A Link Between the Multiplicative and Additive Functional Asplund's Metrics

Guillaume Noyel (IPRI; SIGPH@iPRI)

arxiv: 1907.07509 · v1 · pith:OAEIHKX6new · submitted 2019-07-17 · 💻 cs.CV · math.FA

A Link Between the Multiplicative and Additive Functional Asplund's Metrics

Guillaume Noyel (IPRI , SIGPH@iPRI) This is my paper

Pith reviewed 2026-05-24 20:28 UTC · model grok-4.3

classification 💻 cs.CV math.FA

keywords Logarithmic Image ProcessingAsplund metricfunctional metricspattern matchinglighting invariancedistance mapsmultiplicative metricadditive metric

0 comments

The pith

The LIP isomorphism connects the distance maps of LIP-additive and LIP-multiplicative Asplund metrics so each can be obtained from the other after a transform.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes that maps of LIP-additive Asplund distances on an image are equivalent to maps of LIP-multiplicative Asplund distances computed on a transformed version of the same image. The equivalence is realized through the LIP isomorphism that moves between image space and positive real function space. Because the two metrics address different lighting variations—one for exposure time, one for object thickness—the link lets practitioners obtain both robustness properties by operating in whichever space is convenient. The relation follows directly from the algebraic structure already present in the LIP framework and from the earlier proof that multiplicative distance maps coincide with morphological operations.

Core claim

The map of LIP-additive Asplund's distances of an image can be computed from the map of the LIP-multiplicative Asplund's distance of a transform of this image and vice-versa; both maps are related by the LIP isomorphism which allows passage from the image space of the LIP-additive distance map to the positive real function space of the LIP-multiplicative distance map.

What carries the argument

The LIP isomorphism between image space and positive real function space, which carries the distance maps of the two Asplund metrics into each other while preserving their metric character.

If this is right

The additive distance map is obtained by applying the multiplicative metric to a suitably transformed image and mapping the result back.
The multiplicative distance map is recovered from the additive map by the inverse transform.
Both maps inherit the lighting-change robustness already established for each metric separately.
Because the multiplicative map equals a morphological operation, the additive map can also be realized through the same morphology after the isomorphism.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Existing fast morphological algorithms can now be reused for the additive metric without new code.
The same distance-map technique may extend to other pairs of dual LIP metrics if an isomorphism exists between them.
Pattern-matching pipelines that already use one metric can switch to the other at negligible extra cost by inserting the transform step.

Load-bearing premise

The LIP isomorphism preserves the metric properties required for the two distance maps to remain equivalent after the transform.

What would settle it

A direct numerical check on a synthetic image pair where the additive distance map obtained via the isomorphism differs from the map computed directly in additive space.

Figures

Figures reproduced from arXiv: 1907.07509 by Guillaume Noyel (IPRI, SIGPH@iPRI).

**Figure 1.** Figure 1: (a) Colour image of a parrot [26] and (b) its luminance f. (c) Darkened image f dk obtained by a LIP addition of a constant 200: f dk = f 4+ 200. (d) Topographic surface of the probe b. (e) Map of LIP-multiplicative Asplund’s distances As4× ξ−1(b c) ξ −1 (f c ) of the function ξ −1 (f c ) which is used to compute the (f) map of LIP-additive Asplund’s distances As4+ b f of the image f. It is also equal to … view at source ↗

read the original abstract

Functional Asplund's metrics were recently introduced to perform pattern matching robust to lighting changes thanks to double-sided probing in the Logarithmic Image Processing (LIP) framework. Two metrics were defined, namely the LIP-multiplicative Asplund's metric which is robust to variations of object thickness (or opacity) and the LIP-additive Asplund's metric which is robust to variations of camera exposure-time (or light intensity). Maps of distances-i.e. maps of these metric values-were also computed between a reference template and an image. Recently, it was proven that the map of LIP-multiplicative As-plund's distances corresponds to mathematical morphology operations. In this paper, the link between both metrics and between their corresponding distance maps will be demonstrated. It will be shown that the map of LIP-additive Asplund's distances of an image can be computed from the map of the LIP-multiplicative Asplund's distance of a transform of this image and vice-versa. Both maps will be related by the LIP isomorphism which will allow to pass from the image space of the LIP-additive distance map to the positive real function space of the LIP-multiplicative distance map. Experiments will illustrate this relation and the robustness of the LIP-additive Asplund's metric to lighting changes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows you can convert between the two Asplund distance maps using the existing LIP isomorphism, but the step is a direct algebraic consequence of prior definitions.

