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Warping Peirce Quincuncial Panoramas

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Warping Peirce Quincuncial Panoramas Warping Peirce Quincuncial Panoramas Chamberlain Fong [ [email protected] ] 11/12/2010 Abstract The Peirce quincuncial projection is a mapping of the surface of a sphere to a square. It is a conformal mapping except for four points on the equator. These points of non-conformality cause significant artifacts in photographic applications. In this paper, we propose an algorithm and a user-interface to mitigate these artifacts. We then promote the Peirce quincuncial projection as a viable alternative to the stereographic projection in photographic applications. Keywords: Peirce Quincuncial Mapping, Peirce Quincuncial Projection, Panography, Image Warping, Map Projections, Computational Photography, Stereographic Projection, Equirectangular Images Introduction The Peirce quincuncial projection [1][2] is a mapping of the surface of a sphere to a square. It was originally proposed by Charles Sanders Peirce in 1879 for geographic applications. Indeed, Peirce was working for U.S. Coast and Geodetic Survey when he came up with this projection. In the recent years, the photographic community [3] has adopted this projection for aesthetic purposes. The figure below shows Peirce's projection of the world and an example of its use for photographic purposes. Figure 1: Peirce's quincuncial projection of the the world and a photographic example 1 Spherical Panoramas Anyone who has used street view in Google maps should be familiar with these type of photographs. There is a sphere of light that shines upon us all the time. Advances in photography and computer graphics [4][5] has allowed the photographer to capture and view this sphere. These type of photographs are called spherical panoramas . They can be either be viewed interactively using a computer as is the case with Google street view; or using some sort of projection mapping the surface of the sphere to a planar image. There is a large and mature community of photographers who specialize on spherical panoramas. For the rest of this paper, we shall refer to these type of photographers as panographers and their art as panography . There is a theorem in differential geometry that states that the sphere is not a developable surface. This means that it cannot be flattened onto a plane without distortion. Since the times of Ptolemy circa 100 A.D., geographers and cartographers have come up with various projections of the sphere to the plane. This is a significant problem in geography and map-making that there are whole books [6] devoted to this topic. Needless to say, every type of spherical projection has some sort of compromise in which it distorts distance, area, scale, direction, or angles in order to represent the sphere on a plane. The spherical panorama community [7] has adopted several spherical projection techniques to view their photographs. The standard projection used by panographers is called the equirectangular projection. Most spherical panoramas are stored and viewed as equirectangular images. Equirectangular images always have an image dimension ratio of 2:1. This follows from the fact that the sphere has 360 angular degrees in longitude and 180 angular degrees in latitude. An example of an equirectangular image is show below. We have color-coded some important parts for illustration purposes. The equator is colored red. The north and south poles of the sphere are colored purple and yellow, respectively. In panography jargon, the north pole is known as the zenith and the south pole is known as the nadir . Figure 2: An equirectangular panorama 2 Another popular projection among panographers is the stereographic projection. The stereographic projection is conformal, which means that it preserves angles locally within the image. This property makes stereographic panoramas aesthetically pleasing to the eyes. In fact, there is a large online community [8] of panographers who use the stereographic projection to make artful "little planets" that simulate an overhead aerial view. Figure 3 shows an example of the stereographic projection applied to our equirectangular image in figure 2. There are two types of stereographic projections depending on which pole the projection occurs. Figure 3: Stereographic panorama -- nadir type (left) & zenith type (right) The Peirce Quincuncial Projection The Peirce quincuncial projection is closely related to the stereographic projection. It was devised by Peirce after drawing inspiration from Hermann Schwarz's mathematical result on a conformal mapping of the circle onto a polygon with n sides. This result is called the Schwarz-Christoffel transformation and is an important theorem in the field of Complex Analysis. The Peirce quincuncial projection has several features that make it visually interesting. One pole lies exactly in the center of the square, while the other pole is split among the four corners of the the square. The equator lies on an inner square that is formed by connecting the midpoints of each of the edges in the perimeter of the square. This inner square is tilted by 45 degrees when compared to the outer square. One important feature of the Peirce quincuncial projection is that it is a conformal mapping, except for four points. Conformality is important for aesthetic reasons in photographic applications. We will talk more about the non-conformal points in a latter section of this paper. 