To eleborate on Mikhail's answer a bit further, let's look a worst case scenario: Suppose you have an infinately long room full of objects and one large projection screen upfront and you flatten it by taking a projection orthogonal to the projection screen and look from behind the screen. The information loss will be 100%. Taking projections in other directions will dramatically reduce the loss of information.

I guess it is a function of depth z that depends on visibility at z.

It seems impossible to define it without the knowledge of hidden 3D scene but 3D priors may help to estimate them in specific cases ...

Researcher in vision or image synthesis should have better ideas.

This information lost irreversibly. You must have second camera, mirror as sensor or modelling this scene. This is a philosophical and fundamental question

Philosophically it depends on the purpose of observer. He must determine the importance of the lost and the saved information.

Then we have to start by identifying the initial position of the camera

And to give an initial assessment of the amount of information when the camera is in the current position ...

Depending on the purpose of the observer of the lost information can even be infinite ...

Of cause depth of z.

I agree with the other answers here. It depends on what is in the scene and what kind of information you care about. Sometimes, the "lost" data doesn't matter because you are making single-frame renders that aren't meant to be viewed together. If you have some other goal, like to extract 3D data from a photo (or render), you lose a great deal. If you have enough reference objects of the right type (large cubes are excellent for this, or smaller cubes spaced a known distance from each other) in the image that you can get a good idea what focal length lens is being used to render the scene. With the lens type known dimensions of many of the visible objects in the scene may be approximated fairly accurately, particularly objects that do not have curved surfaces. There are programs like Alias that allow "3D sketching" from 2D images. These can work quite well, but you should keep in mind they are not retrieving 3D data, but are used as a reference to approximate it.

AP

To my mind, it depends on your goals, as said before.

In general it reminds me of the problems having with 3d scans in cities or recording architecture. When you are interested i can send you some recent research work on 3d scans in architecture. But in general you hit a philosophic question as well.

Where are you heading at with your work?

Regards

who

What is your definition of 'information'?

Assuming you mean information in the Shannon sense (rather than some non-technical way). The answer is to set-up some task which can be measured in both 2D and 3D. For example some sort of classification or estimation problem. If it is estimation then one can measure the Fisher information or equivalent and bingo...you have an answer. This sort of approach, for the estimation of Geometric properties was practices by Kenichi Kanatani (his web page contains papers of the great man).

If you think about the equivalent problem in audio "How much extra information does stereo bring?" We would normally attempt to measure the mutual information between the two channels. Of course this raises the interesting question of how best to estimate information. In the case of audio, a good estimate is look at the difference in size between the compressed signal stereo and twice mono. I suspect one could adapt a similar estimate today - what is the payload of a 3D TV signal compared to one consisting of two identical channels. Rough stuff but maybe an estimate.

I hope it is obvious, from the previous remarks that the geometry of the capture system will be significant - wide-baseline stereo will be less compressible than narrow-baseline.

D

If only one 2D image is taken at least 2/3 information is lost. One must go for at least 3 cameras and make use of hyper-spectral imaging for not losing the information.

Not as much as we could expect, because it's usually more the mapping of the 2D surface of objects immerged in the 3D space than a real projection of the 3D space itself in a 2D image.

Now, more seriously, it clearly depends on the complexity of the 3D shape. If you have a simple 3D geometry, such as a cube, for example, you might get away with 2 cameras conveying most of the information (from the direction of 2 opposite corners, for instance). A 3rd camera would give you ALL the information, in this case. If you have concave 3D shapes you will need more than 3 cameras. Surely this has been published and I suggest visiting the library, Stephan! The only way of conveying all the 3D informationis by taking many 2D slices (such as in MRI or CAT X-Ray) with a sampling frequency (in the 'distance' domain) higher than Nyquist (determined by the complexity of the shape), similarly to the Fsam > 2Fmax for time-frequency analysis.

Depth information will be lost, though not fully

what type of data are you looking at? If you are contemplating the loss of a facet/value/dimension on each and every one of your collected data points and the practicality of proceeding with your investigation despite this loss, you need to look at the context .

If these are 3D over time medical imaging sets I would say keep as much data as possible. If this is a 3D animation you wanted a still from I would say use your best judgment.

The scientific context around your question is required for us to give you a useful answer as to how much data(especially relevant data) will be lost.

