Talk:Lambertian reflectance

Untitled
To me, there's a mistake in the very beginning of the article:

"If a surface exhibits Lambertian reflectance, light falling on it is scattered such that the amount of light an observer sees, the surface luminance, is the same regardless of their angle of view."

If exhibiting a lambertian reflectance is the result of lambert's cosine law, then we can't really talk of "amount" of light. While I agree with the use of the term 'luminance', it implies a notion of brightness, and not really of quantity or amount. Lambert's cosine law says that the amount of light decreases with the cosine of the angle between the normal of the surface and the line of sight. But as the apparent surface the observer sees decrease with the same law, the brightness of the source remains constant. However I may be wrong... palleas - 04/03/2006
 * Yes, you're right. I'll try to reword it. What the person who wrote that probably meant was that the amount of light an observer sees per unit area of the surface, as projected on the plane perpendicular to the observer's angle of view. That would be correct, but isn't exactly elegant. The description must not use the term "brightness", though. The use of "brightness" as a description of or measure of light is strongly discouraged, because it tends to lead to confusion between luminance, radiance, luminous and radiant intensity, irradiance, illuminance, etc. As discussed at brightness, the latter term should be used only to describe a person's subjective impression of light, not actual physically measurable "amounts" of light.--Srleffler 17:22, 3 April 2006 (UTC)


 * I guess luminance (or radiance in energy units) is the term to use here, though it doesn't show clearly that there is the cosine factor in it. Maybe the article should have a first part that could be more 'instinctive', and say that a lambertian reflector is as bright whatever the line of view, and then a second part that explains it scientifically. I totally agree that the photometry terms are a hassle, and I'm french, so I have to deal with them in french and english, and as you can well guess, it's often not the same....--Palleas 9:10, 4 April 2006 (UTC)

I still somehow disagree with the definition. The term 'amount' annoys me. I know it's a very difficult definition to write properly, without using ambiguous terms... Or maybe keep amount of light, but make clearer the fact that it's not constant per unit area, but per perceived unit area? Palleas 07:25, 18 April 2006 (UTC)

quit it with the annoying overzealous merging. when i look up lambertian i want to know what people are talking about first, not what lambert's freaking cosine law is. i don't care about his cosine law, i want to know what lambertian means. if i don't make sense to you question yourself, not me.

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Correct me if I'm wrong, but I think the equation is wrong if N and L don't have unit length. Shouldn't it be: ID = cos(alpha) * C * IL = (L dot N) / (|L| * |N|) * C * IL? Update: Sorry, just noticed that the article mentions that L and N have to be normalized.

This statement (first two sentences) is at complete odds with the Lambert Cosine Law that this article is linked to. I would simply refer to the law which clearly explains the effect. This should be repaired now...the other comments are rather old. November, 2007

Doesn't make sense..
The article states that the apparent brightness is the same regardless of the observer's angle of view, and that rotating the object doesn't change its apparent brightness. Then it says the reflection is calculated using the normalized vector of the surface and the vector to the light source. Now if angle has no effect, then why exactly are we taking it into account in our calculation, or what exactly is it that we're calculating? I think the least that can be said is that the article needs to be re-worded. I personally don't know how to interpret it; I don't know if this is the equation I need to use for emulating some sort of reasonable shading. Perhaps he meant that rotating the object does not change its apparent brightness if the light source rotates about the object with it? Inhahe (talk) 06:09, 23 January 2009 (UTC)
 * Your conclusion is correct. The shading of the surface remains the same regardless of the observer's angle to the surface. There shouldn't be any implication that the light or surface are changing with respect to each other. Xot (talk) 20:59, 21 June 2009 (UTC)

The image is irrelevant to this case
Although it is not mentioned in the article, the picture attached in it seems to say that it is Lambert's cosine law that is used to calculate the shading, since it seems to illustrate that the intensity of the reflected light varies with the direction of it (which it does, although it doesn't affect the perceived brightness of a surface – and hence not the shading – but this is due to another phenomenon...). For the picture to say that the intensity of the reflected light varies with the direction of the incident light instead of the reflected light (which it also does; this is actually Labmert's cosine law traced backwards), the arrows should be pointed in the other direction. --Kri (talk) 00:43, 7 February 2011 (UTC)

Because of this I changed the image text; it became very long though. Maybe it's possible to shorten it down a bit or to find a picture which is more relevant to this case. --Kri (talk) 00:45, 7 February 2011 (UTC)


 * The true Lambert's cosine law actually refers to the angle of view, not to the angle of incidence of source light, so the figure is correct to describe Lambertian reflectance.


