# Pattern Measurement Theory of Stochastic Frequency Distribution Analysis (SFDA)

The underlying algorithm for all applications is the Stochastic Frequency Distribution Analysis (SFDA) developed by Verity IA. It is employed with different operation constants for the wide variety of patterns to which it is applied.

The Stochastic Frequency Distribution Analysis measurement algorithm has been applied to:

- Formation
- Transmitted Light
- Formation X-ray
- Prufbau Strips
- IGT Strips
- Calendar Blackening

## What is Stochastic Analysis?

**Stochastic:** Derives from the Greek, stochas, for target.

Imagine a target shoot with nine (9) marksmen. Each marksman has the same number of shots: nine (9). In this match, the marksman’s skill is determined by then Standard Deviation ( σ or Error) of the distance the marksman’s nine shots are from the center of the target.

Statistically, the marksman’s skill is determined by the Standard Deviation of the distance the marksman’s nine shots are from the center of the target.

The σ or Error for each target, arranged as a 2 dimensional data array:

4 | 6 | 5 |

8 | 4 | 2 |

3 | 3 | 3 |

Each target score, σ _{1 to n}, is used to calculate the Standard Deviation, SD_{σ}, and the Mean, X_{σ} (or team score) for the group of targets.

In a digital image the target has 256 imaginary rings determined by the 8 bit luminance value for each picture point, where Black = 0 & White = 255

As each arrow hits this target it strikes one of 256 rings and the score is recorded as a luminance value.

## Stochastic Distribution

- The square target area is moved across the image in a regular pattern of rows and columns to form a uniform 2D matrix
- The Differences among the picture point luminance values (LV) within each target is calculated and saved in a 2D vector
- The Standard Deviation and Average of the target Differences are two of the three terms used compute the Pattern Measurement Number

**The Area of Interest (AOI) is covered with contiguous targets**

Statistical data from each target are saved in a 2D array for subsequent computation of overall Pattern Number.

**Directional Orientation**

The 2D array can also be used to extract the horizontal (CD) and vertical (MD) variations

## Applying Stochastic Frequency Distribution Analysis (SFDA) to Pattern Measurement

Which Image has the most distinctive pattern?

A, B & C have exactly the same number of each target luminance values (LV), but they are distributed differently within the inspection area.

ISO 13660 Mottle (A) = ISO 13660 Mottle (B) = ISO 13660 Mottle (C)

**ISO 13660 provides the same number for each pattern. ISO 13660 fails when determining the most distinctive pattern!
**

## Using SFDA to find the most distinctive pattern

Verity IA SFDA based Pattern Measurement works on a digital image of any size and recognizes each pixel as a separate measurement unit. Luminance Value (LV) is the digital value of the measurement element on a scale of 0 to 255

## What Characteristic Differentiates These Images?

## Pattern Measurement

## Pattern Measurement – Basic Premise 2

### Transitions within the Image

The absolute difference in the luminance values among the four (4) picture elements within a 2 element x 2 element target is an index of the three dimensional rate of change. The standard deviation of these indices and their average are two terms in the Pattern Number calculation.

### Spatial Luminance Variance (LV)

The average LV for these same four (4) picture elements is used to create a new element stored in a new data base ¼ the size of the original image. The standard deviation among these new elements is the spatial distribution component in the Pattern Number calculation.

### Building the Target Size Layer

The four (4) elements within the tile area, averaged together. These averages are then used to create a new virtual image or layer dedicated to the target size.Each target in the new layer is twice the physical width & height of the original but remains 2 elements x 2 elements.