Efficient 3D object classification by using direct Krawtchouk moment invariants

Abstract

In this paper, we present an efficient set of moment invariants, named Direct Krawtchouk Moment Invariants (DKMI), for 3D objects recognition. This new set of invariants can be directly derived from the Krawtchouk moments, based on algebraic properties of Krawtchouk polynomials. The proposed computation approach is effectively compared with the classical method, which rely on the indirect computation of moment invariants by using the corresponding geometric moment invariants. Several experiments are carried out so as to evaluate the performance of the newly introduced invariants. Invariability property and noise robustness are firstly investigated. Secondly, the numerical stability is discussed. Then, the performance of the proposed moment invariants as pattern features for 3D object classification is compared with the existing Geometric, Krawtchouk, Tchebichef and Hahn Moment Invariants. Finally, a comparative analysis of computational time of these moment invariants is illustrated. The obtained results demonstrate the efficiency and the superiority of the proposed method.

Appendices

Appendix A: Proof of proposition 1

With the help of (41), the translated version of Krawtchouk polynomials can be expressed by a linear combination of monomials as

$$ k_{n}(x-x_{0};p,N) =\sum\limits_{i = 0}^{n} C_{n,i} (x-x_{0})^{i}, $$

(59)

According to the binomial theorem, it is possible to expand any power of x − x₀ into a sum of the form:

$$ (x-x_{0})^{i}=\sum\limits_{s = 0}^{i}{i \choose s} (-1)^{i-s} x^{s} x_{0}^{i-s}, $$

(60)

and with the help of Proposition 1 we can express x^s in terms of Krawtchouk polynomials. Hence, (59) can be written as:

$$ k_{n}(x-x_{0};p,N) =\sum\limits_{i = 0}^{n}\sum\limits_{s = 0}^{i} \sum\limits_{u = 0}^{s} {i \choose s}C_{n,i}D_{s,u} (-1)^{i-s} x_{0}^{i-s} k_{u}(x;p,N). $$

(61)

Similarly we can deduce that,

$$ k_{m}(y-y_{0};p,M) =\sum\limits_{j = 0}^{m}\sum\limits_{t = 0}^{j} \sum\limits_{v = 0}^{t} {j \choose t}C_{m,j}D_{t,v} (-1)^{j-t} y_{0}^{j-t} k_{v}(y;p,M) $$

(62)

and

$$ k_{k}(z-z_{0};p,K) =\sum\limits_{e = 0}^{k}\sum\limits_{f = 0}^{e} \sum\limits_{w = 0}^{f} {e \choose f}C_{k,e}D_{f,w} (-1)^{e-f} z_{0}^{e-f} k_{w}(z;p,N). $$

(63)

As a consequence, by substituting (61), (62) and (63) into (49), we can write $KM_{nmk}^{t}$ of a translated image in terms of KM_uvw of the original image as:

$$ \begin{array}{ll} KM_{nmk}^{t}= &\displaystyle \sum\limits_{i = 0}^{n}\sum\limits_{j = 0}^{m}\sum\limits_{e = 0}^{k} \sum\limits_{s = 0}^{i}\sum\limits_{t = 0}^{j}\sum\limits_{f = 0}^{e} \sum\limits_{u = 0}^{s}\sum\limits_{v = 0}^{t}\sum\limits_{w = 0}^{f} {i \choose s} {j \choose t} {e \choose f} \\ &\displaystyle \times C_{n,i}C_{m,j}C_{k,e}D_{s,u}D_{t,v}D_{f,w} (-1)^{i-s+j-t+e-f} x_{0}^{i-s} y_{0}^{j-t} z_{0}^{e-f} KM_{uvw}. \end{array} $$

(64)

Therefore, the proof is completed.

Appendix B: Proof of proposition 2

With the help of (40), the deformed version of Krawtchouk polynomials can be expressed as follows:

$$ k_{n}(a_{11} x + a_{12} y + a_{13} z;p,N) =\sum\limits_{i = 0}^{n} C_{n,i} (a_{11} x + a_{12} y + a_{13} z)^{i}. $$

(65)

By using the trinomial theorem, it is possible to write the (65) as:

$$ k_{n}(a_{11} x + a_{12} y + a_{13} z;p,N) =\sum\limits_{i = 0}^{n}\sum\limits_{s = 0}^{i} \sum\limits_{u = 0}^{s} \frac{i!}{s!u!(i-s-u)!} C_{n,i} a_{11}^{s} a_{12}^{u} a_{13}^{i-s-h} x^{s} y^{u} z^{i-s-h}. $$

(66)

Similarly, we can deduce that,

$$ k_{m}(a_{21} x + a_{22} y + a_{23} z;p,M) =\sum\limits_{j = 0}^{m}\sum\limits_{t = 0}^{j} \sum\limits_{v = 0}^{t} \frac{j!}{t!v!(j\,-\,t\,-\,v)!} C_{m,j} a_{21}^{t} a_{22}^{v} a_{23}^{j-t-v} x^{t} y^{v} z^{j-t-v}, $$

(67)

and

$$ k_{k}(a_{31} x + a_{32} y + a_{33} z;p,K) \,=\,\sum\limits_{e = 0}^{k}\sum\limits_{f = 0}^{e} \sum\limits_{w = 0}^{f} \frac{e!}{f!w!(e\,-\,f\,-\,w)!} C_{k,e} a_{31}^{f} a_{32}^{w} a_{33}^{e-f-w} x^{f} y^{w} z^{e-f-w}. $$

(68)

Hence, by substituting (66), (67) and (68) into (54), we can write $KM_{nmk}^{t}$ of a deformed image in terms of KM_uvw of the original image as:

$$ \begin{array}{ll} KM_{nmk}^{d}= &\displaystyle \sum\limits_{i = 0}^{n}\sum\limits_{j = 0}^{m}\sum\limits_{e = 0}^{k} \sum\limits_{s = 0}^{i}\sum\limits_{t = 0}^{j}\sum\limits_{f = 0}^{e} \sum\limits_{u = 0}^{s}\sum\limits_{v = 0}^{t}\sum\limits_{w = 0}^{f} \sum\limits_{r = 0}^{\delta}\sum\limits_{l = 0}^{\sigma}{\sum}_{d = 0}^{\epsilon} \\ &\displaystyle \frac{i!}{s!u!(i-s-u)!} \frac{j!}{t!v!(j-t-v)!} \frac{e!}{f!w!(e-f-w)!} C_{n,i}C_{m,j}C_{k,e} \\ &\displaystyle \times D_{\delta,r}D_{\sigma,l}D_{\epsilon,d} a_{11}^{s} a_{12}^{u} a_{13}^{i-s-h} a_{21}^{t} a_{22}^{v} a_{23}^{j-t-v} a_{31}^{f} a_{32}^{w} a_{33}^{e-f-w} KM_{rld}~, \end{array} $$

(69)

where δ = s + t + f, σ = u + v + w and 𝜖 = i − s − h + j − t + e − f − w.

The proof is completed.