Robust cost function for optimizing chamfer masks

Baraka Jacob Maiseli; LiFei Bai; Xianqiang Yang; Yanfeng Gu; Huijun Gao

doi:10.1007/s00371-017-1367-8

The Visual Computer

May 2018, Volume 34, Issue 5, pp 617–632 | Cite as

Robust cost function for optimizing chamfer masks

Authors
Authors and affiliations

Baraka Jacob MaiseliEmail author
LiFei Bai
Xianqiang Yang
Yanfeng Gu
Huijun Gao

Original Article

First Online: 06 May 2017

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Abstract

Chamfering, a mask-driven technique, refers to a process of propagating local distances over an image to estimate a reference metric. Performance of the technique depends on the design of chamfer masks using cost functions. To date, most scholars have been using a mean absolute error and a mean squared error to formulate optimization problems for estimating weights in the chamfer masks. However, studies have shown that these optimization functions endure some potential weaknesses, including biasedness and sensitivity to outliers. Motivated by the weaknesses, the present work proposes an alternative difference function, RLog, that is unbiased, symmetrical, and robust. RLog takes the absolute logarithm of the relative accuracy of the estimated distance to compute optimal chamfer weights. Also, we have proposed an algorithm to map entries of the designed real-valued chamfer masks into integers. Analytical and experimental results demonstrate that chamfering based on our weights generate polygons and distance maps with lower errors. Methods and results of our work may be useful in robotics to address the matching problem.

Keywords

Chamfering Euclidean Mean absolute error Optimization

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Notes

Acknowledgements

Funding was provided by China Postdoctoral Science Foundation.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no competing interests.

Appendix 1: Existence of maximum in RLog

Recalling the RLog formula in (2.18), we want to show that $g(\theta )$ contains a global maximum, $g_m$. From the formula, shifted by $+2g(\theta )$ for positivity,

$$\begin{aligned} g'(\theta )=\frac{u\cos \theta -v\sin \theta }{v\cos \theta +u\sin \theta } \end{aligned}$$

(7.1)

and

$$\begin{aligned} g''(\theta )=-\frac{u^2+v^2}{(v\cos \theta +u\sin \theta )^2}\le 0. \end{aligned}$$

(7.2)

Equation 7.2 suggests the existence of a maximum in g. Substituting $g'(\theta )=0$ into (7.1) and solve for $\theta $, we get the location

$$\begin{aligned} \theta _m=\arctan \left( \frac{u}{v}\right) \end{aligned}$$

on the RLog curve that gives the maximum. Substituting $\theta =\theta _m$ into the RLog formula yields the maximum value of

$$\begin{aligned} g_m=\frac{1}{2}\log (u^2+v^2). \end{aligned}$$

(7.3)

Appendix 2: Computation of optimal chamfer weights

For any chamfer mask, optimal weights that control the major error lobe can be computed by considering the conditions $g_m=-g(0^{\circ })$ and $u=(\phi -\varphi )v$, where $\phi $ and $\varphi $ depend on the kernel size (Table 4). Plugging the conditions into the respective formulas gives

$$\begin{aligned} v=\frac{1}{\root 4 \of {(\phi -\varphi )^2+1}}\quad \text { and }\quad u=\frac{\phi -\varphi }{\root 4 \of {(\phi -\varphi )^2+1}}. \end{aligned}$$

(8.1)

Table 4

Values of $\phi $ and $\varphi $ for various kernel sizes

Kernel size	$\phi $	$\varphi $
$3\times 3$	$\sqrt{2}$	1
$5\times 5$	$\sqrt{5}$	2
$7\times 7$	$\sqrt{10}$	3

Appendix 3: Detailed derivations of chamfer weights for kernels of order $2n+3$ ($n=1,2,\ldots $)

This Appendix provides further details on how to derive high-order chamfer kernels. For demonstration purposes, we design the local weights for $5\times 5$ and $7\times 7$ kernels.