The answer has two aspect:

1- If you have no prior information about 3D scene, for example unknown scene, one dimension of three dimension will be lost. It means that you may lose depth information (Z axes), when you want to transform (X,Y,Z) from 3D scene to (x,y) in 2D image dimension.

2- If you have a prior information about 3D scene, for example you want to get image from a cube (say that cube shape is a prior information), then you maybe able to compute extra information from captured 2D image dimension, for example computation of Z (or depth).

There are many scenarios which prior information are available. For example predefined shading, texture, motion and so on.

The question, indeed, lacks of contextual information.

If you think from the point of view of an human visible image, a 3D scene is a function R^3 -> R^3

(each (x,y,z) coordinate map to a (r,g,b) color - this is of course a very simplistic approximation ; and I volontarily skip transparency plus the fact that human vision is more complex than my linear RGB model)

A 2D scene is a function R^2 -> R^3

There is of course far more information in (R^3)^(R^3) than in (R^2)^(R^3).

Now if your question is more pragmatic ; if your question is "How idiot are people who claim to sell you dvd player able to show a 2D DVD in 3D", that is different.

3D scenes are not "any scene". In other words, the set of all possible 3D scenes for a movie is really smaller than the set of all 3D scenes. And probably not that much bigger than the set of all possible 2D images from a movie.

As Payman said : if you know what you are seeing, it is often possible to deduce 3D informations from 2D informations. For example, if you can assume that Leonardo DiCaprio skin is as smooth and defectless as a baby skin, and assume that the light is unidirectional, then you can easily compute from the 2D image color the cosinus of the angle of the normal of his skin with the light ; theoretically, you should be able to create a 3D image of Leonardo's face for Titanic 3D from Titanic 2D images (of course there is occlusion problem ; but if you also assume that the shape of Leonardo's face remains more or less the same during the whole movie...).

So pragmatically, not that much information was lost because James Cameron used 2D camera instead of stereoscopic camera.

James Cameron's problem is more a technological problem : there is no mature technology able to recreate Titanic 3D from Titanic 2D. So yes, those "can play 2D movie in 3D" dvd player are stupid ; but not because of information theory.

"The question, indeed, lacks of contextual information."

More likely, the question lacks of sense.

“Loss” can be a meaningful criterion only if you compare two data sets. For example, a “ground truth” and an output of your camera. What is the GT in the question is completely obscure.

Adrian Egli: For xRay images the lost is more complex to describe. It not only depends on the shape of the 3D scene

You know how to obtain xRay image of THE 3D SCENE!!!!!???? ::a::

The information lost depends on the image. For a picture of a random field it should be 33% because one of the three coordinates is lost (this is what Payman Moallem said.) If the picture is a vertical aerial photo of a flat terrain then almost nothing is lost (only one float coordinate for an image of many millions of pixels).

Antonio Ruiz: For a picture of a random field it should be 33%

You can prove that it is not 31.14159% ?

Antonio Ruiz : again it depends of the definition of information (and "information loss"). With the same simplistic reasonning, I could say 16,67%

If you consider a finite random set of (r,g,b) point in a 3D space, and its projection on a 3D plane,... since this is a finite random set, the probability that two point has exactly the same projection is null. So, what you get at the end is exactly N (r,g,b,x,y) information instead of exactly N (r,g,b,x,y,z) information. ie 16,667% of information less.

I know. My "proof" is plain stupid. I just want to illustrate the fact that you can proove any figure you want with this kind of reasonning (it depends of what is the information, which weight has each piece of "information", what else do you know (cf in the case of the question "is it possible to play a 2D dvd movie in 3D ?" : the answer is far more complex than "you will lack 33% of information" ; the knowledge we have on the field allows to get more information than you would think from a 2D image)