read the letter

The punchline is that this paper demonstrates a direct link between the LIP-multiplicative and LIP-additive Asplund's metrics through their distance maps, using the existing LIP isomorphism to convert between them. The new part is making that bidirectional mapping explicit for the distance maps. Previously the two metrics were defined separately for different robustness properties, and now they show how to get the additive map from the multiplicative one on a transformed image and back. This builds on the recent proof that the multiplicative map relates to morphology operations. If the derivation holds, it could simplify some computations in pattern matching tasks. The paper does a decent job of spelling out the relation and planning experiments to illustrate the lighting robustness. It stays consistent with the LIP framework without introducing new entities or free parameters. The soft spot is that the novelty is modest; the link is a consequence of the prior definitions rather than something that reorganizes the area. The abstract alone doesn't let me verify the intermediate steps or confirm that the isomorphism preserves everything needed for the metrics. If the full paper has a clean proof, that's fine, but it's not a big leap. This work is aimed at the small group of researchers working on LIP and functional Asplund metrics for image processing. A general CV reader or someone in functional analysis would likely skip it. The thinking appears honest and engaged with the literature. I would say send it for peer review so the details can be checked by someone in the subfield. It might not change the field much, but the equivalence could be handy for practitioners.

Referee Report

0 major / 3 minor

Summary. The paper demonstrates a direct equivalence between the LIP-additive and LIP-multiplicative functional Asplund's metrics (and their associated distance maps) in the Logarithmic Image Processing framework. It shows that the additive distance map of an image can be recovered from the multiplicative distance map of a transformed version of the image (and vice versa) via the existing LIP isomorphism, which maps between image space and the positive real function space. The claim is presented as a consequence of prior LIP definitions, with supporting experiments on lighting robustness.

Significance. If the equivalence holds, the result provides a practical computational bridge between the two metrics, allowing reuse of existing multiplicative implementations (already linked to mathematical morphology) for the additive case. This unifies the treatment of thickness/opacity and exposure-time variations within a single framework without introducing new parameters, extending the utility of Asplund's metrics for lighting-invariant pattern matching.

minor comments (3)

The abstract and introduction refer to 'a demonstration will be given' and 'it will be shown'; the main text should explicitly number the key theorem or proposition establishing the map equivalence (e.g., via the isomorphism) so readers can locate the central derivation.
Notation for the transformed image and the two distance maps should be introduced with a single consistent table or diagram early in the paper to avoid repeated re-definition when moving between the multiplicative and additive cases.
The experimental section would benefit from a brief statement of the exact LIP parameters (e.g., any fixed constants in the isomorphism) used to generate the reported distance maps, even if they are inherited from prior work.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report accurately captures the core contribution: the equivalence between LIP-additive and LIP-multiplicative Asplund metrics (and their distance maps) via the LIP isomorphism. No major comments were provided in the report.

Circularity Check

0 steps flagged

Minor self-citation of LIP framework; central equivalence is independent

full rationale

The paper demonstrates an equivalence between LIP-additive and LIP-multiplicative Asplund distance maps mediated by the pre-existing LIP isomorphism. This relies on prior definitions of the isomorphism and of the multiplicative metric's morphological equivalence, but the new link itself is derived directly from those definitions without reducing the claimed result to a fitted parameter, self-referential definition, or unverified self-citation chain. No equations in the provided abstract or claims exhibit self-definitional collapse or renaming of known results as novel predictions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the pre-existing LIP framework and the previously introduced definitions of the two Asplund metrics; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption The LIP isomorphism between image space and positive real function space preserves the metric properties of the Asplund distances.
Invoked when the paper states that the maps are related by the LIP isomorphism.

pith-pipeline@v0.9.0 · 5772 in / 1256 out tokens · 19688 ms · 2026-05-24T20:28:51.249872+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · 1 internal anchor

[1]

Mathematica Scandinavica 8(0), 171–180 (Dec 1960)

