translation values that give the maximum similarity are found.

10. A method which, compares and computes similarity of images in image registration, characterized by, a quadratic (or higher degree) bi-variate polynomial is fitted to each point of the surface height matrices/images using a neighborhood of the point and the use of these polynomial coefficient matrices in similarity computation of two surface height matrices/images, the similarity of which are to be measured.

11. A method, as described in Claim 2 or Claim 6, characterized by, the use of photometric stereo technique in order to get surface normal matrices (SNM) of cartridge cases and/or bullets expelled from firearms in forensic science or marks on the parts thereof (MG).