 * In shading, however, the other cosine involved (that of incidence angle) is more "visible" than Lambert's one. Indeed, Lambert's law produces a uniform brightness, so in shading one "doesn't care it" for an ideal lambertian surface, and works on the other cosine. Which, however, is not Lambert's law: any non-absorbing (and non-transparent) surface, lambertian or not, gives back light per surface area proportionally to the cosine of incidence angle (even a mirror), simply to conserve energy (this may not be true for a colored material).


 * But I agree that the matter is confusing, and that someone calls "lambertian" also the dependence on the incidence angle. So I propose that we clarify more explicitly these facts in the article, rather than in the caption.


 * In the meantime, I have changed slightly the figure and moved it near the lead (to make clear that it does not refers to shading), so I can remove the phrase in the caption. --GianniG46 (talk) 18:27, 10 February 2011 (UTC)


 * I created another section in the article for extra clarification, as well as changing the image text. If it has gotten too long again, feel free to shorten it down. --Kri (talk) 20:28, 10 February 2011 (UTC)

Fixing article
This article is contains errors and is not internally consistent (as previous discussion paragraphs note). Neither the picture nor its caption match Lambert's Cosine Law (aside: the Lambert Cosine Law article is also wrong). Lambertian surfaces reflect light equally in all directions, independent of viewing angle. The reflected magnitude is a function of the cosine between the incident light and the surface normal. I will fix the article to match this and remove the picture. The correct picture would have red arrows of equal length emitting in all directions, but I don't have time to draw a picture nor do I know how to add it to Wikipedia. —Preceding unsigned comment added by 134.223.116.201 (talk) 01:01, 25 February 2011 (UTC)


 * That is simply wrong: Lambertian surfaces do not reflect light equally in all directions. Although Lambertian reflectance implies that the appeared brightness of a surface is equal in all directions, Lambert's cosine law does still hold. Note that the appeared brightness and the reflected intensity is not the same thing; nor are they proportional to each other. The appeared brightness is proportional to the reflected Luminous intensity divided by cosine of the angle between the surface normal and the outgoing reflected light; this is due to a phenomenon not mentioned in the article.


 * The picture is correct and I'm restoring it to the article. --Kri (talk) 15:27, 25 February 2011 (UTC)

Still In Error
The brightness of a differential Lambertian area does not vary with viewing direction (think chalk or clay), and the equation for IsubD matches these words. The brightness of a plate would decrease with the cosine of the viewing angle because the projected area decreases, not because the reflected light is less intense off axis (as shown in the picture). The equation for IsubD implies the red arrows should be equal length in all directions. IsubD depends on alpha, the cosine between the incident light and the surface normal, but for the picture to be correct, it would also have to depend on the observing angle, which it does not.


 * What equation for ID are you talking about? The only equation I can think of is one that can be derived from Lambert's cosine law, which says differently from what you are saying. It feels like you are mixing up what is intended by brightness in this context and intensity; they are not the same thing. Please also in the future sign your comments by putting four tildes in the end of them ( ~ ) so it's possible to know who is writing, thank you. --Kri (talk) 01:19, 26 February 2011 (UTC)


 * I'm talking about the IsubD equation in this article. There is no viewing angle dependence.  I've been looking for a peer reviewed reference for Lambert's Cosine Law but have not yet found one (my physics book doesn't even mention it).  Admittedly, my knowledge on this topic is based on what I've read on the internet.  If you Google "lambertian reflectance", you'll see more discussion on the topic and better diagrams showing reflectance that falls off as the angle of incidence increases, but that is independent of viewing direction.  And sorry about the signature.  Mlnewhart (talk) 21:38, 28 February 2011 (UTC)


 * Don't worry about it, it's not a necessarity, it just makes it easier to know who is writing what. Well, in computer graphics (I guess that is from where the main stream of articles about Lambertian reflectance will come), the viewing angle will not have to be considered, since a Lambertian surface appears equally bright from all viewing directions.


 * When a surface is viewed from the side, rather than directly from the front, it gets tilted in the observer's point of view and hence it will take up less area on the retina in one of the observer's eyes. Since the appeared brightness isn't affected by tilting the surface, the illuminance, that is the luminous flux or the luminous power per area unit, on the part of the retina that is exposed to light from the surface will be unchanged. It is this illuminance that is used in computer graphics to determine the color of the surface in the rendered image. So, the rendered color of a Lambertian surface in computer graphics will not depend on from which angle it is viewed.