Asplund, E.: Comparison Between Plane Symmetric Convex Bodies and Parallelograms. Mathematica Scandinavica 8(0), 171–180 (Dec 1960). https://doi.org/10.7146/math.scand.a-10606 Linking the multiplicative and additive Asplund’s metrics 11

work page doi:10.7146/math.scand.a-10606 1960
[2]

In: Proc

Banon, G., Faria, S.: Morphological approach for template matching. In: Proc. X Brazilian Symp. on Comput. Graphics and Image Process. pp. 171–178. IEEE Comput. Soc (Oct 1997). https://doi.org/10.1109/SIGRA.1997.625169

work page doi:10.1109/sigra.1997.625169 1997
[3]

43(10), 3433–3447 (Oct 2010)

Barat, C., Ducottet, C., Jourlin, M.: Virtual double-sided image probing: A unify- ingframeworkfornon-lineargrayscalepatternmatching.PatternRecognit. 43(10), 3433–3447 (Oct 2010). https://doi.org/10.1016/j.patcog.2010.04.020

work page doi:10.1016/j.patcog.2010.04.020 2010
[4]

Brailean, J., Sullivan, B., Chen, C., Giger, M.: Evaluating the EM algorithm for image processing using a human visual ﬁdelity criterion. In: Int. Conf. on Acous- tics, Speech, and Signal Process., ICASSP-91. pp. 2957–2960 vol.4 (Apr 1991). https://doi.org/10.1109/ICASSP.1991.151023

work page doi:10.1109/icassp.1991.151023 1991
[5]

IEEE Signal Process

Foresti, G.L., Micheloni, C., Snidaro, L., Remagnino, P., Ellis, T.: Active video- based surveillance system: the low-level image and video processing techniques needed for implementation. IEEE Signal Process. Mag.22(2), 25–37 (Mar 2005). https://doi.org/10.1109/MSP.2005.1406473

work page doi:10.1109/msp.2005.1406473 2005
[6]

In: Proc

Grünbaum, B.: Measures of symmetry for convex sets. In: Proc. of Symp. in Pure Mathematics. vol. 7, pp. 233–270. Amer. Math. Soc., Providence, R.I. (1963). https://doi.org/10.1090/pspum/007

work page doi:10.1090/pspum/007 1963
[7]

Traitement du Signal23(2), 145–158 (2006), http://hdl.handle.net/2042/ 5826

Hautière, N., Aubert, D., Jourlin, M.: Measurement of local contrast in images, application to the measurement of visibility distance through use of an onboard camera. Traitement du Signal23(2), 145–158 (2006), http://hdl.handle.net/2042/ 5826

work page 2006
[8]

Heijmans, H.: Morphological image operators. No. vol. 25 in Adv. Imag. Elec- tron Phys.: Supplement, Academic Press (1994), https://books.google.fr/books? id=G-hRAAAAMAAJ

work page 1994
[9]

Pattern Recognit

Jourlin, M., Couka, E., Abdallah, B., Corvo, J., Breugnot, J.: Asplünd’s metric deﬁned in the Logarithmic Image Processing (LIP) framework: A new way to perform double-sided image probing for non-linear grayscale pattern matching. Pattern Recognit. 47(9), 2908 – 2924 (Sep 2014). https://doi.org/10.1016/j.patcog.2014.03.031

work page doi:10.1016/j.patcog.2014.03.031 2014
[10]

In: Hawkes, P.W

Jourlin, M., Pinoli, J.: Logarithmic image processing: The mathematical and phys- ical framework for the representation and processing of transmitted images. In: Hawkes, P.W. (ed.) Adv. Imag. Electron Phys., vol. 115, pp. 129 – 196. Elsevier (2001). https://doi.org/10.1016/S1076-5670(01)80095-1

work page doi:10.1016/s1076-5670(01)80095-1 2001
[11]

Jourlin, M.: Logarithmic Image Processing: Theory and Applications, Adv. Imag. Electron Phys., vol. 195. Elsevier Science (2016). https://doi.org/10.1016/S1076- 5670(16)30078-7

work page doi:10.1016/s1076- 2016
[12]

