A 2­-dimensional image of a 3­-dimensional object can vary significantly according to the distance from the viewer to the object. CSMacLaren described perspective distortion as it relates to the Vader helmet in 2008 in the topic “Doing Better Photographic Analysis.”

There are pitfalls and challenges when using images as a basis for comparison, but often images are all we have to use as source material as a basis for comparison. A better understanding of the sources of optical effects or distortions can assist in making (or not making) use of images to support our opinions or conclusions.

Here I attempt to extend the discussion by quantifying the perspective distortion effect in terms of angular magnification. There's a little bit of math and trigonometry, and a few pictures too; I hope this might be of interest to some.

The following link is to a single pdf document stored in google docs:

https://drive.google.com/file/d/0B13YyeojRHKeUzdLVHFucnV6X1E/edit?usp=sharing

Totally true.

A perfect example of this is this promo pic taken some time before ESB came out (left). The helmet in the picture appears to have a really small, skinny neck, especially when you compare it to the picture in the middle of the screen used ANH helmet.
But a photo of this helmet taken many years later (right), shows a neck that is much more in line with the original screen used ANH.

But this is not just distance. The first picture was also taken with a wide-angle lens. If you look at the dome and compare how much of the side skirts you can see it appears to be almost farther from the camera the middle image, since you can see more
of the side skirts. but the small neck suggest that the helmet is much closer. Lighting can also affect how something appears. Brightly lit objects, often appear broader or fuller than they do under lower light situations.

The ESB poster helmet has a slimmer neck than the screen used helmet, mainly caused by the pulling in of the right side of the neck (left in picture) making it straighter and not flare out as much as it should. In fact, it is one of the more recurring flaws seen on Vader helmets - the pulling in on the right side versus the pulling out on the left side also seen on many helmets. It certainly isn't as "thin" as it appears in most of the ESB poster pictures, as seen in this alternate picture from the same photo session. Would be interesting to know what camera was used for those finished shots and it would be cool to see more from the series.

Yeah, it definitely is pulled in, I can see it. I've never seen this picture before. I'm sure it's definitely taken with a nice big studio camera with fully adjustable focal lengths and stops, used with studio lighting and backdrops, which is why it's so hard to replicate the pictures sometimes, using your home digital camera.

Lens distortion is a separate consideration from perspective distortion caused by distance to the object. Both are important to determine if two images are comparable.

Examples of lens distortion include barrel distortion in a wide-angle fisheye lens and pincushion distortion in a zoom lens. Perspective control (e.g. tilt-shift lenses) can also transform an image.

Rectilinear lenses which have no lens distortion or transformation produce images which are most easily compared. Cropped images taken from the same distance using a wide-angle lens or a telephoto lens can compared if the optics of both lenses are rectilinear.

Image differences due to lens distortion can be resolved through software if the parameters of the lens used are known or can be ascertained from the image. This is accomplished by appropriately stretching or compressing the 2D image.

However, images of an object taken at different distances cannot be corrected through software because it's due to the 3D depth of the object which is not captured in a photo. Features of the object which span depth along the viewing axis will look different based on distance.

It may be possible though to compare a feature from images taken at different distances if that feature is confined to one depth plane (e.g. the nose and breath mask outline if both photos were taken with an optical axis perpendicular to that feature). The eyebrow however would be a feature that could not be compared from images taken at different distances because that feature will always span multiple depth planes.

But in every case, for two photos to be usable for comparison of shape and dimension, the viewing angle to the helmet must be the same.

I tried to replicate the viewpoint (camera distance and angle) of the ESB movie promo picture. In order for the mouth to appear small relative to the rest of the helmet, I knew I'd need to pull back some distance (> 10 ft at least). Best I could come up with was from a distance of about 12 ft. It's also about a far back as the feeble iPhone 5 camera and digital zoom could manage. Angles relative to the front of a level faceplate: elevation angle about 10 degrees (i.e. aiming camera at a top-down angle), azimuth angle about 5 degrees, favoring the left side of Vader. I also tilted the dome so that Vader's right side jutted out a bit more to better match the promo picture.

