Footwear Evidence and 

Modelling Accidentals

Neil Spencer

Jared Murray

Steve Fienberg
2

Motivation: Are these a match?

CRIME SCENE PRINT SUSPECT’S SHOE
CRIME SCENE PRINT SUSPECT’S SHOE

Motivation: Are these a match?

Class Characteristics

Brand, Type, and Size

Step 1

3
4

Motivation: Are these a match?

CRIME SCENE PRINT SUSPECT’S SHOE

Class Characteristics

Brand, Type, and Size

Step 1
5 CRIME SCENE PRINT SUSPECT’S SHOE

Motivation: Are these a match?

Class Characteristics

Brand, Type, and Size

Step 1

 Accidentals (RACS)

Holes, Cuts, Scrapes, etc.

Step 2
6 CRIME SCENE PRINT SUSPECT’S SHOE

Motivation: Are these a match?

Class Characteristics

Brand, Type, and Size

Step 1

Holes, Cuts, Scrapes

Step 2

1

2

6

5

4

3

1

2 6

5

4

3

8 7 8

7

 Accidentals (RACS)
Similarity assessed by likelihood ratio

P( )

MATCH

P( )

NO MATCH

=

= =
Similarity assessed by likelihood ratio

P(Suspect Shoe | RACs)

Σ P(Other Shoes | RACs)

P(RACs | S.S.)P(S.S.)

Σ P(RACs | O.S.)P(O.S.)

=

=

8
Similarity assessed by likelihood ratio

P(Suspect Shoe | RACs)

Σ P(Other Shoes | RACs)

P(RACs | S.S.)P(S.S.)

Σ P(RACs | O.S.)P(O.S.)

=

This is the focus of our work

=
We are developing a statistical model for

P( | )

Probability of Accidentals given the Shoe 

10
We are developing a statistical model for

P( | ) =

Example

Probability of Accidentals given the Shoe 

11
12

The Existing Approach 

Journal of Forensic Identification

65 (4), 2015 \ 673

•	 Orientation: Excluding point characteristics, a defect 

shaped defect that has some different angular orientaorientation, differentiating it from another similarly of any particular shape will have a specific rotational 

-

perpendicular to that line, or rotated to any intermedia sole may be parallel with a line from toe to heel, or tion. For example, a 25 mm-long, straight-sliced cut in 

-

ate angle.

In the course of a shoe print examination, these variables are 

Hypothetical Shoe and Printsimilar characteristics on a suspect shoe. the identity or nonidentity of a shoe print when compared with with the class characteristics, enable an examiner to determine with all the other defects. These individual characteristics, along considered independently for each defect and then in combination 

To begin a probabilistic analysis of these types of individual 

used with the following assumptions:characteristics, a hypothetical shoe and its related shoe print is 

•	 The shoe is

a flat-soled, athletic shoe, men’s size

8

1⁄2

(US).

•	 The surface area of

a print made by the shoe is 16,000

sq mm (Figure 1).

Figure 1

Hypothetical shoe with 16,000 sq mm grid.

l the other defects. These individual characteristics, along 

he class characteristics, enable an examiner to determine 

ntity or nonidentity of a shoe print when compared with 

r characteristics on a suspect shoe. 

hetical Shoe and Print

begin a probabilistic analysis of these types of individual 

teristics, a hypothetical shoe and its related shoe print is 

with the following assumptions:

	 The shoe is a flat-soled, athletic shoe, men’s size 8 1⁄2

(US).

	 The surface area of a print made by the shoe is 16,000

sq mm (Figure 1).

Figure 1

Hypothetical shoe with 16,000 sq mm grid.

