La teta asustada (2009)

Aligning images and stitching them into wide-angle images or panoramas are among the most widely used algorithms in computer vision. A practical stitching algorithm requires robust solutions for image alignment, warping, artifacts removal, etc. Here we focus only on the math- ematical model relating pixel coordinates in one image to the pixel coordinates in another. A pinhole-camera model is also used for the sake of simplicity.

There are various camera motions in real life such as 2-D planar translations, 3-D translations, 3-D camera rotations and even random combinations of these motions. Among these scenarios, the two most-used camera motions to create a photographic panorama as shown in Figure 4.1 are planar translations and rotations.

To be more specific, the camera translation is a planar motion perpendicular to the optical axis, whereas the camera rotation is around the optical center. Without loss of generality, we carry out the analysis with 1-D images, which means both the translation and rotation become 1-D motions. Furthermore, all the motions of the 1-D images are modeled in the corresponding 2-D light fields.

Camera rotations in 2-D light fields

We first define a 2-D light field L(x, p) that is created by moving a pinhole camera on the axis x. A 1-D image taken at position x₀ is then denoted with L(x₀, p) where p denotes the coordinates on the image sensor. In the 2-D light field, a 3-D camera rotation around the optical center is reduced to a tilt of the pinhole camera around its optical center.

The camera rotation is usually used to extend the field of view that is defined by the range of the pixel coordinate p. Here we give a definition of the 1-D image I(p) that is captured by the pinhole camera as follows:

I(p) = L(x₀, p), p∈  −S 2f, S 2f  ,

where x₀ represents the location of the camera, S represents the size of the camera sensor and f represents the focal length of the pinhole camera. By rotating the camera, we actually extend the range of the sensor such that even light rays with angles larger than S

2f or angles smaller

than−_2fS can also be recorded as shown in Figure 4.2. In the light field, we can clearly see how several 1-D photos merge to one image with larger field of view.

The relationship between the original image I(p) and the image I(p) after a camera rotation of θ can be formulated

Figure 4.1: Creating a wide view photo by translating a mobile phone on the top and a 360 deg panorama photo camera rotations on the bottom.

In conclusion, the rotation of the 1-D camera corresponds to a shift in the p dimension in the 2-D light field, where no constraints have to be posed on the scene. By rotating the camera, acquired images are shifted in the p dimension and a wider field of view is achieved by merging these images directly.

Camera translations in 2-D light fields

When registering images by camera translation, we usually assume the observed scene to be far enough to be modeled as a plane. This assumption is also widely used in image super- resolution by camera translation.

This problem can be better understood in the 2-D light field. The camera translation is a shift in the x direction, as the 2-D light field itself is defined as a stack of 1-D images taken by a moving camera on the x axis. Taking a photo captured at x₀ as an example, we can represent

Ɖ

dž

Ɖ

Figure 4.2: Image stitching by camera rotations and their representations in the 2-D light field. The red slices on the right denote the 1-D images by the camera rotations. Then the green areas are used to represent overlapping areas among the observed scenes.

the image Ix0(p) as Ix0(p) = L(x0, p) p∈  −S 2f, S 2f  . Another image Ix0+Δx(p) is taken by a translation of Δx on the x axis.

Ɖ

Ɖ

dž

Figure 4.3: Image stitching by camera translation and their representations in 2-D light field on the right. The red slices on the right denote the 1-D images by the camera translations. Then the green areas are used to represent overlapping areas among the observed scenes.

The registration between different images is achieved by matching the overlapping areas in each image. The success of the matching process relies on the fact that this particular area should keep the same appearance for different viewing angles. Unlike the matching pixels for camera rotation, pixels from different images by translations correspond to different light rays. Only when the surface in the scene satisfies the Lambertian property, we can successfully match pixels that correspond to the same location in the scene.

Furthermore, creating the panorama photo requires the scene to have a constant depth value. The creation of the panorama photo is to extend the field of view that is the p dimension in the light field. As shown in the 2-D light field, the translation in the x dimension is equivalent to shift in p direction given that the surface is flat and obeys the Lambertian assumption.

be formulated as

I_x0(p) = I_x₀(p + Δp), Δp = Δxf z.

In conclusion, the translation of the 1-D camera corresponds to a shift in the x dimension in the 2-D light field. Furthermore, under the assumption that the captured scene is a planar surface with Lambertian-reflectance properties, these 1-D images by translations are equivalent to the images captured by camera rotations. Thus, a wider field of view can also be achieved by camera translations.