Talk:Image moment

Mistake
Can anyone check: I think the second central moments are wrong and M10 or M01 in the equation should not be squared. — Preceding unsigned comment added by 192.44.38.255 (talk) 07:21, 6 December 2017 (UTC)

Suggestions
This article needs a more general introductory section to give better context to the layman.--Hooperbloob 05:11, 12 August 2005 (UTC)

It would be nice also to have a history section about how moments were discovered to have uses in computer vision and also how they have been used in practice (eg OCR etc).--AlexSpurling 23:16, 24 May 2006 (UTC)

Raw Moments - unreferenced/plagiarism/fair use?
I just read the first paragraph, word-for-word, of the Raw Moments section out of the popular book "Digital Image Processing, Second Edition" by Rafael C. Gonzalez - pg 672-673. I'm not sure on the legalities but should this not at least be referenced? Yani 03:05, 20 November 2006 (UTC)
 * Word for word? That would indeed be an astonishing coincidence, as I made some small tweaks to the first paragraph a couple of months ago, without ever having read Gonzalez, making the changes for readability, accuracy, and logical flow.  But I've heard great minds think alike [smile].  That said, if you think that book is a good reference for this page, by all means add it.  I don't have access to it now, but I you think there are fair use issues, why don't you quote the relevant parts here on the talk page and we can deal with what to do next. Baccyak4H (talk) 03:28, 20 November 2006 (UTC)
 * I'm not going to quote the relevant parts simply because I'd be retyping the whole section! I agree this is astonishing ...but it is unbelievable that such a large correspondence is a coincidence. The only other explanations are that either the book copied Wikipedia, or both have been copied some third party source. Yani 14:38, 12 December 2006 (UTC)

Covariance matrix
I changed the definition of the covariance matrix because the multiple use of $$\mu$$ was confusing. I also added a clue how the matrix elements are calculated (according to the covariance matrix definition).

Streamlining and background info
I reorganized the article to make its style more consistent throughout, and showed more explicitly its outliine structure both in sectioning and language. I kept pretty much all the previous material, just moved it around a bit and reworded / elaborated slightly. And I added a little more explanatory material in the intro per Hooperbloob's good request.

I am considering writing an article on moment invariants which would apply to moments in general (e.g., physics) and cover the translation, scale, and rotation types, as well as (possibly) invariance to certain well-defined types of "blurring" (useful for image processing obviously). It would also describe analogs in 3D and perhaps 1D, and it would hopefully update 2D rotation invariant moments to sets which are superior to Hu's in some ways. If I did this, I may move some of this material to that article. Any comments? Baccyak4H 03:38, 19 August 2006 (UTC)

Orientation angle formula
Could someone please check the formula for the orientation angle, obtained from the covariance matrix ? I implemented this formula and it yielded partially incoherent results. I instead used the formula on this page (3rd slide), worked fine. But I don't want to change the original page if it's correct.Mr Bam (talk) 10:21, 16 January 2008 (UTC)

the formula is correct: it yields exact values for the orientation as long as the object has μ20 != μ02. it yields hazardous results for squares, hexagons, octogons, ..., circles. —Preceding unsigned comment added by 217.148.242.166 (talk) 10:30, 18 June 2008 (UTC)


 * Well, anyone who uses a single-argument arctan gets what they deserve, eh? 70.234.244.77 (talk) 05:40, 6 July 2010 (UTC)Michael

Rotation invariant moments
Is it correct to state that I1 is invariant to scale? I've been playing around with these formulas, and the invariant moments I1 and I2 are indeed invariant wrt rotation and translation (And I2 is invariant wrt scale) but I1 is not invariant wrt scale. Indeed, all other things being equal, if I1 roughly corresponds to the image's inertia around its centroid, I would expect it to be NOT invariant wrt scale, as a larger (i.e. more massive) object has greater resistance to a change in angular velocity. Of course, I'm not a mathematician, so correct me if I'm wrong. Jeroen (talk) 14:30, 24 February 2009 (UTC)
 * If I understand correctly, I1 and I2 are functions of the &eta;s of the preceding section, which are themselves scale invariant. If you really found that one of the Is was not scale invariant, it could only be if the &eta; formulas were wrong.  Baccyak4H (Yak!) 16:08, 24 February 2009 (UTC)

I7 seems to be incorrect: other sources state (Note the plus operator):

\begin{align} I_7 =\ & (3\eta_{21} - \eta_{03})(\eta_{30} + \eta_{12})[(\eta_{30} + \eta_{12})^2 - 3(\eta_{21} + \eta_{03})^2] + \\ \ & (\eta_{30} - 3\eta_{12})(\eta_{21} + \eta_{03})[3(\eta_{30} + \eta_{12})^2 - (\eta_{21} + \eta_{03})^2]. \end{align} $$

Can we derive this?
 * I think the old version was correct. I rotated an image in MATLAB, and your new version (with the plus operator) is not rotation invariant, but, if I use a minus operator, it is rotation invariant.  I would suggest reverting the edit. 70.234.244.77 (talk) 05:47, 6 July 2010 (UTC)Michael

The result of the I7 version currently on the page agrees with the results I've obtained with OpenCV, so probably correct. However, on the I2 calculation, the 4 should be outside the parenthesis to agree with the results on OpenCV. Other online sources seem to agree with this. Someone should check this further and correct it if needed. — Preceding unsigned comment added by 200.17.202.190 (talk) 16:56, 30 November 2012 (UTC)

Scale invariance vs Size invariance
Maybe it is useful to note that there is a difference between scale invariance and size invariance. What the article talks about, is scale invariance, which means the features are the same if the size of the object changes, but not the size of the embedding space (or the digital image). When however we have a digital image, and the size of this image changes (but not the size of the object relative to the image), we need another invariance measure. I'd call it size invariance because we're dealing with image size. Think of it as calculating the moments of an image, and of the same image, resized. When you want to obtain the same features from both images, these are the invariants I found:


 * $$\eta'_{ij} = \frac{\mu_{ij}}

{width^{i+1}\, height^{j+1}}\!$$ where i + j ≥ 2.

The +1's are there because the features, besides dividing by the respective amounts of width and height, should also be divided by the area of the image.

Similarly, we can obtain size invariance for the important raw moments $$M_{00}\!$$, $$\bar{x}\!$$ and $$\bar{y}\!$$:

$$\frac{M_{00}}{width\, height}, \frac{\bar{x}}{width}, \frac{\bar{y}}{height}$$

--Zom-B (talk) 14:09, 20 November 2009 (UTC)

New source
This relatively new book just came on my radar:



Baccyak4H (Yak!) 19:02, 10 June 2010 (UTC)

Scale invariance possibly only approximate
I am new to this and cannot provide definitive proof for this, but it appears that the scale invariant moments η are just approximately invariant to scale.

Please see: http://math.stackexchange.com/questions/1218329/scale-invariant-image-moments-not-scale-variant for an example.

This should maybe be pointed out in the article.

Max Krichenbauer (talk) 23:26, 3 April 2015 (UTC)

Invariances are only exact in the continuous domain. Note that both scaling and rotation are ill-defined on a discrete grid. crisluengo (talk) 22:38, 9 February 2017 (UTC)