I'm using my old standby DP DLX (left) to the ESB movie poster (right). An apples to oranges comparison I know, but only trying to use for feature match for viewpoint estimation. No stretching or distortion of the images, just scaling and exposure adjustments. I mainly was trying to match the eyebrow shape, dome orientation (particularly having the bottom level), and breathmask size. This was the best I could come up with:



The narrowness of the neck of the ESB promo appears to be inherent trait. The emphasis of the ESB promo poster is the bottom of the dome which you see more of if the neck is narrower. This narrowing of the neck could be accomplished using a software distortion effect. But I don't think Photoshop for the Apple ][ or TRS-80 was available in 1979 ;)

One of the tricks to the ESB pic is that it is taken at the height of Vader's widow peak. But I would also say it was taken at about 12 feet with a wide angle distortion lens.



The closest I could get to the original image, that is, the size of the dome forehead vs the face vs the neck. This only about 3 1/2 feet.



At about 8 - 9 feet the bottom of the skirt levels out, with the dome forehead and face size is similar, but the neck is much wider, almost as wide as the face



To get the closest result, I used a distort program on a pic taken at 8-9 feet. Obviously this was not available in 1979, but the program is meant to replicate a distortion lens which did exist.

So, I'm not really sure why you started this thread. I keep kind of wondering that. But it is definitely important. Especially if you're going to use photo analysis it to try to say that something is or isn't what is purported to be. From what I've seen, there are three people on this forum who regularly use visual comparison, myself, Mac, and Thomas. There may be others but not as frequent.

Thomas, if you're still out there (haven't seen you lately), I know you like to use transparent overlays to show matches or mismatches in castings. But I submit, that unless you are in control of both subjects being compared, i.e. exact same distance to object, same camera/ lens, exact orientation to object and lighting, such data it is haphazard at best and could be easily wrong. As an example, I submit one of your comparisons that used an overlay.

While it appears to show a difference in the Christie's facemask and your ESB promo cast, I can tell just looking at the two different pictures, that they are not taken from exactly the same distance and exactly the same orientation. The Christie's is taken at a slightly higher angle (Note the distance between the tusks/tubes) and from somewhat further away (Note how far the notched bridge between the eyes projects from forehead, just as when you get closer to the ground, objects in the distance start to go over the horizon). It should be noted that the Christie's was also a prepared helmet that you're comparing to an unfinished casting.

I also suspect, that the picture of the Christie's was obtained, and the picture of the ESB promo with a digital camera, attempting to replicate the Christie's photo. If this is the case, I suggest anyone, that before you submit, that it may be necessary to take multiple photos and do a match comparison. I know this is easier for some harder for others. But if you cannot do this then you have no business doing such comparisons.


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Note: it is not my intention to pick on you Thomas, as you have a solid compassion that I welcome and you trying hard to get your viewpoints across and I respect that, as opposed to those who come along and say, no, it isn't or this is this way, period.


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John

I started this thread to expand upon the topic of perspective distortion. I have no other agenda except to expand on the understanding and knowledge base. It may lead to more flexible methods for performing photographic analysis, or it may not. But it's worthy of investigation.

In prior threads the effect was demonstrated to be significant, and it was dependent on distance. The recommendation was to take photos from a standard recommended distance 6 ft and from same camera angle as a basis for comparison, and that's a good suggestion. But it did not quantify the effect.

By quantifying the effect, we *may* be able to do things like:

1) Knowing the actual linear distance (or ratio) between two visible features in a photograph, we could determine the distance from camera to the item.

or

2) If you know the depth separation between two features and the linear measurement of one feature, and the distance from camera to the item, you could calculate the actual linear measurement of the second feature.

I believe the perspective distortion effect as it concerns us can be characterized as the difference in angular magnification between front and back of the photographed item. In essence, it's merely a matter of geometry. In fact a better name would be "distorted perception" because what's captured in the 2D image is accurate in that it is the same as what you'd see with a one eye. But it looks strange with lack of other depth cues and when the image is zoomed and/or cropped.

In order to prove anything, it's important to demonstrate it using controlled, measurable test cases. So I'll start by comparing an item with itself at different distances. Then we can compare what is predicted through calculations with what is observed in actual images to see if they are the same.

