At right is a rendering of the head model in an

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Unformatted text preview: ring of the head model in an incorrect pose. or non-shiny surface. Lambert's law states that the visible intensity of a surface patch is related to the dot product between the surface normal and the lighting. For a Lambertian object the imaging equation is: X~ vT x = 1.3 i li  ux ; i where the model value ux is the normal vector of a surface patch on the object, li is a vector pointing toward light source i, and i is proportional to the intensity of that light source Horn, 1986 contains an excellent review of imaging and its relationship to vision. 13 Paul A. Viola CHAPTER 1. INTRODUCTION Drawing the explicit parallel between 1.1 and 1.3 we can see that the imaging function is X~ F ux; q = 1.4 i li  ux ; i where q = f i; ~ig. As the illumination changes the functional relationship between the model l and image will change. Since we can not know beforehand what the imaging function will be, aligning a model and image can be quite di cult. These di culties are only compounded if the surface properties of the object are not well understood. For example, many objects can not be modeled as having a Lambertian surface. Di erent surface nishes will have di erent re ectance functions. In general re ectance is a function of lighting direction, surface normal and viewing direction. The intensity of an observed patch is then: X vT x = R i; ~i;~; ux ; lo 1.5 i where ~ is a vector pointing toward the observer from the patch and R is the re ectance o function of the surface. For an unknown material a great deal of experimentation is necessary to completely categorize the re ectance function. Since a general vision system should work with a variety of objects and under general illumination conditions, overly constraining assumptions about re ectance or illumination should be avoided. Let us examine the relationship between a real image and model. This will allow us to build intuition about the alignment process. Data from the real re ectance function can be obtained by aligning a model to a real image. An alignment associates points from the image with points from the model. If the alignment is correct, each pixel of the image can be interpreted as a sample of the imaging function R. The imaging function could be displayed by plotting intensity against lighting direction, viewing direction and surface normal. Unfortunately, because intensity is a function of so many di erent parameters the resulting plot can be prohibitively complex and impossible to visualize. Signi cant simpli cation will be necessary if we are to detect any structure in this data. In a wide variety of real images we can assume that the light sources are far from the object at least in terms of the dimensions of the object. When this is true and there are no shadows, each patch of the object will be illuminated in the same way. Furthermore, we will 14 1.1. AN INTRODUCTION TO ALIGNMENT AI-TR 1548 assume that the observer is far from the object, and that the viewing direction is therefore constant throughout the image. The resulting relationship between normal and intensity is three dimensional: the normal vector has unit length and is determined by two parameters, its x and y components; the intensity is a third parameter. A three dimensional scatter plot of normal versus intensity is really a slice through the high dimensional space in which R is de ned. Though this graph is much simpler than the original, three dimensional plots are still quite di cult to interpret. We will slice the data once again so that all of the points have a single value for the y component of the normal. Figure 1.3 contains a graph of the intensities along a single scan-line of the image of Ron. Figure 1.4 shows similar data for the correctly aligned model of Ron. Model normals from this scan-line are displayed in two graphs: the rst shows the x component of the normal while the second shows the y component. Notice that we have chosen this portion of the model so that the y component of the normal is almost constant. As a result the relationship between normal and intensity can be visualized in only two dimensions. Figure 1.5 shows the intensities in the image plotted against the x component of the normal in the model. Notice that this relationship appears both consistent and functional. Points from the model with similar surface normals have very similar intensities. The data in this graph could be well approximated by a smooth curve. We will call an imaging function like this one consistent. Interestingly, we did not need any information about the illumination or surface properties of the object to determine that there is a consistent relationship between model normal and image intensity. Figure 1.6 shows the relationship between normal and intensity when the model and image are no longer aligned. The only di erence between this graph and the rst is that the intensities come from a scan-line 3 centimeters below the correct alignment i.e. the model is no longer aligned with the image, it is 3...
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