$5\times 5$ chamfer mask

Consider portion of a chamfer polygon, represented by Fig. 4, which contains two sides, namely $\overline{AB}$ and $\overline{BC}$. The polygon is inscribed in a disk of radius r, and our goal is to find the maximum possible errors that each side makes relative to its corresponding arc. For the $5\times 5$ mask, we should compute errors due to $\overline{AB}$ and due to $\overline{BC}$; note that errors are zeros when the sides are (ideally) perfectly aligned with the arcs Open image in new window and Open image in new window , respectively. Adopting notations from the earlier sections, $f_1(\theta )$ and $f_2(\theta )$ correspond to the lengths of $\vec {p}$ from $\overline{AB}$ and from $\overline{BC}$, respectively.

The following items give the systematic procedures to compute optimal errors from each polygon’s side generated by a $5\times 5$ chamfer kernel (refer to Fig. 4 for definitions of variables):

(a)

Use the sine rule to find $f_1(\theta )$ and $f_2(\theta )$:
$$\begin{aligned} f_1(\theta )=\frac{r\sin \theta _1}{a\sin (\theta +\theta _1)} =\frac{r\sin \theta _1}{a(\sin \theta \cos \theta _1+\cos \theta \sin \theta _1)}. \end{aligned}$$

(9.1)
and
$$\begin{aligned} f_2(\theta )=\frac{r\sin \theta _2}{a\sin (\theta +\theta _2)}=\frac{r\sin \theta _2}{a(\sin \theta \cos \theta _2+\cos \theta \sin \theta _2)}. \end{aligned}$$

(9.2)
(b)

Use equations of the lines $l_1$ and $l_2$, which, respectively, extend $\overline{AB}$ and $\overline{BC}$, to define $\sin \theta _1$, $\cos \theta _1$, $\sin \theta _2$, and $\cos \theta _2$ in (9.1) and (9.2):
$$\begin{aligned} \sin \theta _1=\frac{a}{\sqrt{(c-2a)^2+a^2}},~~ \cos \theta _1=\frac{c-2a}{\sqrt{(c-2a)^2+a^2}}; \end{aligned}$$

(9.3)
and
$$\begin{aligned} \sin \theta _2= & {} \frac{c-b}{\sqrt{(2b-c)^2+(c-b)^2}},\nonumber \\ \cos \theta _2= & {} \frac{2b-c}{\sqrt{(2b-c)^2+(c-b)^2}} \end{aligned}$$

(9.4)
(c)

Substitute (9.3) into (9.1) and (9.4) into (9.2):
$$\begin{aligned} f_1(\theta )=\frac{r}{(c-2a)\sin \theta +a\cos \theta } \end{aligned}$$

(9.5)
and
$$\begin{aligned} f_2(\theta )=\frac{r}{(2b-c)\sin \theta +(c-b)\cos \theta }. \end{aligned}$$

(9.6)
(d)

Use the RLog formula to find $g_1(\theta )$ and $g_2(\theta )$, which are the respective errors described by $\vec {p}$ on $\overline{AB}$ and $\overline{BC}$:
$$\begin{aligned} g_1(\theta )=\log \left( \frac{1}{r}f_1(\theta )\right) =-\log ((c-2a)\sin \theta +a\cos \theta ) \end{aligned}$$

(9.7)
and
$$\begin{aligned} g_2(\theta )= & {} \log \left( \frac{1}{r}f_2(\theta )\right) \nonumber \\= & {} -\log ((2b-c) \sin \theta +(c-b)\cos \theta ). \end{aligned}$$

(9.8)
(e)

Identify the polygon’s longest side, which corresponds to the edge that produces the maximum RLog value. From Fig. 4,
$$\begin{aligned} \overline{AB}=\frac{r}{ac}\sqrt{5a^2-4ac+c^2}=r\sqrt{5\left( \frac{a}{c}\right) +\left( \frac{c}{a}\right) -4} \end{aligned}$$

(9.9)
and
$$\begin{aligned} \overline{BC}= & {} \frac{r}{bc}\sqrt{5b^2-6bc+2c^2}\nonumber \\= & {} r\sqrt{5\left( \frac{b}{c}\right) +2\left( \frac{c}{b}\right) -6}. \end{aligned}$$

