A Pixel Is Not A Little Square, A Pixel Is Not A Little Square, A Pixel Is Not A Little Square!

My purpose here is to, once and for all, rid the world of the misconception that a pixel is a little geometric square. This is not a religious issue. This is an issue that strikes right at the root of correct image (sprite) computing and the ability to correctly integrate (converge) the discrete and the continuous. The little square model is simply incorrect. It harms. It gets in the way. If you find yourself thinking that a pixel is a little square, please read this paper. I will have succeeded if you at least understand that you are using the model and why it is permissible in your case to do so (is it?). Everything I say about little squares and pixels in the 2D case applies equally well to little cubes and voxels in 3D. The generalization is straightforward, so I won’t mention it from hereon1. I discuss why the little square model continues to dominate our collective minds. I show why it is wrong in general. I show when it is appropriate to use a little square in the context of a pixel. I propose a discrete to continuous mapping—because this is where the problem arises—that always works and does not assume too much. I presented some of this argument in Tech Memo 5 ([Smith95]) but have encountered a serious enough misuse of the little square model since I wrote that paper to make me believe a full frontal attack is necessary. The Little Square Model The little square model pretends to represents a pixel (picture element) as a geometric square2. Thus pixel (i, j) is assumed to correspond to the area of the plane bounded by the square {(x, y) | i-.5 ≤ x ≤ i+.5, j-.5 ≤ y ≤ j+.5}. 1 Added November 11, 1996, after attending the Visible Human Project Conference 96 in Bethesda, MD. 2 In general, a little rectangle, but I will normalize to the little square here. The little rectangle model is the same mistake. A Pixel Is Not a Little Square! Microsoft Tech Memo 6 Alvy 2 I have already, with this simple definition, entered the territory of controversy—a misguided (or at least irrelevant) controversy as I will attempt to show. There is typically an argument about whether the pixel “center” lies on the integers or the half-integers. The “half-integerists” would have pixel (i, j) correspond instead to the area of the plane {(x, y) | i ≤ x ≤ i+1., j ≤ y ≤ j+1.}. This model is hidden sometimes under terminology such as the following— the case that prompted this memo, in fact: The resolution-independent coordinate system for an image is {(x, y) | 0. ≤ x ≤ W/H, 0 .≤ y ≤ 1.}, W and H are the width and height of the image. The resolution dependent coordinate system places the edges of the pixels on the integers, their centers on the edges plus one half, the upper left corner on (0., 0.), the upper right on (W., 0.), and the lower left on (0., H). See the little squares? They would have edges and centers by this formulation. So What Is a Pixel? A pixel is a point sample. It exists only at a point. For a color picture, a pixel might actually contain three samples, one for each primary color contributing to the picture at the sampling point. We can still think of this as a point sample of a color. But we cannot think of a pixel as a square—or anything other than a point. There are cases where the contributions to a pixel can be modeled, in a low-order way, by a little square, but not ever the pixel itself. An image is a rectilinear array of point samples (pixels). The marvelous Sampling Theorem tells us that we can reconstruct a continuous entity from such a discrete entity using an appropriate reconstruction filter3. Figure 1 illustrates how an image is reconstructed with a reconstruction filter into a continuous entity. The filter used here could be, for example, a truncated Gaussian. To simplify this image, I use only the footprint of the filter and of the reconstructed picture. The footprint is the area under the non-0 parts of the filter or picture. It is often convenient to draw the minimal enclosing rectangle for footprints. They are simply easier to draw than the footprint—Figure 1(d). I have drawn the minimal rectangles as dotted rectangles in Figure 1. 3 And some assumptions about smoothness that we do not need to worry about here. A Pixel Is Not a Little Square! Microsoft Tech Memo 6 Alvy 3 (a) A 5x4 image. (b) The footprint of a reconstruction filter. A truncated Gaussian, for example. (c) Footprint of image under reconstruction. (d) Footprint of reconstructed image. Medium quality reconstruction. (e) Reconstruction translated (.5,.5), then resampled into a 6x5 image. FIGURE 1 Dotted line is minimally enclosing rectangle Fixed reference point A Pixel Is Not a Little Square! Microsoft Tech Memo 6 Alvy 4