*2.2.3 Top hat transform*

The concept of Top-Hat, due to F. Meyer, is a residue intended to eliminate the slow variations of the signal, or to amplify the contrasts. It therefore applies mainly to functions (**Figure 5**).

We call Top-Hat the residual between the identity and an invariant opening under vertical translation [43]:

*Mathematical Morphology and the Heart Signals DOI: http://dx.doi.org/10.5772/intechopen.104113*

$$
\rho(f) = f - (f \circ B) \tag{6}
$$

We define in the same way a dual top-hat, residue between a closure and the identity:

$$
\rho(f) = (f \bullet B) - f \tag{7}
$$

#### *2.2.4 Multi-scale morphology*

*f* and *s*, represent respectively, a discrete signal and the structuring element (SE) for a morphological analysis. The morphology operator *R*, based on multi-scale analysis [44, 45], can be defined as a set *Tλ*j*λ*>0, *λ*∈ *N*, where *Rλ*ð Þ¼ *f λR f* ð Þ *=λ* .

Multi-scale erosion and dilation are defined by:

$$((f \ominus s)\_\lambda = \lambda [(f/\lambda) \ominus s] = f \ominus \lambda \text{s} \tag{8}$$

$$((f \oplus \mathfrak{s})\_\lambda = \lambda[(f/\lambda) \oplus \mathfrak{g}] = f \oplus \lambda \mathfrak{s} \tag{9}$$

and *λg* ¼ *s* ⊕ *s* ⊕ … … *:* ⊕ *s*ð Þ *λ* � 1 *times* .

The original purpose of multi-scale morphology analysis is based on the morphology composition of the structuring element g is to improve the speed of morphology

#### **Figure 4.**

*a) Opening signal (lime draw* f original*). b) Closing signal.*

**Figure 5.** *Opening and top hat.*

analysis by a large scale of structuring element and to expand the domains of application in signal processing.

The original purpose of multi-scale morphology analysis is based on the morphology composition of the structuring element g is to improve the speed of morphology analysis by a large scale of structuring element and to expand the domains of application in signal processing.