(9.10)
Let $\phi _1=\frac{c}{a}$ and $\phi _2=\frac{c}{b}$. If a, b, and c are selected within the allowable limits, then $\phi _1\ge \phi _2$. Substituting $\phi _1$ into (9.9) and $\phi _2$ into (9.10), we have
$$\begin{aligned} \overline{AB}=r\sqrt{\frac{5}{\phi _1}+\phi _1-4} \end{aligned}$$

(9.11)
and
$$\begin{aligned} \overline{BC}=r\sqrt{\frac{5}{\phi _2}+2\phi _2-6}. \end{aligned}$$

(9.12)
Evidently, $\overline{AB}\ge \overline{BC}$ (a typical test can be done using the Euclidean local weights $a=1$, $b=\sqrt{2}$, and $c=\sqrt{5}$).
(f)

$\overline{AB}\ge \overline{BC}$ implies that $g_{1m}\ge g_{2m}$, where $g_{1m}$ and $g_{2m}$ are the respective peaks for the curves $g_{1}$ and $g_{2}$. One strategy to minimize $g_{1m}$ is to push $\overline{AB}$ outward in the direction of $\vec {p}$ such that a triangle extended from $\triangle OAB$ is isosceles, or $c=\sqrt{5}a$ and $-g_1(0^{\circ })=g_{1m}$. Solving the equations gives $a_{\mathrm {opt}}=0.9865$ and $c_{\mathrm {opt}}=2.2059$. The value of b should be chosen such that the maximum RLog stays in the range $0\le g_1(\theta )\le g_{1m}$, and a natural choice is the Euclidean value $b=\sqrt{2}$.

$7\times 7$ chamfer mask

We use Fig. 7 to optimize the local weights in the $7\times 7$ chamfer mask. Following similar strategies as those used in the case of the $5\times 5$ mask, the steps are itemized below:

(a)

The derived lengths of $\vec {p}$ on $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DE}$ are, respectively,
$$\begin{aligned} f_1(\theta )= & {} \frac{r\sin \theta _1}{a\sin (\theta +\theta _1)}= \frac{r\sin \theta _1}{a(\sin \theta \cos \theta _1+\cos \theta \sin \theta _1)}, \end{aligned}$$

(9.13)
$$\begin{aligned} f_2(\theta )= & {} \frac{r\sin \theta _2}{a\sin (\theta +\theta _2)}= \frac{r\sin \theta _2}{a(\sin \theta \cos \theta _2+\cos \theta \sin \theta _2)}, \end{aligned}$$

(9.14)
$$\begin{aligned} f_3(\theta )= & {} \frac{r\sin \theta _3}{a\sin (\theta +\theta _3)}=\frac{r\sin \theta _3}{a(\sin \theta \cos \theta _3+\cos \theta \sin \theta _3)},\nonumber \\ \end{aligned}$$

(9.15)
and
$$\begin{aligned} f_4(\theta )=\frac{r\sin \theta _4}{a\sin (\theta +\theta _4)}=\frac{r\sin \theta _4}{a(\sin \theta \cos \theta _4+\cos \theta \sin \theta _4)}. \end{aligned}$$

(9.16)
(b)

The trigonometric functions with respect to $\theta _i$ ($i=1,2,3,$ and 4) are
$$\begin{aligned} \sin \theta _1= & {} \frac{a}{\sqrt{(d-3a)^2+a^2}}, \end{aligned}$$

(9.17)
$$\begin{aligned} \cos \theta _1= & {} \frac{d-3a}{\sqrt{(d-3a)^2+a^2}}, \end{aligned}$$

(9.18)
$$\begin{aligned} \sin \theta _2= & {} \frac{d-c}{\sqrt{(3c-2d)^2+(d-c)^2}}, \end{aligned}$$

(9.19)
$$\begin{aligned} \cos \theta _2= & {} \frac{3c-2d}{\sqrt{(3c-2d)^2+(d-c)^2}}, \end{aligned}$$

(9.20)
$$\begin{aligned} \sin \theta _3= & {} \frac{2c-e}{\sqrt{(2e-3c)^2+(2c-e)^2}}, \end{aligned}$$