Jl f FiIdtifiti

Reprinted from Journal of Forensic Identification, 2006, 56 (4), 577–599

Article

Footwear Examinations: Mathematical 

Probabilities of Theoretical Individual 

Characteristics

Rocky S. Stone

Albuquerque Metropolitan Crime Laboratory (retired)

Albuquerque, NM

Abstract: The trend in the forensic sciences favors objectivity 

over subjectivity. Courts in the United States are becoming increasingly hesitant to accept the opinion of an examiner who states, “It’s 

a ‘match’ because I say it’s a ‘match’”. Objectivity, in most cases, is 

reinforced by quantification. The individual characteristics that appear 

on a shoe print or shoe impression can be quantified using two primary variables. Their location on the print and their configuration and 

orientation yield measurable, discriminating data values. Theoretical 

types of individual characteristics that are found on shoe prints are 

described and discussed, and a hypothetical model is presented with 

probability estimates applied to quantify the likelihood of occur- rence of the characteristics. With marks or combinations of marks 

of reasonable complexity, the magnitudes of the resultant numbers, 

though entirely abstract and based upon conservative assumptions, 

are remarkable.

Introduction

The presence of accidental, random defects on a shoe leaving 

a print* may allow an examiner to “positively identify” that 

particular shoe to the exclusion of all other shoes as having 

created the print. The assumed underlying premise is that 

nature never repeats itself. When physical entities, both natural 

and man-made, are examined in sufficiently fine detail, the 

* Throughout this discussion, unless specifically differentiated, the 

term “print” will be used to refer to both three-dimensional shoe 

impressions and two-dimensional prints.

(Stone 2006) Model Set-up

• Uniform density

• Independently distributed

• on hypothetical contact
13

The Existing Approach 

Journal of Forensic Identification

65 (4), 2015 \ 673

•	 Orientation: Excluding point characteristics, a defect 

shaped defect that has some different angular orientaorientation, differentiating it from another similarly of any particular shape will have a specific rotational 

-

perpendicular to that line, or rotated to any intermedia sole may be parallel with a line from toe to heel, or tion. For example, a 25 mm-long, straight-sliced cut in 

-

ate angle.

In the course of a shoe print examination, these variables are 

Hypothetical Shoe and Printsimilar characteristics on a suspect shoe. the identity or nonidentity of a shoe print when compared with with the class characteristics, enable an examiner to determine with all the other defects. These individual characteristics, along considered independently for each defect and then in combination 

To begin a probabilistic analysis of these types of individual 

used with the following assumptions:characteristics, a hypothetical shoe and its related shoe print is 

•	 The shoe is

a flat-soled, athletic shoe, men’s size

8

1⁄2

(US).

•	 The surface area of

a print made by the shoe is 16,000

sq mm (Figure 1).

Figure 1

Hypothetical shoe with 16,000 sq mm grid.

l the other defects. These individual characteristics, along 

he class characteristics, enable an examiner to determine 

ntity or nonidentity of a shoe print when compared with 

r characteristics on a suspect shoe. 

hetical Shoe and Print

begin a probabilistic analysis of these types of individual 

teristics, a hypothetical shoe and its related shoe print is 

with the following assumptions:

	 The shoe is a flat-soled, athletic shoe, men’s size 8 1⁄2

(US).

	 The surface area of a print made by the shoe is 16,000

sq mm (Figure 1).

Figure 1

Hypothetical shoe with 16,000 sq mm grid.

Jl f FiIdtifiti

Reprinted from Journal of Forensic Identification, 2006, 56 (4), 577–599

Article

Footwear Examinations: Mathematical 

Probabilities of Theoretical Individual 

Characteristics

Rocky S. Stone

Albuquerque Metropolitan Crime Laboratory (retired)

Albuquerque, NM

Abstract: The trend in the forensic sciences favors objectivity 

over subjectivity. Courts in the United States are becoming increasingly hesitant to accept the opinion of an examiner who states, “It’s 

a ‘match’ because I say it’s a ‘match’”. Objectivity, in most cases, is 

reinforced by quantification. The individual characteristics that appear 

on a shoe print or shoe impression can be quantified using two primary variables. Their location on the print and their configuration and 

orientation yield measurable, discriminating data values. Theoretical 

types of individual characteristics that are found on shoe prints are 

described and discussed, and a hypothetical model is presented with 

probability estimates applied to quantify the likelihood of occur- rence of the characteristics. With marks or combinations of marks 

of reasonable complexity, the magnitudes of the resultant numbers, 

though entirely abstract and based upon conservative assumptions, 

are remarkable.

