Irregularity Index for Vector-Valued Morphological Operators

Abstract

Mathematical morphology is a valuable theory of nonlinear operators widely used for image processing and analysis. Although initially conceived for binary images, mathematical morphology has been successfully extended to vector-valued images using several approaches. Vector-valued morphological operators based on total orders are particularly promising because they circumvent the problem of false colors. On the downside, they often introduce irregularities in the output image. This paper proposes measuring the irregularity of a vector-valued morphological operator by the relative gap between the generalized sum of pixel-wise distances and the Wasserstein metric. Apart from introducing a measure of the irregularity, referred to as the irregularity index, this paper also addresses its computational implementation. Precisely, we distinguish between the ideal global and the practical local irregularity indexes. The local irregularity index, which can be computed more quickly by aggregating values of local windows, yields a lower bound for the global irregularity index. Computational experiments with natural images illustrate the effectiveness of the proposed irregularity indexes.

Appendix A Supervised and Unsupervised Morphological Approaches

Let us briefly review the supervised and unsupervised vector-valued morphological approaches, whose details can be found in [34, 35].

In a supervised ordering, the surjective mapping $\rho :{\mathbb {V}} \rightarrow {\mathbb {L}}$ is defined using a set $F \subset {\mathbb {V}}$ of foreground values and a set $B \subset {\mathbb {V}}$ of background values such that $F \cap B = \emptyset$. Given the sets F and B, the mapping $\rho$ is expected to satisfy the inequality $\rho ({\varvec{f}}) > \rho ({\varvec{b}})$ for ${\varvec{f}}\in F$ and ${\varvec{b}}\in B$. Considering ${\mathbb {V}} \subset {\mathbb {R}}^d$ and ${\mathbb {L}} \subset {\mathbb {R}}$, the decision function of a SVM can be used to accomplish this goal [26, 32, 34]. Precisely, consider sets $F = \{{\varvec{f}}_1,\ldots ,{\varvec{f}}_K\} \subset {\mathbb {R}}^d$ and $B = \{{\varvec{b}}_1,\ldots ,{\varvec{b}}_M\} \subset {\mathbb {R}}^d$ of foreground and background values, respectively. An SVM-based morphological approach is obtained by considering the mapping $\rho _S:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$ given by

$$\begin{aligned} \rho _{S}({\varvec{x}})= \sum _{{\varvec{f}}\in F} \alpha _i \kappa ({\varvec{x}},{\varvec{f}}) - \sum _{{\varvec{b}}\in B} \beta _j \kappa ({\varvec{x}},{\varvec{b}}), \quad \forall {\varvec{x}}\in {\mathbb {R}}^d, \end{aligned}$$

(33)

where $\kappa :{\mathbb {R}}^d \times {\mathbb {R}}^d \rightarrow {\mathbb {R}}$ is a Mercer kernel [26]. For example, Gaussian radial basis function kernel is given by

$$\begin{aligned} \kappa ({\varvec{x}},{\varvec{y}}) = e^{-\frac{1}{2\sigma }\Vert {\varvec{x}}-{\varvec{y}}\Vert _2^2}, \quad \forall {\varvec{x}}, {\varvec{y}}\in {\mathbb {R}}^d, \end{aligned}$$

(34)