(9.21)
$$\begin{aligned} \cos \theta _3= & {} \frac{2e-3c}{\sqrt{(2e-3c)^2+(2c-e)^2}}, \end{aligned}$$

(9.22)
$$\begin{aligned} \sin \theta _4= & {} \frac{e-2b}{\sqrt{(3b-e)^2+(e-2b)^2}}, \end{aligned}$$

(9.23)
and
$$\begin{aligned} \cos \theta _4=\frac{3b-e}{\sqrt{(3b-e)^2+(e-2b)^2}}. \end{aligned}$$

(9.24)
(c)

Substitutions of ${\sin \theta _{i}}'s$ and ${\cos \theta _{i}}'s$ into the respective equations for ${f_{i}}'s$ ($i=1,2,3$ and 4) yield
$$\begin{aligned} f_1(\theta )= & {} \frac{r}{(d-3a)\sin \theta +a\cos \theta }, \end{aligned}$$

(9.25)
$$\begin{aligned} f_2(\theta )= & {} \frac{r}{(3c-2d)\sin \theta +(d-c)\cos \theta }, \end{aligned}$$

(9.26)
$$\begin{aligned} f_3(\theta )= & {} \frac{r}{(2e-3c)\sin \theta +(2c-e)\cos \theta }, \end{aligned}$$

(9.27)
and
$$\begin{aligned} f_4(\theta )=\frac{r}{(3b-e)\sin \theta +(e-2b)\cos \theta }. \end{aligned}$$

(9.28)
(d)

The corresponding RLog formulae are
$$\begin{aligned} \overline{AB}:~g_1(\theta )= & {} \log \left( \frac{1}{r}f_1(\theta )\right) = -\log ((d-3a)\sin \theta \nonumber \\&+a\cos \theta ), \end{aligned}$$

(9.29)
$$\begin{aligned} \overline{BC}:~g_2(\theta )= & {} \log \left( \frac{1}{r}f_2(\theta )\right) = -\log ((3c-2d)\sin \theta \nonumber \\&+(d-c)\cos \theta ), \end{aligned}$$

(9.30)
$$\begin{aligned} \overline{CD}:~g_3(\theta )= & {} \log \left( \frac{1}{r}f_3(\theta )\right) = -\log ((2e-3c)\sin \theta \nonumber \\&+(2c-e)\cos \theta ), \end{aligned}$$

(9.31)
and
$$\begin{aligned} \overline{DE}:~g_4(\theta )= & {} \log \left( \frac{1}{r}f_4(\theta )\right) =-\log ((3b-e)\sin \theta \nonumber \\&+(e-2b)\cos \theta ). \end{aligned}$$

(9.32)
(e)

Comparing the RLog formulae, $g_1(\theta )$ contains the maximum possible error of $g_{1m}=-\frac{1}{2}\log ((d-3a)^2+a^2)$, suggesting that $\overline{AB}$ is the longest side.
(f)

Considering the optimality conditions for $\triangle OAB$, we solve the equations $d=\sqrt{10}a$ and $-g_1(0^{\circ })=g_{1m}$. Therefore, the optimal local weights for the $7\times 7$ chamfer mask are $a_{\mathrm {opt}}=0.9935$ and $d_{\mathrm {opt}}=3.1418$. The remaining non-redundant weights should be carefully selected on the condition that RLog on either error lobe stays within the bounds dictated by $g_{1m}$. And, an obvious selection is the Euclidean local weights $b=\sqrt{2}$, $c=\sqrt{5}$, and $e=\sqrt{13}$.