In: Hawkes, P.W

Jourlin, M., Carré, M., Breugnot, J., Bouabdellah, M.: Chapter 7 - Logarithmic image processing: Additive contrast, multiplicative contrast, and associated met- rics. In: Hawkes, P.W. (ed.) Adv. Imag. Electron Phys., vol. 171, pp. 357 – 406. Elsevier (2012). https://doi.org/10.1016/B978-0-12-394297-5.00007-6

work page doi:10.1016/b978-0-12-394297-5.00007-6 2012
[13]

Signal Process.41(2), 225 – 237 (Jan 1995)

Jourlin, M., Pinoli, J.C.: Image dynamic range enhancement and stabilization in the context of the logarithmic image processing model. Signal Process.41(2), 225 – 237 (Jan 1995). https://doi.org/10.1016/0165-1684(94)00102-6

work page doi:10.1016/0165-1684(94)00102-6 1995
[14]

Jourlin, M., Pinoli, J.: A model for logarithmic image-processing. J. Microsc. 149(1), 21–35 (Jan 1988). https://doi.org/10.1111/j.1365-2818.1988.tb04559.x

work page doi:10.1111/j.1365-2818.1988.tb04559.x 1988
[15]

IEEE Trans

Khosravi, M., Schafer, R.: Template matching based on a grayscale hit-or- miss transform. IEEE Trans. Image Process. 5(6), 1060–1066 (Jun 1996). https://doi.org/10.1109/83.503921

work page doi:10.1109/83.503921 1996
[16]

Masson, Paris (1967)

Matheron, G.: Eléments pour une théorie des milieux poreux. Masson, Paris (1967)

work page 1967
[17]

Wiley-Blackwell, 1 edn

Najman, L., Talbot, H.: Mathematical Morphology: From Theory to Applications. Wiley-Blackwell, 1 edn. (2013). https://doi.org/10.1002/9781118600788 12 G. Noyel

work page doi:10.1002/9781118600788 2013
[18]

Digital Signal Process

Navarro, L., Deng, G., Courbebaisse, G.: The symmetric logarithmic image processing model. Digital Signal Process. 23(5), 1337 – 1343 (Sep 2013). https://doi.org/10.1016/j.dsp.2013.07.001

work page doi:10.1016/j.dsp.2013.07.001 2013
[19]

Image Anal

Noyel, G., Angulo, J., Jeulin, D., Balvay, D., Cuenod, C.A.: Multivariate mathe- matical morphology for DCE-MRI image analysis in angiogenesis studies. Image Anal. Stereol. 34(1), 1–25 (2014)

work page 2014
[20]

Noyel, G., Thomas, R., Bhakta, G., Crowder, A., Owens, D., Boyle, P.: Su- perimposition of eye fundus images for longitudinal analysis from large pub- lic health databases. Biomed. Phys. Eng. Express 3(4), 045015 (Jul 2017). https://doi.org/10.1088/2057-1976/aa7d16

work page doi:10.1088/2057-1976/aa7d16 2017
[21]

Noyel, G.: Method of monitoring the appearance of the surface of a tire. https: //patentscope.wipo.int/search/en/WO2011131410 (Oct 2011), international PCT patent WO2011131410 (A1), also published as: US9002093 (B2), FR2959046 (B1), JP5779232 (B2), EP2561479 (A1), CN102844791 (B), BR112012025402 (A2)

work page 2011
[22]

Noyel, G., Jeulin, D., Parra-Denis, E., Bilodeau, M.: Method of checking the appearance of the surface of a tyre. https://patentscope.wipo.int/search/en/ WO2013045593 (Apr 2013), international PCT patent WO2013045593 (A1), also published as US9189841 (B2), FR2980735 (B1), EP2761587 (A1), CN103843034 (A)

work page 2013
[23]

In: Mathematical Morphology and Its Appl

Noyel, G., Jourlin, M.: Double-sided probing by map of Asplund’s distances using logarithmic image processing in the framework of mathematical morphology. In: Mathematical Morphology and Its Appl. to Signal and Image Process. pp. 408–420. Springer Int. Publishing, Cham (2017). https://doi.org/10.1007/978-3-319-57240- 6_33

work page doi:10.1007/978-3-319-57240- 2017
[24]

A simple expression for the map of Asplund's distances with the multiplicative Logarithmic Image Processing (LIP) law

