K-harmonic means clustering algorithm using feature weighting for color image segmentation

Abstract

This paper mainly proposes K-harmonic means (KHM) clustering algorithms using feature weighting for color image segmentation. In view of the contribution of features to clustering, feature weights which can be updated automatically during the clustering procedure are introduced to calculate the distance between each pair of data points, hence the improved versions of KHM and fuzzy KHM are proposed. Furthermore, the Lab color space, local homogeneity and texture are utilized to establish the feature vector to be more applicable for color image segmentation. The feature group weighting strategy is introduced to identify the importance of different types of features. Experimental results demonstrate the proposed feature group weighted KHM-type algorithms can achieve better segmentation performances, and they can effectively distinguish the importance of different features to clustering.

Appendices

Appendix A

In this Appendix, the detailed derivations for obtaining the update equation of c _j and w _q are provided. At first, the partial derivation by c _j of L is calculated as follows.

$$ \frac{{\partial L}}{{\partial {{\boldsymbol{c}}_{j}}}} = - kp\sum\limits_{i = 1}^{n} {\frac{{{{\left[d_{ij}^{({\boldsymbol{w}})}\right]}^{-p-2}}{\text{diag}}({{\boldsymbol{w}}^{2}})({{\boldsymbol{x}}_{i}} - {{\boldsymbol{c}}_{j}})}} {{{{\left( {\sum\nolimits_{j = 1}^{k} {{{\left[d_{ij}^{({\boldsymbol{w}})}\right]}^{- p}}}} \right)}^{2}}}}} $$

(A.1)

As we can see from the above equation, diag(w ²) is not related to variable i and $\frac {{{{\left [d_{ij}^{({\boldsymbol {w}})}\right ]}^{- p - 2}}}}{{{{\left ({\sum \nolimits _{j = 1}^{K} {{{\left [d_{ij}^{({\boldsymbol {w}})}\right ]}^{- p}}}} \right )}^{2}}}} {{ \;=\;} }{m_{WKHM}}({{{{\boldsymbol {c}}_{j}}} \left / {{{\boldsymbol {x}}_{i}}}\right .}) \cdot {w_{WKHM}}({{\boldsymbol {x}}_{i}})$, thus the equation of c _j is obtained as (2) by letting (A.1) be equal to 0. But it should be noted that the Euclidian distance d _{i j} is replaced by $d_{ij}^{(\boldsymbol {w})}$ in m _{W K H M} (c _j /x _i ) and w _{W K H M} (x _i ).

$$ {m_{WKHM}}({{{{\boldsymbol{c}}_{j}}} \left/ {{{\boldsymbol{x}}_{i}}}\right.}) = \frac{{{{\left[d_{ij}^{({\boldsymbol{w}})}\right]}^{- p - 2}}}}{{\sum\nolimits_{j = 1}^{k} {{{\left[d_{ij}^{({\boldsymbol{w}})}\right]}^{- p - 2}}}} } $$

(A.2)

$$ {w_{WKHM}}({{\boldsymbol{x}}_{i}}) = \frac{{\sum\nolimits_{j = 1}^{k} {{{\left[d_{ij}^{({\boldsymbol{w}})}\right]}^{- p - 2}}}} }{{{{\left( \sum\nolimits_{j = 1}^{k} {{{\left[d_{ij}^{({\boldsymbol{w}})}\right]}^{- p}}} \right)}^{2}}}} $$

(A.3)

Then, letting the partial deviation by w _q of L, which is denoted as (A.4), to be equal to 0, hence the update equation of w _q is obtained as (A.5), where the Lagrange multiplier λ should be eliminated.

$$ \frac{{\partial L}}{{\partial{w_{q}}}}=kp{w_{q}}\sum\limits_{i=1}^{n} {\frac{{\sum\nolimits_{j=1}^{k} {\left( {{{\left[d_{ij}^{({\boldsymbol{w}})}\right]}^{-p-2}}\cdot {{({x_{iq}}-{c_{jq}})}^{2}}} \right)}}}{{{{\left( {\sum\nolimits_{j = 1}^{k} {{{\left[d_{ij}^{({\boldsymbol{w}})}\right]}^{- p}}}} \right)}^{2}}}}}-\lambda $$