 * However, the total luminous flux from the surface that hits the retina, is the illuminance of the retina times the area that is hit. Since the area on the retina is proportional to cosine of the angle between the surface normal and the reflected light, then so is the luminous flux.


 * I guess that in this article and in the article about Lambert's cosine law it is a bit unclear what is meant by the intensity; in this article, the intensity ID of the light simply seems to be the color that is put on the surface when it's rendered, or the measured intensity of the light, which is independent of the viewing angle. The other article however speaks about the radiant intensity, which is proportional to cosine of the angle (this is what the cosine law says; it is this intensity that is depicted in the image in this article), as well as about the measured, or observed intensity IO, which is not proportional to cosine of the angle, a conclusion that they also make in the article. --Kri (talk) 23:55, 28 February 2011 (UTC)

By the way, I removed the picture. While it does match Lambert's cosine law, it doesn't match the context of this article. --Kri (talk) 00:15, 1 March 2011 (UTC)


 * How does it not match the context of this article? It is a useful 2D representation of what is happening in a slice of the hemisphere above the Lambertian surface. I think it would be useful to reinstate it, see my note further down on a picture being worth 1000 words. Jack Hogan (talk) 07:57, 3 March 2023 (UTC)

Constant Factor in Formula for Outgoing Light
Shouldn't there be a factor of $$1/\pi$$ involved in the formula for $$I_D$$? Suppose that an equal amount of light was coming in from all directions, with $$I_L = 1$$. Then, integrating the formula for $$I_D$$ given in the article over the hemisphere, you get that a total of $$2\pi^2$$ light is going out, while only $$2\pi$$ came in. Note that the extra factor of $$2\pi$$ for the outgoing light comes from the fact that $$I_D$$ light shines out in all directions equally, and that the surface area of a unit hemisphere is $$2\pi$$. A similar argument gives the $$2\pi$$ for incoming light. Gereeter (talk) 17:42, 21 January 2013 (UTC)

More examples, a simple measure of deviation from Lambert's law
It would be nice to have more examples of approximately Lambertian surfaces. I've added snow and charcoal, with templates. Though non-Lambertian reflectance is a complicated, many-parameter property, depending on angles of incidence, reflection, not to mention polarization, it would be good to know if there is a rough measure in common use, a single number, to characterize the degree of deviation from Lambertian emission or reflectance. Does anyone know of one? CharlesHBennett (talk) 07:43, 8 November 2017 (UTC)

See Also Links
An example might be placed in the See Also links. Here is one suggestion: https://en.wikipedia.org/wiki/Deep_lambertian_networks which is used for image recognition as part of mapping. CvS 18:08, 3 July 2018 (UTC) — Preceding unsigned comment added by Chrisvonsimson (talk • contribs)

Backscattering yields a uniform full moon image.
urila — Preceding unsigned comment added by Urila (talk • contribs) 02:27, 31 July 2018 (UTC)

There is no true photo of light scattered from a surface that obeys Lambert's cosine law.

The mean Backscattering from the full moon is directed back to the sun because the scattering dipoles on the moon oscillate in a plane perpendicular to the coming sunlight.

Any calculation that does not take the direction of the polarizing dipoles into account will not be correct.

Backscattering yields a uniform moon image, as well as the earth's image and the images of all the planets and their moons.

Back-scattering from any surface will not depend on the surface inclination angle to the coming light, and a curved surface will look uniform. It is, therefore, not surprising that true images, that obey Lambert's cosine law, are difficult to find.

In the calculation of Lambert's cosine law, there is a hidden assumption that the scattering dipoles oscillate in all random directions in space. This is not the case with the moon, and probably with other examples. Urila (talk) 13:07, 29 October 2018 (UTC) [2]

urila — Preceding unsigned comment added by Urila (talk • contribs) 00:37, 31 July 2018 (UTC)

Urila (talk) 13:26, 29 October 2018 (UTC)


 * Wikipedia does not publish original research. --Srleffler (talk) 03:26, 30 October 2018 (UTC)