where $\sigma >0$ is a parameter. Moreover, $\alpha _1,\ldots ,\alpha _K$ and $\beta _1,\ldots ,\beta _M$ solve the quadratic optimization problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \text{ maximize } &{} \displaystyle \sum _{i=1}^{K}\alpha _{i}+\sum _{j=1}^{M}\beta _{j} -\dfrac{1}{2}\sum _{i, l =1}^{K} \alpha _{i}\alpha _{l} \kappa ({\varvec{f}}_i, {\varvec{f}}_l) \\ &{}\quad - \, \dfrac{1}{2}\sum _{j, l =1}^{M} \beta _{j}\beta _{l} \kappa ({\varvec{b}}_j, {\varvec{b}}_l) \\ &{} \quad \displaystyle {+ \, \dfrac{1}{2}\sum _{i =1}^{K} \sum _{j=1}^{M}\alpha _{i}\beta _{j} \kappa ({\varvec{f}}_i, {\varvec{b}}_j)}\\ \text{ subject } \text{ to } &{} \displaystyle {\sum _{i=1}^{K}\alpha _i - \sum _{j=1}^{M}\beta _j = 0}, \\ &{} 0 \le \alpha _i, \beta _j \le C, \end{array}\right. } \end{aligned}$$

(35)

where the parameter $C>0$ controls the trade-off between the classification error and the margin of separation between background and foreground values [12, 26].

In an unsupervised morphological approach, the mapping $\rho :{\mathbb {V}} \rightarrow {\mathbb {L}}$ is determined using a set of unlabeled values. The statistical depth projection-based approach, for example, determines the mapping $\rho$ based on “anomalies” with respect to a background composed of the majority of pixel values of an image [35]. Formally, suppose ${\mathbb {V}} \subset {\mathbb {R}}^d$ and ${\mathbb {L}} \subset {\mathbb {R}}$. Given a training sample represented by a matrix $\mathbf{X } = [{\varvec{x}}_1, \ldots , {\varvec{x}}_n] \in {\mathbb {R}}^{d \times n}$, the projection depth function $\rho _P^*:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$ is defined by

$$\begin{aligned} \rho _P^*({\varvec{x}}) = \sup _{{\varvec{u}}\in {\mathbb {S}}^{d-1}} \dfrac{|{\varvec{u}}^{T}{\varvec{x}}- \text {MED}({\varvec{u}}^{T}\mathbf{X })|}{\text {MAD}({\varvec{u}}^T\mathbf{X })}, \quad \forall {\varvec{x}}\in {\mathbb {R}}^d, \end{aligned}$$

(36)

where ${\mathbb {S}}^{d-1} = \{{\varvec{x}}\in {\mathbb {R}}^d:||{\varvec{x}}||_{2} = 1\}$, $\text {MED}:{\mathbb {R}}^n \rightarrow {\mathbb {R}}$ is the median operator, and $\text {MAD}:{\mathbb {R}}^n \rightarrow {\mathbb {R}}$ is the median absolute deviation from the median operator. Recall that the median absolute deviation from the median is given by

$$\begin{aligned} \text {MAD}({\varvec{t}}) = \text {MED}(|{\varvec{t}}- {\varvec{1}}_n\text {MED}({\varvec{t}})|), \end{aligned}$$

(37)

where ${\varvec{1}}_n \in {\mathbb {R}}^n$ denotes the vector of ones and the absolute value $|\cdot |$ is computed in a component-wise manner. In practice, we compute the depth projection function by replacing the supremum with the maximum on a finite set of elements in the hypersphere ${\mathbb {S}}^{d-1}$. Formally, the function $\rho _P:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$ given by

$$\begin{aligned} \rho _P({\varvec{x}}) = \max _{{\varvec{u}}\in {\mathbb {U}}} \dfrac{|{\varvec{u}}^{T}{\varvec{x}}- \text {MED}({\varvec{u}}^{T}\mathbf{X })|}{\text {MAD}({\varvec{u}}^T\mathbf{X })}, \quad \forall {\varvec{x}}\in {\mathbb {R}}^d, \end{aligned}$$

(38)

where ${\mathbb {U}} = \{{\varvec{u}}_1, {\varvec{u}}_2, \ldots , {\varvec{u}}_k \} \subset {\mathbb {S}}^{d-1}$, is taken as an approximation of the theoretical depth projection function $\rho _P^*$. The projection depth morphological approach is defined by ranking the vector-values according to the mapping $\rho _P:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$ given by (38) together with a look-up table.