References

1.
Song, C., Pang, Z., Jing, X., Xiao, C.: Distance field guided L_1-median skeleton extraction. Vis. Comput. 1–13 (2016). doi: 10.1007/s00371-016-1331-z
2.
Saha, P.K., Borgefors, G., di Baja, G.S.: A survey on skeletonization algorithms and their applications. Pattern Recognit. Lett. 76, 3–12 (2016)CrossRefGoogle Scholar
3.
Coeurjolly, D., Montanvert, A.: Optimal separable algorithms to compute the reverse Euclidean distance transformation and discrete medial axis in arbitrary dimension. arXiv:0705.3343 (2007)
4.
Maurer, C.R., Qi, R., Raghavan, V.: A linear time algorithm for computing exact Euclidean distance transforms of binary images in arbitrary dimensions. IEEE Trans. Pattern Anal. Mach. Intell. 25(2), 265–270 (2003)CrossRefGoogle Scholar
5.
Bailey, D.G.: An efficient Euclidean distance transform. In: International Workshop on Combinatorial Image Analysis, pp. 394–408. Springer (2004). doi: 10.1007/978-3-540-30503-3_28
6.
Breu, H., Gil, J., Kirkpatrick, D., Werman, M.: Linear time Euclidean distance transform algorithms. IEEE Trans. Pattern Anal. Mach. Intell. 17(5), 529–533 (1995)CrossRefGoogle Scholar
7.
Zhang, H., Xu, M., Zhuo, L., Havyarimana, V.: A novel optimization framework for salient object detection. Vis. Comput. 32, 31–41 (2016)CrossRefGoogle Scholar
8.
Weng, Y., Xu, W., Wu, Y., Zhou, K., Guo, B.: 2D shape deformation using nonlinear least squares optimization. Vis. Comput. 22(9–11), 653–660 (2006)CrossRefGoogle Scholar
9.
Ding, S., Sheng, B., Xie, Z., Ma, L.: Intrinsic image estimation using near-L_0 sparse optimization. Vis. Comput. 33, 1–15 (2016). doi: 10.1007/s00371-015-1205-9
10.
Muñoz, A., Gutierrez, D., Serón, F.J.: Optimization techniques for curved path computing. Vis. Comput. 23(7), 493–502 (2007)CrossRefGoogle Scholar
11.
Wang, Z., Bovik, A.C.: Mean squared error: love it or leave it? A new look at signal fidelity measures. IEEE Signal Process. Mag. 26, 98–117 (2009)CrossRefGoogle Scholar
12.
Tofallis, C.: A better measure of relative prediction accuracy for model selection and model estimation. J. Oper. Res. Soc. 66(8), 1352–1362 (2015)CrossRefGoogle Scholar
13.
Butt, M.A., Maragos, P.: Optimum design of chamfer distance transforms. IEEE Trans. Image Process. 7(10), 1477–1484 (1998)CrossRefGoogle Scholar
14.
Grevera, G.J.: Distance transform algorithms and their implementation and evaluation. In: Deformable Models , pp. 33–60. Springer (2007). doi: 10.1007/978-0-387-68413-0_2
15.
Liu, W., Jiang, H., Bai, X., Tan, G., Wang, C., Liu, W., Cai, K.: Distance transform-based skeleton extraction and its applications in sensor networks. IEEE Trans. Parallel Distrib. Syst. 24(9), 1763–1772 (2013)CrossRefGoogle Scholar
16.
Xu, D., Li, H., Zhang, Y.: Fast and accurate calculation of protein depth by Euclidean distance transform. In: Research in Computational Molecular Biology, pp. 304–316. Springer (2013). doi: 10.1007/978-3-642-37195-0_30
17.
Mishchenko, Y.: A fast algorithm for computation of discrete Euclidean distance transform in three or more dimensions on vector processing architectures. Signal Image Video Process. 9, 19–27 (2015)CrossRefGoogle Scholar
18.
Salvi, D., Zheng, K., Zhou, Y., Wang, S.: Distance transform based active contour approach for document image rectification. In: Applications of Computer Vision (WACV), 2015 IEEE Winter Conference on IEEE , pp. 757–764 (2015)Google Scholar
19.
Elizondo-Leal, J.C., Parra-González, E.F., Ramírez-Torres, J.G.: The exact Euclidean distance transform: a new algorithm for universal path planning. Int. J. Adv. Robot. Syst. 10, 266 (2013)CrossRefGoogle Scholar
20.