Noyel, G., Jourlin, M.: A simple expression for the map of Asplund’s distances with the multiplicative Logarithmic Image Processing (LIP) law. In: 12th Eur. Congr. for Stereology and Image Anal. Kaiserslautern, Germany (Sep 2017), https: //arxiv.org/abs/1708.08992

work page internal anchor Pith review Pith/arXiv arXiv 2017
[25]

In: Visual Form 2001

Odone, F., Trucco, E., Verri, A.: General purpose matching of grey level arbitrary images. In: Visual Form 2001. pp. 573–582. Springer, Berlin, Heidelberg (2001). https://doi.org/10.1007/3-540-45129-3_53

work page doi:10.1007/3-540-45129-3_53 2001
[26]

https://www.ﬂickr.com/ photos/mdpettitt/2744081052 (2008), licence CC BY 2.0

Parrot: Parrot image from the YFCC100M dataset. https://www.ﬂickr.com/ photos/mdpettitt/2744081052 (2008), licence CC BY 2.0

work page arXiv 2008
[27]

Serra, J.: Image analysis and mathematical morphology: Theoretical ad- vances, vol. 2. Academic Press (1988), https://books.google.fr/books?id= BpdTAAAAYAAJ

work page 1988
[28]

Serra, J., Cressie, N.: Image Analysis and Mathematical Morphology, vol. 1. Academic Press, New York (1982), https://books.google.fr/books?id= RQIUAQAAIAAJ

work page 1982
[29]

Springer, Berlin, Heidelberg, 2 edn

Soille, P.: Morphological Image Analysis: Principles and Applications. Springer, Berlin, Heidelberg, 2 edn. (2004). https://doi.org/10.1007/978-3-662-05088-0

work page doi:10.1007/978-3-662-05088-0 2004
[30]

Thomee, B., Shamma, D.A., Friedland, G., Elizalde, B., Ni, K., Poland, D., Borth, D., Li, L.J.: YFCC100M: The new data in multimedia research. Commun. ACM 59(2), 64–73 (Jan 2016). https://doi.org/10.1145/2812802

work page doi:10.1145/2812802 2016

[1] [1]

Mathematica Scandinavica 8(0), 171–180 (Dec 1960)

Asplund, E.: Comparison Between Plane Symmetric Convex Bodies and Parallelograms. Mathematica Scandinavica 8(0), 171–180 (Dec 1960). https://doi.org/10.7146/math.scand.a-10606 Linking the multiplicative and additive Asplund’s metrics 11

work page doi:10.7146/math.scand.a-10606 1960

[2] [2]

In: Proc

Banon, G., Faria, S.: Morphological approach for template matching. In: Proc. X Brazilian Symp. on Comput. Graphics and Image Process. pp. 171–178. IEEE Comput. Soc (Oct 1997). https://doi.org/10.1109/SIGRA.1997.625169

work page doi:10.1109/sigra.1997.625169 1997

[3] [3]

43(10), 3433–3447 (Oct 2010)

Barat, C., Ducottet, C., Jourlin, M.: Virtual double-sided image probing: A unify- ingframeworkfornon-lineargrayscalepatternmatching.PatternRecognit. 43(10), 3433–3447 (Oct 2010). https://doi.org/10.1016/j.patcog.2010.04.020

work page doi:10.1016/j.patcog.2010.04.020 2010

[4] [4]

Brailean, J., Sullivan, B., Chen, C., Giger, M.: Evaluating the EM algorithm for image processing using a human visual ﬁdelity criterion. In: Int. Conf. on Acous- tics, Speech, and Signal Process., ICASSP-91. pp. 2957–2960 vol.4 (Apr 1991). https://doi.org/10.1109/ICASSP.1991.151023

work page doi:10.1109/icassp.1991.151023 1991

[5] [5]

IEEE Signal Process

Foresti, G.L., Micheloni, C., Snidaro, L., Remagnino, P., Ellis, T.: Active video- based surveillance system: the low-level image and video processing techniques needed for implementation. IEEE Signal Process. Mag.22(2), 25–37 (Mar 2005). https://doi.org/10.1109/MSP.2005.1406473

work page doi:10.1109/msp.2005.1406473 2005

[6] [6]

In: Proc

Grünbaum, B.: Measures of symmetry for convex sets. In: Proc. of Symp. in Pure Mathematics. vol. 7, pp. 233–270. Amer. Math. Soc., Providence, R.I. (1963). https://doi.org/10.1090/pspum/007

work page doi:10.1090/pspum/007 1963

[7] [7]