3 Figure 4: Overview of the Peirce quincuncial projection An overview of the Peirce Quincuncial projection is shown in figure 4. In this overview, we start from a sphere and produce a square projection. The first step splits the sphere into northern and southern hemispheres. The next step does a stereographic projection of the two hemispheres. This is followed by using the Schwarz-Christoffel transformation on these two stereographic discs to produce two squares. These two squares are then reassembled as a bigger square in a quincuncial fashion. The main equation provided by Peirce for his quincuncial projection is: In this equation, every point in the stereographic disc represented by the complex variable w with polar coordinates (p ,θ ) is mapped to complex variable z in the Peirce square with (x,y) coordinates. In practice, it suffices to know that Jacobi elliptic function cn( z,1/ √2) can be used in mapping the circle to a square in the complex plane, and vice versa. Note that the complex Jacobi elliptic function is just a special case of the more general Schwarz-Christoffel transformation. Figure 5 shows a detailed example of the Peirce quincuncial projection steps applied to a real spherical panorama. We start with our equirectangular image split into the northern and southern hemisphere and proceed with intermediate images corresponding to steps in the Peirce quincuncial projection. The projection is called quincuncial because the two small squares are rearranged in a quincuncial shape to form a larger square. 4 Figure 5: Visual example of the Peirce quincuncial projection steps 5 Figure 6: Peirce quincuncial panorama Limitations and Previous Work As the panorama in figure 6 shows, the Peirce quincuncial projection has a serious inherent flaw. It is not conformal on four points. These four points fall on the midpoints in the perimeter of the square. For photographic applications, these non-conformal points cause ugly discontinuities in the image. This flaw has limited the use of the Peirce quincuncial projection in the spherical panorama community. For example, in figure 6, the non-conformal flaw causes the basketball ring to be split in half into left and right parts of the image. It also causes the lower ground to bulge unnaturally. During the authorship of this paper, there were only less than a dozen Peirce quincuncial panoramas showing up in a Google image search. In contrast, there are over 46,000 results for stereographic panoramas. This is true despite the fact that there is a Photoshop plugin called Flexify [9] by Flaming Pear Software for converting equirectangular images to Peirce quincuncial. Moreover, there is free source code available for a full implementation of the Peirce quincuncial projection in the Mathmap scripting plugin of the GIMP photo-editing application, written by flickr.com user Breic. It is likely that panographers who tried the Peirce quincuncial projection for their spherical panoramas found the non-conformal flaws unappealing and abandoned this projection. In this paper, we will present a proposed modification of the Peirce quincuncial projection to mitigate its non-conformal flaw. It is our hope that our modification will make the Peirce quincuncial just as popular as the stereographic projection among panographers. 6 Our proposed modification improves on previous ideas by people who have used the Peirce quincuncial projection for panoramas. For example, flickr.com member Breic [10] allows for the user to move the equator up and down in latitude within his Mathmap script. He also gives the user freedom to spin the spherical panorama along its main axis to move the four non-conformal points together around the equator before applying the projection. Our proposed modification gives the user more freedom by allowing the user to choose any latitude and longitude for each of the four non-conformal points. We will discuss a user-interface and an algorithm for warping the projection in the next section. Warping the Equator The main idea of this paper is to warp the equator of the input spherical panorama so that the non-conformal points fall on "uninteresting" areas of the image. This warping will hide the image discontinuities and make the result more aesthetically pleasing. The resulting image is a warped quincuncial panorama that is no longer conformal, but still generally preserves much of the characteristic look of a overhead aerial image. We developed an interactive program where a user works on an input equirectangular image and moves four control points in the user-interface. These four control points always fall on the equator and will, in fact, redefine the equator after they have been selectively manipulated and moved.
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