Linnér, E., Strand, R.: Anti-aliased Euclidean distance transform on 3D sampling lattices. In: Discrete Geometry for Computer Imagery, pp. 88–98. Springer (2014). doi: 10.1007/978-3-319-09955-2_8
21.
Dong, J., Sun, C., Yang, W.: An improved method for oriented chamfer matching. In: Intelligence Science and Big Data Engineering, pp. 875–879. Springer (2013)Google Scholar
22.
Tzionas, D., Gall, J.: A comparison of directional distances for hand pose estimation. In: Pattern Recognition, pp. 131–141. Springer (2013)Google Scholar
23.
Kaliamoorthi, P., Kakarala, R.: Directional chamfer matching in 2.5 dimensions. IEEE Signal Process Lett 20(12), 1151–1154 (2013)CrossRefGoogle Scholar
24.
Nguyen, D.T.: A novel chamfer template matching method using variational mean field. In: 2014 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 2425–2432. IEEE (2014)Google Scholar
25.
Paglieroni, D.W.: Distance transforms: Properties and machine vision applications. CVGIP Graph. Models Image Process. 54, 56–74 (1992)CrossRefGoogle Scholar
26.
Ma, T., Yang, X., Latecki, L.J.: Boosting chamfer matching by learning chamfer distance normalization. In: Computer Vision–ECCV 2010, pp. 450–463. Springer (2010). doi: 10.1007/978-3-642-15555-0_33
27.
Thiel, E., Montanvert, A.: Shape splitting from medial lines using the 3–4 chamfer distance. In: Visual Form, pp. 537–546. Springer (1992). doi: 10.1007/978-1-4899-0715-8_51
28.
Cuisenaire, O., Macq, B.: Fast Euclidean distance transformation by propagation using multiple neighborhoods. Comput. Vis. Image Underst. 76(2), 163–172 (1999)CrossRefGoogle Scholar
29.
Saito, T., Toriwaki, J.I.: New algorithms for Euclidean distance transformation of an n-dimensional digitized picture with applications. Pattern Recognit. 27(11), 1551–1565 (1994)CrossRefGoogle Scholar
30.
Shih, F.Y., Wu, Y.T.: Fast Euclidean distance transformation in two scans using a 3$\times $ 3 neighborhood. Comput. Vis. Image Underst. 93(2), 195–205 (2004)CrossRefGoogle Scholar
31.
Verwer, B.J.: Local distances for distance transformations in two and three dimensions. Pattern Recognit. Lett. 12(11), 671–682 (1991)CrossRefGoogle Scholar
32.
De Myttenaere, A., Golden, B., Le Grand, B., Rossi, F.: Mean absolute percentage error for regression models. Neurocomputing 192, 38–48 (2016)CrossRefGoogle Scholar
33.
Franses, P.H.: A note on the mean absolute scaled error. Int. J. Forecast. 32, 20–22 (2016)CrossRefGoogle Scholar
34.
Majidpour, M., Qiu, C., Chu, P., Pota, H.R., Gadh, R.: Forecasting the EV charging load based on customer profile or station measurement? Appl. Energy 163, 134–141 (2016)CrossRefGoogle Scholar
35.
Foss, T., Stensrud, E., Kitchenham, B., Myrtveit, I.: A simulation study of the model evaluation criterion MMRE. IEEE Trans. Softw. Eng. 29(11), 985–995 (2003)CrossRefGoogle Scholar

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Authors and Affiliations

Baraka Jacob Maiseli
- 1
- 2
Email authorView author's OrcID profile
LiFei Bai
- 3
Xianqiang Yang
- 3
Yanfeng Gu
- 1
Huijun Gao
- 3

1.School of Electronics and Information EngineeringHarbin Institute of TechnologyHarbinPeople’s Republic of China
2.Department of Electronics and Telecommunication Engineering, College of Information and Communication TechnologiesUniversity of Dar es SalaamDar es SalaamTanzania
3.Research Institute of Intelligent Control and SystemsHarbin Institute of TechnologyHarbinPeople’s Republic of China

Kernel size	\(\phi \)	\(\varphi \)
\(3\times 3\)	\(\sqrt{2}\)	1
\(5\times 5\)	\(\sqrt{5}\)	2
\(7\times 7\)	\(\sqrt{10}\)	3

Robust cost function for optimizing chamfer masks

Abstract

Keywords

Notes

Acknowledgements

Compliance with ethical standards

Conflict of interest

References

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