The 3D distribution of matter in natural scenes are to a large extent created by processes where matters is distributed along multiple surfaces. And so multiple surfaces enclose the objects of interest. The surface enclosing a tree is extremely complicated if consider in all its details but it can be analysed at different spatial scales and at coarse 3D spatial scale, the surface surrounding a tree become relatively simple and it this one that we imagine and perceived the tree. If the 3D world would be made of random matter distribution, the imaging process would be a highly information degenerative process but since most 3D distributions already contraint on 2D surfaces in 3D space, the degeneracy is minimal, especially at coarse spatial scales. Most of the topological differential surface information are conserved during the imaging process. I have spent thousand of hours observing images at different spatial scales as topographical surface using visualization tools and the topography of the luminance surfaces are very similar to the actual object surfaces. There are deformations introduced by lighting effects for sure but the main topographical elements, the dales, the mountains, the valleys and the ridges correspond mostly to the actual surface feature of the objects. The topography of the image of a human body is particularly interesting to observe once a bit of spatial guaussian filtering is done on the image. What is even more interesting to observe is the sequence of filter images from very coarse spatial scale to the fine spatial scale. The sequence of the appearance of the topographical element of the human body closely resemble the morphogenesis development of a human body from an embryo. The facial features appears in the same sequence that those of a human embryo. The reason for that is that guassian filtering is the mathematical optimal method to remove information from the image surface and so the sequence of information disapearance is closely related to the sequence of information creation during the ontogeny process.

As I can see the most thinks, “information” of a 3D scene is composed by the reflectance of the discrete points in the scene. Thus, we can simplify the problem and consider only a 2D slice of the full 3D scene.

Then we can project this subspace of the 3D scene onto 1D image and estimate the “loss”. The next iteration of the desired simplification could be a ray in the 3D scene and its projection, which is just one point. One point in general can occlude only one point. Thus, the loss cannot be more than 50% :)

To eleborate on Mikhail's answer a bit further, let's look a worst case scenario: Suppose you have an infinately long room full of objects and one large projection screen upfront and you flatten it by taking a projection orthogonal to the projection screen and look from behind the screen. The information loss will be 100%. Taking projections in other directions will dramatically reduce the loss of information.

Summing up: It depends on the 3D geometry and the 'point of view' of the 2D camera relative to that ;-)

Going the other way from Joel's limit case a 2D projection of an evenly lit empty rectangular room (of known dimensions) painted uniformly with no contrast would have zero loss. Of course the entire room can then be represented with one pixel. If you then paint a known width uniform horizontal and vertical stripe around the room you will be able to reconstruct the room dimensions from the orthogonal 2D projection with no loss. Now suppose you add more elements to the room. When features begin to degrade due to loss of resolution, hidden surfaces, lighting and lack of scale hints then information is lost. For a simple example lets but a ball in the room. If the ball has a scene painted on it we will only get a clear view of the part of the painting on the center of the ball facing us. We will only get a hint of what is on the top, bottom and sides of the ball and no information from the back. If lighting is directional then half the ball will be in a shadow. However, if this ball is recognized and is in our library of known balls then the scene can be reconstructed in 3D with little loss.

Finally, the discussion is completely embedded in the framework of scholasticism.

The question: how many balls can overwhelm information of the scene? Is equivalent to "how many angels can dance on the head of a pin?"

One item missing in 2D from 3D perception is the surround effect diminishing the impact of 3D scene when it does not fill the field of view. Huge distortions were tolerated in 3D when the display bezel(frame) was viewed in tens of thousands of measurements reported by Bell Labs 1983 Symposium Natioanl Academy of Science.

My wonder is the cost/benefit to add 3D to display for the vision impaired via quasi-retina skin. Would photogating/range-gating reveal said "loss" obscured by foreground that is more of a threat to approaching blind person.

I must admit first that information theory is not my strongest area, but couldn't an entropy approach be taken in order to bound the information loss question? For example, take a 3D heat map of a scene, then project down to a 2D heat map. For each map, sample the scene and estimate entropy. Then could we not call the change in entropy an estimate of the information lost regardless of the context, or information the observer desires?

It depends on the geometry and the point of view of the 2D camera . There is no way to account for amount of "loss" information unless you have the ground truth. Also the foreshortening enters the game and can enduce large errors.

If the question were "WHAT information is lost in the 3D to 2D projection" then the answer would be "the depth".

Since the question is "HOW MUCH information is lost..." then there is no quantitative answer.

suppose the depth of the 3-D image is 256x256x256 then the information lost is 256x256x255. because when you project 3-D data on to the plane then only that plane information is sent to the frame buffer.

For x-ray tomography theory this question is more easily quantified. According to the Fourier Slice Theorem, a single projection creates a line in fourier space. By filtering of this projection we can effectively simulate this becoming a thin wedge in Fourier space. To recover all the data, we would need PI/2*N of these filtered projections. Where N is one dimension of the reconstructed NxNxN volume

Thomas Dufresne has a good suggestion, namely, quantifying information loss going from 3D to 2D by considering x-ray tomography image slices in terms of projections to lines in Fourier space. My suggestion is consider feature vectors that describe points in a 3D object and that describe points in an x-ray tomography slice. Then it is possible to quantify the descriptive nearness of pairs of local sets A, B, where A is a set of points in a 3D object and B is a set of points in a slice.

