Panorama Equivalence

# Panorama Equivalence

When using a narrower-angle lens and panoramic stitching to get a higher-resolution photo with the field of view of a wider-angle lens, there are some things that change in the resulting photo.

## ​1. Overview

Panoramic stitching can be used to obtain wider-angle photos or higher resolution photos. But going from a lens to a lens with a longer focal length changes the image in ways other than zooming in. Therefore, the resulting panorama, while having the same angle of view as the single photo from the original lens, may also change in other ways.

## ​2. Equivalent Lens and Sensor

To describe how the image changes it is convenient to talk about an "equivalent lens" and an "equivalent sensor": the equipment that, if used, would produce the same output image as the stitched panorama, but in one exposure. This allows us to intuitively grasp the changes and also allows us to use existing photographic calculators to make predictions about the output panorama.

Throughout this article, I will refer to the "lens" as the actual lens that is attached to the actual camera and through which the exposures that make up the panorama are captured. The "sensor" is the actual sensor sitting in the camera. The "equivalent lens" is an imaginary lens, which if mounted on the camera whose sensor has been replaced with the "equivalent sensor" would, when exposed, produce the output panorama in a single exposure.

The calculations presented here are not supposed to be exact, but rather be easy to use without any tools and to provide a way to quickly get a feel for what the finished panorama will look like when you are in the field.

### ​2.1. Panorama Size

For simplicity we will restrict ourselves to panoramas that have the same aspect ratio as the sensor and therefore consist of the same integer number of rows and columns. For example, a panorama with three rows and three columns, or two rows with two photos in each. If you are capturing a panorama with a different aspect ratio then you can use the larger of the number of rows and columns and then pretend that your output panorama is cropped from a larger panorama. For example, a two-row by four-column panorama can be treated as a four by four panorama that you then crop down to half height.

We will refer to the number of rows and columns as N.

We also assume that the panorama photos overlap a little; that is, about 20-30%.

### ​2.2. Equivalent Sensor

When computing the values of an equivalent sensor it is possible to either scale up the sensor and its resolution, or just scale up the resolution. For simplicity we'll do the latter. The size of the equivalent sensor is therefore the same as the size of the sensor.

The resolution of the equivalent sensor is simply the resolution of the sensor times the number of photos in the panorama, or the resolution of the sensor times the square of N.

### ​2.3. Equivalent Focal Length

The focal length of the equivalent lens is the focal length of the lens divided by N, the number of rows or columns of the panorama. If you shoot a three-by-three panorama, the equivalent lens will have a focal length that is a third of the lens's. For example, a 45mm lens will get you a 15mm equivalent lens; a 60mm lens will get you a 20mm equivalent lens.

### ​2.4. Equivalent Aperture

The equivalent aperture as expressed as an f-number is the f-number of the lens divided by N, the number of rows or columns of the panorama. If you shoot a three-by-three panorama, the equivalent lens will have an f-number that is a third of the lens's. For example, an f/6 lens will get you an f/2 equivalent lens; a f/9 lens will get you an f/3 equivalent lens.

## ​3. Application

Now that we have the parameters of our equivalent lens the question is how to apply it in the field.

### ​3.1. Framing

By using a lens with the same focal length as the equivalent lens you can frame the shot. We will call this lens the "framing lens", and the f-number of the lens when it is set to produce the depth of field that you want for the final panorama the "framing f-number". We will call the result of an exposure the "framing shot".

### ​3.2. Depth of Field

If you are fine with a lens with the larger (smaller f-number) aperture of the equivalent lens you don't have to do anything. This is the technique used in the Brenizer method or "bokehrama"[a], where you use the huge aperture to get a razor-thin depth of field. But if you are shooting landscapes you may want to figure out what the f-number for the lens is where you get the same depth of field as with the framing lens set at the framing f-number.

The first thing you need to decide is if by "the same depth of field" mean: (1) a panorama that is equally sharp when viewed at the same size as the framing shot, or (2) a panorama that is equally sharp when viewed at 100%.

The difference is subtle: in the first case you want a circle of confusion for the equivalent lens that is the same size as the circle of confusion of the framing lens, in the second case you want a circle of confusion that covers as many pixels on the equivalent sensor (which has much higher resolution) as the circle of confusion of the framing lens does on the sensor. That is, if you print the framing shot and the panorama at 8x10 inches, the first option will have them look the same, but if you view them both at 100% the panorama will be blurrier because it has so many more pixels; the second option will make the 8x10 panorama look sharper, and when viewed at 100% they will look equally sharp with the panorama having much more detail.

To get the first result, you need to set the lens to N times the aperture of the framing lens. That way, the equivalent lens will have the same aperture as the framing lens. To get the second result you need to set the lens to N^2 times the aperture of the framing lens.

For the latter case you may end up with some very small apertures. For example, when using a 45mm lens and a three by three panorama, you may want the output of a 15mm at f/9 - but that would require you to set your 45mm lens to f/81, which simply isn't possible using ordinary lenses.

The end result is therefore that you probably have to do a tradeoff. One option is to focus farther into the scene. For example, while a 45mm at f/81 is impossible, a 45mm at f/11 is not a problem if you can accept a near focus limit of eight meters instead of half a meter. In landscape photography you typically have a landscape that stretches into the distance, in which case a near focus limit of eight meters is not a problem, or you have some object close to you, in which case the slightly blurry background isn't a problem in the first place. Another option is to do focus stacking of the panoramas, but this can get really boring real fast.

## ​4. Worked Example

This is how I use these formulas:

1. I arrive at location and put my 15mm lens on the camera and frame the shot.

2. Next, I decide on the panorama size. Typically, I do a three-by-three panorama. This means I'll use a Nikon 24-85/3.5-4.5 set to 45mm, and my output panorama will be roughly 200 megapixel.

3. Since I want as much depth as possible, I set the lens to f/13 which is about as much as the lens can handle. Being the 100% pixel-peeper that I am, the equivalent lens is a 15mm@f/1.3.

4. The hyperfocal distance is six meters, so everything from three meters to infinity will be in focus using a circle of confusion of 0.03 mm. This is acceptable. Alternatively, I would now focus on the foreground object that I needed in focus and let the background go blurry.

5. I switch the lens from auto-focus to manual focus and capture the panorama.

2019-09-14