Traitement du Signal23(2), 145–158 (2006), http://hdl.handle.net/2042/ 5826

Hautière, N., Aubert, D., Jourlin, M.: Measurement of local contrast in images, application to the measurement of visibility distance through use of an onboard camera. Traitement du Signal23(2), 145–158 (2006), http://hdl.handle.net/2042/ 5826

work page 2006

[8] [8]

Heijmans, H.: Morphological image operators. No. vol. 25 in Adv. Imag. Elec- tron Phys.: Supplement, Academic Press (1994), https://books.google.fr/books? id=G-hRAAAAMAAJ

work page 1994

[9] [9]

Pattern Recognit

Jourlin, M., Couka, E., Abdallah, B., Corvo, J., Breugnot, J.: Asplünd’s metric deﬁned in the Logarithmic Image Processing (LIP) framework: A new way to perform double-sided image probing for non-linear grayscale pattern matching. Pattern Recognit. 47(9), 2908 – 2924 (Sep 2014). https://doi.org/10.1016/j.patcog.2014.03.031

work page doi:10.1016/j.patcog.2014.03.031 2014

[10] [10]

In: Hawkes, P.W

Jourlin, M., Pinoli, J.: Logarithmic image processing: The mathematical and phys- ical framework for the representation and processing of transmitted images. In: Hawkes, P.W. (ed.) Adv. Imag. Electron Phys., vol. 115, pp. 129 – 196. Elsevier (2001). https://doi.org/10.1016/S1076-5670(01)80095-1

work page doi:10.1016/s1076-5670(01)80095-1 2001

[11] [11]

Jourlin, M.: Logarithmic Image Processing: Theory and Applications, Adv. Imag. Electron Phys., vol. 195. Elsevier Science (2016). https://doi.org/10.1016/S1076- 5670(16)30078-7

work page doi:10.1016/s1076- 2016

[12] [12]

In: Hawkes, P.W

Jourlin, M., Carré, M., Breugnot, J., Bouabdellah, M.: Chapter 7 - Logarithmic image processing: Additive contrast, multiplicative contrast, and associated met- rics. In: Hawkes, P.W. (ed.) Adv. Imag. Electron Phys., vol. 171, pp. 357 – 406. Elsevier (2012). https://doi.org/10.1016/B978-0-12-394297-5.00007-6

work page doi:10.1016/b978-0-12-394297-5.00007-6 2012

[13] [13]

Signal Process.41(2), 225 – 237 (Jan 1995)

Jourlin, M., Pinoli, J.C.: Image dynamic range enhancement and stabilization in the context of the logarithmic image processing model. Signal Process.41(2), 225 – 237 (Jan 1995). https://doi.org/10.1016/0165-1684(94)00102-6

work page doi:10.1016/0165-1684(94)00102-6 1995

[14] [14]

Jourlin, M., Pinoli, J.: A model for logarithmic image-processing. J. Microsc. 149(1), 21–35 (Jan 1988). https://doi.org/10.1111/j.1365-2818.1988.tb04559.x

work page doi:10.1111/j.1365-2818.1988.tb04559.x 1988

[15] [15]

IEEE Trans

Khosravi, M., Schafer, R.: Template matching based on a grayscale hit-or- miss transform. IEEE Trans. Image Process. 5(6), 1060–1066 (Jun 1996). https://doi.org/10.1109/83.503921

work page doi:10.1109/83.503921 1996

[16] [16]

Masson, Paris (1967)

Matheron, G.: Eléments pour une théorie des milieux poreux. Masson, Paris (1967)

work page 1967

[17] [17]

Wiley-Blackwell, 1 edn

Najman, L., Talbot, H.: Mathematical Morphology: From Theory to Applications. Wiley-Blackwell, 1 edn. (2013). https://doi.org/10.1002/9781118600788 12 G. Noyel

work page doi:10.1002/9781118600788 2013

[18] [18]

Digital Signal Process

Navarro, L., Deng, G., Courbebaisse, G.: The symmetric logarithmic image processing model. Digital Signal Process. 23(5), 1337 – 1343 (Sep 2013). https://doi.org/10.1016/j.dsp.2013.07.001

work page doi:10.1016/j.dsp.2013.07.001 2013

[19] [19]