It won’t be just about the pixels as the question suggests the 3D "scene"!!! The art-snapshot 2D picture will be missing other information such as time, broader location or context, etc. However, in my opinion, more than 3/4 information will not be in any 2D picture; following on what Manasi mentioned, yes, we are definitely missing 2/3 of information and more…

If I take your Question in naive terms then you are losing the depth in your picture.

Although, as was pointed out, the question is not properly phrased, I believe the main loss is related to the qualitative difference between the 2D and 3D object representations.

There were many good answers already. What about the extreme case of an empty 3D volume? It projects into an empty 2D image: loss=0% ! It seems that the answer must relate the information content in 3D to the information content in the projection. A measure for this would be entropy. The higher the information content in 3D the more can be lost in the projection. However also the (human) viewer should not be neglected: we are often able to interpret partial information because our knowledge restricts the interpretation vocabulary, sometimes it gives us wrong interpretations: i.e. optical illusions, which would be the strange case of an information gain.

How much? then:

1 you need to get the entropy H1(x ) of your 3D signal,

2 calculate the entropy H2(x ) of your 2D signal

3 By comparison you can evaluate the information loss that should be proportional the bit rate or bit loss

This is the unique & objective way or method to obtain the information loss as is defined by the information theory.

Several subjective methods exist that depend on the context and domain.

The amount of lost information equals to the whole information of 3D scene. The reason should be:

1) We can observe a 3D scene from infinite number of view points, and the observed contents are certainly not the same from different view points. However, a 2D photo is captured from only one view. Since (1/infinite)=zero, the information preserved by one photo can be negligible compared to the information of the original 3D scene.

2) Some body may argue that the information from other view points may be inferred from the captured photo. For a regular shape (e.g., a sphere or a cube), even the whole 3D information can be reconstructed from one picure. However, in these "reconstruction" or "inference" process, some context or prior knowledge has been employed. In other word, the information is "guessed" or "inferred" by us rather than kept by the photo.

It depends on your 3d data. suppose you have a plane so if you project or fit a plane you didn't miss anything. you can use measures like entropy or methods like pca and compute eigen values of cov matrix. the ratio of the smallest eigen value and the sum of them is a measure.

If you assume the scene contains only non-transparent objects, you can approximately quantify the loss of information as the ratio between the area of surfaces not seen from a given viewpoint and the total area of surfaces in the scene. This is actually the loss by 3D to 2.5D transformation. The subsequent 2D image by an ordinary camera ignores the depth (i.e. one of three coordinates) so that the loss from 2.5D to 2D is 33%. I hope these ideas can help you.

I partly disagree with Andrzej: visibility of a white object in front of a white background makes the object visually disappear although every pixel of the object is (in principle) visible. Another example is the famous hidden Dalmatian dog (e.g.http://www.moillusions.com/2006/05/hidden-dalmation-dog.html).

Consequently the loss 3D -> 2D cannot be exclusively decided by considering surface patches, the occlusion boundary must be visible as well to "see" the object's shape. Counting all the pixels that have the "correct" color may be misleading in the above examples because ALL background pixels have the correct color but also all the foreground pixels are correct leading potentially to higher number of correct pixels than there are pixels in the image...

Confirming what others have said, I must say that it depends on:

1- What are the direction and angle used to produce the 2D image

2- Where the target parts of the image are?

At least we know that everything is white :)

BTW, only one side of the object is visible in 2.5D or 2D images. What is on the other side is lost in 3D->2.5D. Maybe this is not object+background but a single object (see the picture)?

Let me discuss with my team members

I think a good way to think about the question is by considering a standard z-buffer renderer. When you render a scene, 3D geometry is projected to a 2D framebuffer. Hereby, loss occurs in two cases:

- overdraw, when a surface pixel is behind another pixel

- projection, when a large patch of a surface is projected to a small region in screen space

The first phenomenon can be quantified by counting the overdraw per pixel. If you transfer the idea to the real-world camera setup, it tells you that everything hidden is lost.