Introduction

The presence of accidental, random defects on a shoe leaving 

a print* may allow an examiner to “positively identify” that 

particular shoe to the exclusion of all other shoes as having 

created the print. The assumed underlying premise is that 

nature never repeats itself. When physical entities, both natural 

and man-made, are examined in sufficiently fine detail, the 

* Throughout this discussion, unless specifically differentiated, the 

term “print” will be used to refer to both three-dimensional shoe 

impressions and two-dimensional prints.

(Stone 2006) Model Set-up

Drawbacks

• Not based on data

• No use of contact surface

• Uniform density

• Independently distributed

• on hypothetical contact
14

386 marked, aligned, and normalized impressions

of men’s shoes from real cases by the Israeli Police

Data

Two parts: Accidentals and Contact Surface
15

Newly Proposed Model

x

y

f(x, y) = X

k1

i=1

X

k2

j=1

⇡ijBeta(x|i, k1  i)Beta(y|j, k2  j)
16

Newly Proposed Model

x

y

f(x, y) = X

k1

i=1

X

k2

j=1

⇡ijBeta(x|i, k1  i)Beta(y|j, k2  j)
17

Newly Proposed Model

x

y

• Contact Surface

• Location

f(x, y) = X

k1

i=1

X

k2

j=1

⇡ijBeta(x|i, k1  i)Beta(y|j, k2  j)

Controls height of basis function

Depends on:
18

Newly Proposed Model

x

y

f(x, y) = X

k1

i=1

X

k2

j=1

⇡ijBeta(x|i, k1  i)Beta(y|j, k2  j)

• Contact Surface

• Location

Controls height of basis function

Depends on:
P( | )

The “Foot” Component

P( | ) P( | )

19
P( | ) P( | ) P( | )

20

The “Foot” Component
P( | ) P( | ) P( | )

=

==

21

The “Foot” Component
Contact ij ⇥ ⇥ Foot ij

Model for Weights

⇡ij =

22

Shoe Specific

Noise ij
Contact ij ⇥ ⇥ Foot ij

Model for Weights

⇡ij =

23

i

j

=0

=1

Six levels

depending

on nearby

contact

intensity

Shoe Specific

Noise ij

=2

=3

=4

=5
Contact ij ⇥ ⇥ Foot ij

Model for Weights

⇡ij =

24

i

j

Six levels

depending

on nearby

contact

intensity

Shoe Specific

Noise ij

j

i

can explain 

clusters =0

=1

=2

=3

=4

=5
Contact ij ⇥ ⇥ Foot ij

Model for Weights

⇡ij =

25

i

j

Six levels

depending

on nearby

contact

intensity

Shoe Specific

Noise ij

j

i

can explain 

clusters

Common across

all Shoes

j

i

=0

=1

=2

=3

=4

=5
Results of Fit (for 386 shoes)

26

Contact Surface Variables

=5

=4

=3

=2

=1

=0
Results

The “Foot” Component
Results

The “Foot” Component

with an example shoe
Results

Example Predictive Distribution for Shoe
Results

Example Predictive Distribution for Shoe

Actual Accidental Locations
Results

Example Predictive Distribution for Shoe
Results

Example Predictive Distribution for Shoe

Actual Accidental Locations
Conclusion

P( | ) =

We developed a model for accidentals given contact surface

It features the contact surface variables and a “foot” variable

~0.88 ~0.35 ~0.02 <0.01 <0.01 <0.01
Thank you

34

Sarena Wiesner Yoram Yekutieli Yaron Shor