And since this is a generic phenomenon, not specific to any particular photographed item, I'll use a stormtrooper helmet I have in-hand rather than a vader helmet. I've taken photos from 1ft to 10ft distance in 1ft increments and will compare the measurements 1) ear-to-ear and 2) between the tube openings.

https://drive.google.com/file/d/0B13YyeojRHKeYjBKRDlLRWtkUGFFU29tRzZhU0tmS2dHWktN/edit?usp=sharing

...to be continued in my next post

Start with making accurate linear measurements on the helmet for features to compare. I've chosen features that are separated in depth, parallel to each other, and perpendicular to our viewpoints. Feature 1 is the distance between the outsides of each ear. Feature 2 is the distance between the outside edges of the tube openings in front:



Length between tube openings 149mm
Length between ears 308mm
Depth from ears to tubes 136mm

Photos were taken pulling back from the front of the helmet in increments of 1 ft maintaining constant height and cropped to contain only the helmet.

From each viewpoint we can calculate the subtended angle of each feature. Note that for a given viewpoint there will be 2 known distances: from viewpoint to tubes and from viewpoint to ears.



The following equation relates subtended angle, distance, and feature length (h=feature width in our case):



We calculate the angle subtended by each feature and plot them as a function of distance to the front of the helmet:



The angles become smaller as we pull back. But what may not be so apparent is that the ratios of the angles change with distance. With increasing distance we find that the ratio of the angles subtended converges to the ratio of the actual linear measurements of the features. This is what creates the perspective distortion along the depth of the object at different distances.



In the captured images we can set the length of the tube-to-tube feature to its known linear measurement (=14.9cm) and then measure length of the other feature (ear-to-ear) in proportion. These image measurements were taken at distances of 2ft, 6ft, and 10ft:





Ironically, a more distant image creates a truer correspondence to the actual feature lengths when compared to each other. In effect there is a compression of depth with greater distance such that the back depth plane and front depth plane are virtually "squashed together".



Measurement errors taken into consideration, these image measurements (points on the above plot) are very close to the calculated predicted values.

So for this controlled experiment the ability to predict and quantify the effects of perspective distortion have been demonstrated, though the conditions were constrained to select features contained entirely in separate depth planes. But given a similar scenario where 3 of 4 values is known among Distance, Depth between Features, Feature length 1, and Feature length 2, the equations provided could be used to predict the unknown value.

A google document containing a spreadsheet of values and calculations presented is available at:
https://docs.google.com/spreadsheets/d/16owHfgKkfjNcMJvXJdzJe1f3lSgiGfCDq4I1YdTON-w/edit?usp=sharing

Perspective distortion can be quantified, but the reality is we often do not have sufficient information to solve for missing value(s) and make accurate calculations. Making use of geometry and trigonometry to manually solve for simple, constrained cases does work, but to support real-world scenarios a more general approach is needed. That more general approach would be one that allows for a variety of image source material which a computer can automatically correlate and process.

Web searching led me to Photogrammetry and more specifically to "Structure From Motion" which can take a set of input images and solve for camera viewpoints (distance and angle) as well as create a 3D model of photographed object(s). I'll created a new forum thread entitled "Structure from Motion" to continue the discussion.

Here's a teaser... using only a set of photos as inputs, "VisualSFM" running on a Windows PC solved for the camera viewpoints as well as a 3D object model of the photographed object :

Dispensing with all the nerdy gobbledeegoop, some great examples of perspective distortion can be found from ESB screenshots of Vader's dinner party at Cloud City.

In the first shot note how large the dome looks and how large the window behind Vader looks. Then as the camera moves closer in the second shot how the relative size of the mask and dome changes, and how the window behind Vader now looks smaller. And in the last picture how small the dome looks, the dome now in the foreground.

The helmet appears to be stationary across all 3 of these photos, yet it looks much different in each of them, because of camera distance and which part of the helmet is in the foreground/background.





:!: Inevitable reversion to gobbledeegoop... :!:
The same phenomenon that causes the window size to change is the same one that causes the dome size to change, a consequence of differences in angular magnification between two objects in different depth planes perpendicular to the viewing axis. The differences are proportionally greater at short range, lesser at long range.

If you were to take an actual linear measurement of the diameter of the window and the height of the dome or width of the chest control box etc. you would find their ratio corresponds closely to what you would measure of the same features from the long range photo, less accurate from the short range photo. This is the result of axial magnification along the depth of an object, proportional to the square of lateral or transverse magnification. Axial magnification is greater in the long range photo which effectively makes Vader appear that his back is right against the window, hence the closer correspondence of linear measurement of the real objects to those in the image.