The second phenomenon can be quantified by computing the mapping of world-space size of an object to screen space. For a perspective projection, this is worse if

a) the object is further away or

b) the object surface normal does not point to the camera

Transfered to a real camera scenario this means that you loose more information the further you are from the object and or look close-to-parallel to the object surface.

Now consider a setup where you take not one picture but pictures from all (or many) directions. The projection is invertible (you can reconstruct the 3D scene) if (and only if) the scene is convex. In this case, for every point in the scene there exists a camera position where the point is visible, and the point surface normal points to the camera. However, this only applied if you assume the camera is outside the scene.

In short: if you assume you can caputure many images, quantifying the information loss means quantifying a degree of concavity for the object. If you only have one picture, loss is big in generally, but probably per-pixel-overdraw is a good measure.

I did not consider ambiguity because of the shading. But if you want to dig deeper, check if "convex hull reconstruction", which is a popular problem in computer graphics.

What information? (You need to design this.)

If you are mostly interested in points at the corners of rectangular objects, then you could compare a known count of corner-points with the visible number of corner points.

Or it could be far more complex;

* texture (this could include behaviour of the illuminated surface w.r.t light reflectance and transmission. Google "BRDF shading" for more info)

* color

* shape (curves)

* size (are big objects/details more important than smaller ones?)

* text

Tim has a good point; you can easily investigate this problem using simple 3D computer graphics and experimenting with different scenes and approaches. Since you are in control of the scene, you will know exactly how much 'information' is in the full scene vs the 2D projection.

Occlusion is a big problem though. Consider looking at a full book-shelf from behind. All the titles/authors or the books, their size, shape, and position are all occluded by the back of the bookshelf (and facing the other way).

There is potentially also a fractal-like recursion here if you are basing this on real-world scenes; the closer you look, the more you will see. It's a bit like the "how long is the coastline" problem.

All listed caracteristics by Greg Ruthenbeck and others are included in the allowed bit rate in the case of objective evaluation:

Considering only fundamental image information (ie excluding depth information which might be inferred from prior knowledge of objects in the scene, information loss is between 33% (ie 1 dimension of 3) and nearly 100% (depending on the degree of obscuration - as discussed earlier)

Easy! 3D-2D=D

:-)

Couldn't resist it...

Appart from the answers you are getting, you can sometimes assume that the 3D scene is composed by 2D planes. When 3D points are coplanar, you can compute homographies between the image plane and each world plane.

In that (and only in that) case, the loss is 0.

Marcos,

I believe your homographies require information that's external to the 2D image plane, to reconstruct the full 3D information. An infinite number of coplanar sets of points in 3D, project to the same set of points in the 2D image plane.

Any image of a set of coplanar points in 3D gives the same inforrmation, it does not depend on the point of view. Projectivelly there is not lost of information taking an image of a planar scene. However, if the 3D metric information is search, the internal calibration of the camera is required, even if you have many images from different points of view of a planar scene.

The question is about a 2D image, not about a 2D projection of the 3D data.

With a 2D _image_ it is absolutely not true, that the information retained in the image, is viewpoint independent. This is trivially demonstrable by considering the points lying on any line in a plane. Any viewpoint that lies on the same line, obliterates all knowledge of those points, with the exception that at least one point is on it, somewhere.

Dear Stephan Irgenfried, The link below may help you.

http://courses.cs.washington.edu/courses/cse576/book/ch12.pdf

This example is a singularity. The exception of previous comment is when the point of view is coplanar with the scene.

Any image of a set of coplanar points in 3D gives the same information, it does not depend on the point of view. Other images, and the full set of coplanar points are reproducible by a simple transformation of coordinates. So, there is not lost of information taking an image of a planar scene, except the point of view is on the planar scene

That's better. I find it a bit peculiar to label a smaller infinite subset of possibilities, a singularity, but yes, the coplanar subset of all possible viewpoints, and especially those co-linear with any pairs of points in the plane, lose extra information.

If you know the 2D image, and the transform, and, if there are multiple planar scenes, the labeling of the loci in the 2D scene onto the planes, then you've lost no information. If you know only the 2D image, then you've lost the information of the transform, and potentially the labeling.

You can read some information about 'structure from motion' to understand the information loos from 3d to 2d. Some books to comment this topic: Multiple view geometry (by R. Hartley and A. Zisserman) and An invitation to 3d vision (by Yi Ma, S. Soatto, et al.) In addition, this topic explains how you can recover information 3d from 2d.