(A.4)

$$ {w_{q}} = \frac{\lambda}{{kp\sum\limits_{i=1}^{n} {\frac{{\sum\nolimits_{j=1}^{k} {\left( {{{\left[d_{ij}^{({\boldsymbol{w}})}\right]}^{-p-2}}\cdot{{({x_{iq}}-{c_{jq}})}^{2}}}\right)}}}{{{{\left( {\sum\nolimits_{j=1}^{k} {{{\left[d_{ij}^{({\boldsymbol{w}})}\right]}^{-p}}}} \right)}^{2}}}}}} } $$

(A.5)

In terms of the constraint of feature weights ${w_{q}} \in [0,1],\;\sum \limits _{q = 1}^{d} {{w_{q}} = 1}$, in which the (A.5) is substituted and the calculation of λ is obtained as follows.

$$ \lambda = \sum\limits_{l = 1}^{d} {kp\sum\limits_{i = 1}^{n} {\frac{{\sum\nolimits_{j = 1}^{k} {\left( {\left[d_{ij}^{({\boldsymbol{w}})} \right]^{- p - 2} \cdot (x_{iq} - c_{jq} )^{2}} \right)}} }{{\left( {\sum\nolimits_{j = 1}^{k} {\left[d_{ij}^{({\boldsymbol{w}})} \right]^{- p}} } \right)^{2}} }}} $$

(A.6)

Therefore, the (A.6) is substituted in (A.5) to obtain the update equation of w _q (q = 1, 2, …, d) shown as (9).

Appendix B

In this Appendix, the detailed derivation for obtaining the update equation (20) is provided. First, the partial derivation by c _j (j = 1, 2, …, K) of L ₂ is calculated and the result is set to be 0, then the update equation of cluster centers can also be obtained with the same form as (2), where $d_{ij}^{(g\boldsymbol {w})}$ is utilized in m _{W K H M} (c _j /x _i ) and w _{W K H M} (x _i ). For the computation of feature weights, we firstly analyze the case of q ∈ G(1), the partial deviation by w _q of L ₂ is calculated as follows.

$$ \frac{{\partial {L_{2}}}}{{\partial {w_{q}}}} = kp{w_{q}}\sum\limits_{i = 1}^{n} {\frac{{{v_{1}}\sum\nolimits_{j = 1}^{k} {\left( {{{\left[d_{ij}^{(g{\boldsymbol{w}})}\right]}^{- p - 2}} \cdot {{({x_{iq}} - {c_{jq}})}^{2}}} \right)}} } {{{{\left( {\sum\nolimits_{j = 1}^{k} {{{\left[d_{ij}^{(g{\boldsymbol{w}})}\right]}^{-p}}}} \right)}^{2}}}}}-{\lambda_{1}} $$

(B.1)

Then, letting the value of (B.1) to be 0 and the equation of w _q is obtained as (B.2), which is substituted in $\sum \limits _{q \in G(1)} {{w_{q}} = 1}$, the constraint of feature group G(1), then the calculation of λ ₁ is obtained and substituted in (B.2) again, therefore the update equation of feature weights of G(1) is shown as (20).

$$ {w_{q}} = \frac{{{\lambda_{1}}}}{{kp{v_{1}}\sum\limits_{i = 1}^{n} {\frac{{\sum\nolimits_{j = 1}^{k} {\left( {{{\left[d_{ij}^{(g{\boldsymbol{w}})}\right]}^{- p - 2}} \cdot {{({x_{iq}} - {c_{jq}})}^{2}}} \right)}} }{{{{\left( {\sum\nolimits_{j = 1}^{k} {{{\left[d_{ij}^{(g{\boldsymbol{w}})}\right]}^{- p}}}} \right)}^{2}}}}}} } $$

(B.2)