The Scattered light is considered in the literature as a diffusive light, light that passed a number of scattering events before it left the scattering material. Diffusely scattered light must obey Lambert's Cosine scattering law. In the case of unidirectional light scattered backward from a surface of a sphere, the meaning is maximum scattering intensity in the middle of the sphere, and a decline to zero toward the periphery by the cosine law. The full moon looks uniform and people continue to assume that the light is diffusely scattered from it. More than that. The nearly uniform sphere image is common to all the planets and their moons, including the earth as observed from the moon. Out of thousands upon thousands of true photos, there is no single true photo that obeys Lambert's Cosine law. The only photos that do obey the law are rendered photos, photos that are at least partly simulated. Contrary to all that, if the scattering is assumed to be mainly a single event, then all the scattering dipoles are directly stimulated by the light radiation on the illuminated scattering material. Then scattering by them must be coherent, and then the full moon and all the other illuminated bodies, with similar illumination geometry, must be uniform, at least approximately. The full moon tells us that single event scattering is dominant. Maybe with small corrections of multiple scattering. Why is the single event dominant? It seems that the effect is geometrical and statistical. If we consider one event scattering, two event scattering, multiple event scattering, then the event probability will decline with an increasing number of scatterings. The single event has a probability of at least 50% and it is the strongest event. Nearly all the background that surrounds us is a singly scattered light. A true diffusely scattered light is rather rare.

Urila (talk) 09:42, 16 May 2020 (UTC)

Pictures = 1000 Words
It would seem that a simple picture showing how incoming photons are reflected would be very useful, especially given the confusion seen in the notes above. The easiest picture to show is the one in here that has been removed, I would suggest it be reinstated. It shows a 2D right slice of the hemisphere above the Lambertian surface, with incoming photons from an arbitrary direction heading to a small area on the surface, and the relative distribution of photons reflected from it. The lengths of the red arrows represent the proportion of photons (or Joules or Watts) reflected in each direction, according to Lambert's Cosine Law. I find it useful, nay necessary, to visualize what is happening. Jack Hogan (talk) 07:51, 3 March 2023 (UTC)

Lambertian Reflectance Is Used in Many Fields
The description in the introduction is currently about photometry, but such surfaces star in radiometry, so it would be useful to start the article talking about flux density (E), which can be (Watts, lm or photons/s) per unit area - and can be called irradiance or illuminance - and take it from there. The symbol for radiance/luminance is L. Also, reflectance is not isotropic by definition, a microscopic Lambertian surface reflects just in the hemisphere above it. Jack Hogan (talk) 08:26, 3 March 2023 (UTC)

Figure at top
The figure at the beginning of the page is a misleading and erroneous artist view. Scattering from solids is "Mie" scattering. It has a sharp maximum in the back direction of the incident ray. There is no true scattering that looks like that in the figure. urila — Preceding unsigned comment added by Urila (talk • contribs) 18:21, 29 March 2023 (UTC)
 * That is kind of beside the point. The figure illustrates what Lambertian reflectance is. Whether any real material exhibits true Lambertian scattering is irrelevant.--Srleffler (talk) 16:17, 1 April 2023 (UTC)

As previously noted, the figure is wrong. Lambertian reflectance has (for any fixed incident light) the same radiance in all directions (above the surface), so all the arrows should be drawn with the same length. Of course, the radiance depends on the incident light, and there is a cosine factor for the radiance as a function of the incident light direction, but for fixed incident direction, the radiance is independent of the direction for the reflected light. The statement above is correct: "Lambertian surfaces reflect light equally in all directions, independent of viewing angle.  The reflected magnitude is a function of the cosine between the incident light and the surface normal." However, it is not reflected (pun intended) in the diagram, and the last part of the last sentence of the caption is also nonsense. — Preceding unsigned comment added by Donpage (talk • contribs) 23:56, 3 August 2023 (UTC)
 * The red arrows show radiant intensity, not radiance, as is explained in the caption. The image is correct. The fact that radiant intensity varying with angle as shown produces radiance that is independent of angle is important. One cannot understand Lambertian reflectance without understanding this.
 * The statement "Lambertian surfaces reflect light equally in all directions" is at best so vague as to be useless. You have to be clear about whether you are referring to radiant intensity, radiance, or something else. They are not the same, and are not proportional to one another. Lambertian surfaces are perceived as being equally bright from any viewing angle. That is not the same thing as "reflecting light equally in all directions". --Srleffler (talk) 16:17, 6 August 2023 (UTC)
 * I was also confused when I saw the figure, at first. I've edited the figure caption to clarify why the radiant intensity is different but the radiance is the same in all directions. 73.15.130.165 (talk) 20:02, 1 October 2023 (UTC)
 * I like your addition to the caption.--Srleffler (talk) 22:52, 1 October 2023 (UTC)