For kinematic purposes (angles, distances, velocity, ... ) it depends what you want to measure.

2D versus 3D in the kinematic analysis of the horse at the trot.

http://www.ncbi.nlm.nih.gov/pubmed/19082755

'Information', i.e. negentropy, is a property of probability distributions, not images. So one cannot even address loss of information without specifying probability distributions for the two objects concerned, i.e. codifying the knowledge we have of them. If the knowledge we have changes, or if two people have different knowledge, then the information loss will change. In particular, the more we know about the scene a priori, the less information loss there will be.

Once this has been done, the answer is simply a question of calculation (in principle, although perhaps very difficult in practice). Here is one example of how one might proceed.

Let S be the space of discretized 3d scenes, and I the space of discretized 2d images. (Let's not worry about the type of discretization.) The question of how much information is lost can only really mean something like the following. On the one hand, suppose that the 3d scene is known. This means that our probability distribution for the scene is concentrated on one point in S. This distribution has zero entropy. On the other hand, suppose that someone gives us a 2d image i \in I of the scene. What do we know about the original 3d scene s \in S? This information is contained in the probability distribution P(s | i, K), where K represents all the knowledge we may have about the 3d scene and the image formation process, excluding the 2d image. This probability distribution has an entropy, representing our uncertainty about s, i.e. our loss of information in having only the 2d image rather than full knowledge of the scene. The probability distribution can be rewritten using Bayes' theorem, as proportional to P(i | s, K) P(s | K), and its entropy calculated. Suppose that the discretization of the 3d scene represented by S is sufficient to determine the 2d image completely, i.e. P(i | s, K) = delta(i, f(s)), where f: S -> I is the image formation function. Then the resulting entropy is the entropy of the prior probability of s conditioned on it being a member of f^{-1}(i); P(s | i = f(s), K). Naturally this depends on the knowledge that we have of the scene before we see the current image, i.e. on P(s | K), on the image formation function f, and on the current image i.

We may assume, although it will never be true, that we have no prior information, i.e. that P(s | K) is uniform. Then the above entropy will simply be ln(n(i, f)), where n(i, f) = |f^{-1}(i)| is the cardinality of the set f^{-1}(i), i.e. the number of scenes that could have produced the given image. This will of course depend greatly on the space S, as well as i and f. In a very simple picture, in which the scene is determined by the image up to some 'surface' with depth d_{p} at each pixel p, but is undetermined 'behind' this surface, and in which there are n possibilities for each 'voxel' of the scene, the entropy is V(d)ln n, where V(d) is the volume 'hidden' behind the surface.

One can also try to average over possible images, to get an image independent measure of information loss, the conditional entropy H[S | I, K]. When the image is determined by the scene, we have H[S | I, K] = H[S | K] - H[I | K], and when P(S | K) is uniform, this gives H[S | I, K] = ln N - H[I | K], where N = |S|. The entropy of the image probability is simply the entropy of P(i | K) = n(i, f) / N, but this is hard to calculate even for the simple depth model, unless we specifically relate d to i.

The information lost will also naturally depend on details of the discretization, although the behaviour should be reasonable if the probability distributions for different discretizations are coherent.

Projection of 3-D scenery to 2 D image will lose depth information of the real 3D scene, however, quantify how much the information lost is not straightforward. It is depended on the information needed on specific application.

How much structural information is lost obviously depends on the shape of the 3D-object and its opacity.

Another problem with capturing 3D-scenes in 2D -images is scaling, as not only effective voxel resolution and image pixel resolution, but also camera distance and field of view come into play. Without this additional information, precise object dimensions might not be easily reconstructible, for example in biological applications, when anatomical structures are segmented and rendered from tomographies. Instead of placing a scalebar in the 2D-image, which lacks depth information, it is sometimes better to have a scalebar rendered as a 3D-object in the scene prior to image capture, provided the relative positions of scalebar and object are easily and unequivocally recognisable.

I agree with Karel, in some situations almost 100% data (I mean You think something but get nothing of it) loss will be there. So Orthogonal Sensors are used in some applications.