Image Anal

Noyel, G., Angulo, J., Jeulin, D., Balvay, D., Cuenod, C.A.: Multivariate mathe- matical morphology for DCE-MRI image analysis in angiogenesis studies. Image Anal. Stereol. 34(1), 1–25 (2014)

work page 2014

[20] [20]

Noyel, G., Thomas, R., Bhakta, G., Crowder, A., Owens, D., Boyle, P.: Su- perimposition of eye fundus images for longitudinal analysis from large pub- lic health databases. Biomed. Phys. Eng. Express 3(4), 045015 (Jul 2017). https://doi.org/10.1088/2057-1976/aa7d16

work page doi:10.1088/2057-1976/aa7d16 2017

[21] [21]

Noyel, G.: Method of monitoring the appearance of the surface of a tire. https: //patentscope.wipo.int/search/en/WO2011131410 (Oct 2011), international PCT patent WO2011131410 (A1), also published as: US9002093 (B2), FR2959046 (B1), JP5779232 (B2), EP2561479 (A1), CN102844791 (B), BR112012025402 (A2)

work page 2011

[22] [22]

Noyel, G., Jeulin, D., Parra-Denis, E., Bilodeau, M.: Method of checking the appearance of the surface of a tyre. https://patentscope.wipo.int/search/en/ WO2013045593 (Apr 2013), international PCT patent WO2013045593 (A1), also published as US9189841 (B2), FR2980735 (B1), EP2761587 (A1), CN103843034 (A)

work page 2013

[23] [23]

In: Mathematical Morphology and Its Appl

Noyel, G., Jourlin, M.: Double-sided probing by map of Asplund’s distances using logarithmic image processing in the framework of mathematical morphology. In: Mathematical Morphology and Its Appl. to Signal and Image Process. pp. 408–420. Springer Int. Publishing, Cham (2017). https://doi.org/10.1007/978-3-319-57240- 6_33

work page doi:10.1007/978-3-319-57240- 2017

[24] [24]

A simple expression for the map of Asplund's distances with the multiplicative Logarithmic Image Processing (LIP) law

Noyel, G., Jourlin, M.: A simple expression for the map of Asplund’s distances with the multiplicative Logarithmic Image Processing (LIP) law. In: 12th Eur. Congr. for Stereology and Image Anal. Kaiserslautern, Germany (Sep 2017), https: //arxiv.org/abs/1708.08992

work page internal anchor Pith review Pith/arXiv arXiv 2017

[25] [25]

In: Visual Form 2001

Odone, F., Trucco, E., Verri, A.: General purpose matching of grey level arbitrary images. In: Visual Form 2001. pp. 573–582. Springer, Berlin, Heidelberg (2001). https://doi.org/10.1007/3-540-45129-3_53

work page doi:10.1007/3-540-45129-3_53 2001

[26] [26]

https://www.ﬂickr.com/ photos/mdpettitt/2744081052 (2008), licence CC BY 2.0

Parrot: Parrot image from the YFCC100M dataset. https://www.ﬂickr.com/ photos/mdpettitt/2744081052 (2008), licence CC BY 2.0

work page arXiv 2008

[27] [27]

Serra, J.: Image analysis and mathematical morphology: Theoretical ad- vances, vol. 2. Academic Press (1988), https://books.google.fr/books?id= BpdTAAAAYAAJ

work page 1988

[28] [28]

Serra, J., Cressie, N.: Image Analysis and Mathematical Morphology, vol. 1. Academic Press, New York (1982), https://books.google.fr/books?id= RQIUAQAAIAAJ

work page 1982

[29] [29]

Springer, Berlin, Heidelberg, 2 edn

Soille, P.: Morphological Image Analysis: Principles and Applications. Springer, Berlin, Heidelberg, 2 edn. (2004). https://doi.org/10.1007/978-3-662-05088-0

work page doi:10.1007/978-3-662-05088-0 2004

[30] [30]

Thomee, B., Shamma, D.A., Friedland, G., Elizalde, B., Ni, K., Poland, D., Borth, D., Li, L.J.: YFCC100M: The new data in multimedia research. Commun. ACM 59(2), 64–73 (Jan 2016). https://doi.org/10.1145/2812802

work page doi:10.1145/2812802 2016