2-D shape analysis can be applied to the TFR of the ultrasonic A-scans for material characterization. The motivation of this idea was based on the observation that the mean scatterer diameter, , affects TFR shapes (see Figure 1). It is expected that using shape-related parameters applied to the TFR, we will be able to classify materials with different scatterer sizes. Additionally, if a process of binarization of TFR is employed, previous to the extraction of geometrical parameters, the obtained parameters should be less affected by noise.

The application of geometrical parameters to TFR diagrams provides an innovative way to complement classical techniques based on one dimensional US and an alternative way of obtaining similar meaning and less variance estimators (as we will show in Section 5).

After computing the TFR of the ultrasonic A-scan, binarization with an adequate threshold is done. If we assume that the ultrasonic signal recorded is contaminated with additive white Gaussian noise (AWGN), the binarized TFR will exhibit some sort of two dimensional "jitter". This jitter will affect the shape of the binarized TFR and, of course, the geometrical parameters derived from it. What we propose in this work is to choose a variable with depth-adaptive threshold located at the maximum slope of the Gaussian pulse for the region of the TFR where Gaussian pulse is higher than AWGN. This will minimize the effect of noise in the binarized shape. In the zone of the TFR where amplitude of the Gaussian pulse is comparable to AWGN power we use a constant threshold. An example of a binarized TFR using the adaptive threshold is shown in Figure 2.

Shape-related parameters are obtained from the binarized TFR matrix. These geometrical parameters depend on physical properties of the inspected material. An example of how this can be mathematically modeled is given as follows. Being Figure 2 the binarized TFR generated with the mixed threshold previously described. Let us call the binarized TFR of Figure 2. This representation can be mathematically formulated, in a first approximation, as in (13) if the threshold is properly selected. The parameters and take into account material-related parameters (, , , and ) as derived by (7) and (8). If we take into account (for simplicity) only the Stochastic scattering, we can obtain that the area of the binarized TFR, up to a given depth (), is given by equation (14). Note that to compute the area, the shift term can be omitted

For an arbitrary threshold selection, equality of (13) and (14) does not hold. However, proportionally relationship makes these expressions equally valid for classification purposes. Equation (14) shows that the higher or are, the lower the final area of the binarized response is. This simple demonstration confirms that basic geometrical descriptors can provide important information to compare material characteristics such us attenuation or mean scatterer size

We are going to see in the next section the set of geometrical parameters that allow us to classify materials according to scatterer size.

### 4.1. Geometrical Descriptors

From , the binarized TFR generated with the mixed threshold, we can calculate many geometrical descriptors [8, 9]. Our contribution, at this point, is to work with shape or geometrical parameters having a physical meaning related to the expected changes produced in the TFR. It is expected that geometrical descriptors will provide a most intuitive representation of the model in comparison with the classical signal-processing parameters. For example, we can establish visual relations between orientation parameters and physical variations of attenuation or frequency along depth.

The most representative geometrical descriptors that have proven to give good classification results are given below.

#### 4.1.1. Area

For a generic discrete function in two variables, the moments are defined as

where is the binarized TFR, at coordinates .

The area is related to attenuation parameters and mean scatterer size as it was demonstrated in (14). The area can be obtained as the zero-order moment .

If we use the area parameter to distinguish between materials with similar attenuation coefficient but including scatterers with different sizes, it is expected that materials with higher scatterer sizes get lower value of the area descriptor.

#### 4.1.2. Center of Gravity

By using first-order moments, the center of gravity or centroid of a binary representation can be calculated.

Being and , the center of mass can be defined as where

By dividing the binary representation in smaller regions, along horizontal -axis, we will be able to study the central frequency evolution with depth. Moreover, the center of gravity is used in the definition of the second-order moments as described in (17), note the invariance with respect to response scaling

#### 4.1.3. Orientation

Object orientation () can be calculated using second-order moments. It is geometrically described as the angle between the major axis of the object and the -axis. By minimizing the function , we get the next expression for the orientation

#### 4.1.4. Eccentricity

An important parameter, which is also dependent on TFR shape, is the eccentricity . The eccentricity allows to estimate how similar to a circle an object is. The value ranges from 0 to 1 (). For circular objects and for elliptical objects . To compute the eccentricity we have used the next expression

where and are, respectively, the maior and minor axis of the object, as it is depicted in Figure 3.

We can use the eccentricity parameter to distinguish between materials with similar attenuation coefficient but including scatterers with different sizes. For lower scatterer sizes eccentricity is expected to be higher than for higher scatterer size materials. Seeing TFR diagrams depicted in Figure 1, we can notice how materials with higher value of have more circular shapes than those having lower . So, it is expected that the higher the lower the eccentricity value is.

#### 4.1.5. Boundary Signature (BS)

The BS is a 1-D representation of an object boundary. One of the most simple ways to generate the BS of a region is to plot the distance from the center of gravity of the region to the boundary as a function of the angle, . Figure 3 illustrates this concept. The changes in size of binarized TFR result in changes in the amplitude values of the corresponding BS. It is expected that the higher the value of is, the lower the amplitude of the corresponding BS. Moreover, the BS not only provides information about area changes but also provides the angular direction of such changes. To compute the BS we need to compute for each angle, , the Euclidean distance between the center of gravity and the boundary of the region. As will be demonstrated, it is expected that different values of , for the model or material under test, will correspond with different BSs for the binarized TFR.

#### 4.1.6. Frequency-Derived Parameters

Some frequency-derived parameters have been also tested such as centroid frequency, central frequency, or bandwidth.

To compute the central frequency we work with the TFR in gray scale (not binarized). We divide the TFR diagram into small rectangles (see Figure 4) along the -axis (horizontal), and then we compute the maximum of each rectangle. The final result is the evolution of central frequency along depth.

To compute the centroid frequency evolution with depth we divide the binzarized TFR into small rectangles along the -axis and then we compute the center of gravity (described above) for every rectangle, the final result is the evolution of centroid frequency along horizontal direction.

The process to compute the bandwidth evolution is similar to centroid frequency computation. In the case of bandwidth the width of each rectangle is computed, thus obtaining the evolution of bandwidth with depth.