MINE is not a mathematical answer. But 3d is always more informative of the object and it's context than 2d. In 2d renderings, the brain has to guess what is happening to the object's other sides. Everyone's idea of that answer is whatever seems to be 'sensible' - The answer as to the nature of these "other dimensions' is going to be different for every viewer. Therefore it is definitely problematic and it is not measurable. Research and hypothesis based on this situation can be radically different - it depends on the imagination of the viewer. So projecting three dimensions from a 2d render should be used for purposes requiring an limited impression because it can only allow the viewer an limited understanding of the appearance of the object. "Understanding the 2d object in it's 'correct' context is difficult if not very imprecise, even in a 2d context. Everyone has a different idea of the object's dimensions and context. Projecting it's shape can give an inaccurate idea of it's actual shape. The eye and the brain act to add what is not there. And what is projected there by the mind is a cultural construction, not a physical attribute.

A non-scientific answer would be depth-of-field. The only difference in a 3 dimensional scene vs a 2 dimensional scene is how depth-of field is articulated. All other information should still be present provided you have an accurate depiction presented by the final image. Other variable that can effect the accuracy of the final image includes: aperture settings, focal length, lighting, contrast, movement of object and quality of the paper & film. All of the fore mentioned photographic accouterments can also play a part in the loss of accurate information. Personally, I think the biggest loss of information comes from how the final image is interpreted.

Naturaly there will be loss of information when we project 3D to 2D. Now it depends on application and ones intent. See, it is possible that the information I seek is available in 3D as well as 2D, so it solves my purpose, even with loss of information which is redundant for my case. Similarly, more dimension means more computation and more information and vice versa. Thats why one trades-off getting his purpose solved.

IMHO the question implies that we are posing the question in a definite deterministic way viz 1. we know the information in 3 d then

2 .we map the 3d onto 2d (projective or topologically ?)

Now the question is what is the mapping that retains max information in 2d .

Interestingly if we reverse map 2d to 3d(?) eg Riemann sphere we see

that although all points are mapped we do lose information .. well known

Euclid triangle (180deg) and non Euclidean (>180 or

I'm sure that it is possible mathematically to 'reconstruct' an area that is not visible to the viewer - eg. something hidden by the perspective of the object/s in the image. I would suggest that it is NOT possible to project what the mind cannot imagine. Seeing is very dependent on the categories of the 'possible' that exist in the mind. It could be that whatever exists 'on the other unseen side' of something may not be immediately understandable or imagineable. The mind and the eye work together to provide content for missing information. But the mind only provides what the mind already knows about.

The question implies and requires two answers -

1 the answer t0 the maths required to project a model of the unseen side/s

and

2. As human beings differ from one another in what they know and their expectations, We can merely speculate on what the mind 'sees'. It will differ. How can one possibly 'accurately' project that? ..and be 'correct'.

I agree with Vijay Mankar. Perhaps, defining what is "information" in the context of this question could help. I found the following definition that might be appropriate for the question in hand: "Information is any kind of event that affects the state of a dynamic system that can interpret the information." from: http://en.wikipedia.org/wiki/Information

IMHO..It is indeed a fascinating problem especially if we expand to n dimensions and in engineering , we often do this, eg second order representation of high order systems etc..and in many other situations of idealizing phenomena...we remove the distractions. and reduce the dimensions..

Here there are possibly two perceptions of the problem as follows.. I am reminded of Niels Bohr..

" The opposite of a profound truth could be another profound truth..".

1. Strictly Mathematical : We know that it is an inanimate device that captures the images viz the

the digital camera . It is also well known that engineering drawing ..

relies on projections viz Elevation..Plan .. side view..etc

and we can actually build the part using these images which

was originally a 3d object in the designer's mind. so the question

is how can we innovate a mathematically efficient algorithm

that uses the minimal combinations of 2d views..here it would seem

a perspective view would be appealing..

2. Strictly Philosophical :We could take a completely subjective view.. then we are not relying on mathematics but a holistic world view .. and it does seem highly intractable.. as stated by Jean..." We can merely speculate on what the mind 'sees'.

and How can one possibly 'accurately' project that? ..and be 'correct'. "

but further thought shows . that scientific theories rely precisely

on such extrapolation by the human mind and if it is correct

we get new technologies and knowledge , else it is back to the drawing board.!.

One definitely must keep trying to imagine new technologies or we cease being human. However, that said, it is important to keep the philosophical views in the back of one's mind. None the less, it is extremely useful to be able to project mathematically what might be present in a specific situation. It helps if the person trying to project something that is not immediately visable actually has some knowledge of the larger context.