# Panoramic Resolution

## Introduction

Some panoramas look sharper than others. If two panoramas are represented in the same format, we can simply compare the number of pixels in their respective images in order to compare the resolution of the panoramas. The matter is not quite so simple if the panoramas are represented in different formats.

Some panographers use terms like dots per inch (dpi), pixels per inch (ppi), or dots per centimeter (dpc) to describe resolution. This is a meaningless number, because the dpi can be changed to an arbitrary number without changing the number of pixels in the image, yet the image still retains its sharpness and field of view.

### Resolution Calculators

Resolution Formulas

Perspective Resolution from Focal Length

Perspective Resolution from Field of View

Fisheye Resolution from Field of View

Panorama Resolution

Dimensions from Resolution

Window Field of View from Resolution

Window Height from Resolution

## A Standard of Panoramic Resolution: Angular Pixel Density

It is the purpose of this paper to introduce a well-defined method to compare the resolutions of various panorama formats:

angular pixel density.

First, we need to define what we mean by angular pixel density, because no image format is uniform in angular pixel density across the whole image.

We use the focal length as the basis of the definition.

## Definition

The focal length is defined as the distance to the imaging surface at the center of projection.

This definition is consistent with that for perspective and fisheye lenses.

We interpret this for various image formats of interest:

• For a perspective image, this is the distance to the imaging plane at the center.
• For a cubic panorama, this is the distance to the center of one of the faces.
• For a cylindircal panorama, this is the cylindrical radius (at the equator).
• For an equirectangular spherical panorama, this is the radius of the imaging sphere.
• For a fisheye image, this is the radius of the imaging sphere.

The pixel is used as the unit of measurement for image resolution and focal length. In these units, the focal length is identical to the angular pixel density at the center of projection. We prefer to use the focal length to define panorama resolution because it is a parameter that is central to the mathematics of projection, and is a well-defined for any panoramic image format.

Panorama resolution defined in terms of the focal length happens to be the minimum angular pixel density for the panorama formats we consider here, when "square" pixels are used. This is not necessarily true for other panorama formats, though (e.g. annular formats).

The units for focal length, when interpreted as angular pixel density, are pixels per radian.

It is more convenient to speak of angular pixel density in terms of pixels per degree instead of pixels per radian, for a couple of reasons:

• Degrees are used almost exclusively to quantify measurement of angle, outside of mathematics; and
• The magnitude of currently published panoramas is of the order of a single decimal digit (1-7) when expressed in pixels per degree.

We then make the following definition:

## Definition

Panoramic resolution is the angular pixel density expressed in units of pixels per degree as determined by the focal length.

## Resolution of Various Panorama Formats

Here, we give the formulas to convert between resolution in pixels per linear dimension of various panorama formats and the equivalent dimension in terms of pixels per degree.

Dimension Resolution (pixels/degree) small dimension (s) focal length in mm (f) f * (s - 1) / 24 * pi / 180 face dimension (w) (w - 1) * pi / 360 circumference (c) c / 360 circumference (c) c / 360 diameter (d) (d - 1) / 180 small dimension (s) sqrt(13 * s * s - 20 * s + 8) / 360 ~ (s-1) * sqrt(13) / 360 ~ s / 100

For perspective and fisheye images, we assume rectangular images in a 3:2 aspect ratio. The long dimension can be used instead of the short by simply scaling it first by 2/3.

The formula for the perspective image assumes that a standard 35 mm camera is used, with a frame size of 36 mm x 24 mm.

## Resolution of a Perspective Lens

This calculator computes the resolution in pixels per degree from the focal length.

The image and objective dimensions should correspond, i.e. since a 35 mm objective frame is 24 mm on the short side, the short dimension of the image should be used. If the focal length is given in 35 mm camera equivalents, then the objective dimension should be that of the 35 mm frame (24 mm for the short side and 36 mm for the long side).

This calculator computes the resolution in pixels per degree from the field of view.

Either the long or short dimension can be used, as long as the field of view is specified for the same side.

## Resolution of a Fisheye Lens

This calculator computes the resolution in pixels per degree of a fisheye lens.

## Resolution of a Given Panorama

This calculator converts from resolution in pixels per image linear dimension to resolution in pixels per degree.

Panorama Type circumference side length

## Field of View of a Window Viewed at a Given Angular Resolution

This calculator computes the field of view for a window of a given size and angular resolution.

## Size of a Window Needed to View a Given Angular Resolution

This calculator computes the height of a window needed to show a given field of view at a given angular resolution.

## Pixels Needed for a Given Angular Resolution

This calculator converts from resolution in pixels per degree to resolution in pixels per image linear dimension.

Enter the desired resolution in pixels per degree, and press the button to see the number of pixels needed in each of the different formats.

Panoramic Resolution ==> pixels per degree Perspective (12 mm) small dimension Perspective (15 mm) small dimension Perspective (18 mm) small dimension Perspective (20 mm) small dimension Perspective (24 mm) small dimension Perspective (28 mm) small dimension Perspective (35 mm) small dimension Perspective (40 mm) small dimension Cubic face dimension Cylindrical circumference @124.8°FOV @90°FOV Equirectangular Spherical circumference Single Round 180° Fisheye diameter Double Round 180° Fisheye diameter Round 360° Fisheye diameter Full Frame 180° Diagonal Fisheye small dimension

The total number of pixels in area includes border pixels needed to make each image rectangular. Since these can be chosen to compress well (e.g. solid color), these should add little to the size of the image. In the case of round fisheye images, a factor of pi÷4 = 79% can account for these. When determining pixel area, 180° vertical FOV is assumed for all except the cylindrical panoramas. For the perspective images, a 3:2 aspect ratio is assumed.

The initial value for the resolution, 5.57 pixels/degree, was chosen to match the CD standard used in QuickTime VR authoring: a 15 mm lens producing 768x512 images. A resolution of 2.78 pixels/degree is more the minimum standard used on the web, starting with 384x256 images from a 15 mm lens, when viewed in a 320x240 window.

For a demonstration of what different resolutions of cubic QTVR previews looks like, look at the

### Panoramic Preview Resolution Comparison

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last updated: 10/2/05