Chromatic Aberration, Field Curvature, Distortion, Astigmatism
In optics, aberrations are properties of an optical system, such as lenses, that cause the spread over of light which are supposed to be focused (whether converged or diverged) into one point in the ideal model of paraxial optics, it can be seen as a departure of the performance of an optic from the theoretical predictions of paraxial optics. The optical aberrations result in the blurring and distortion of images and the deterioration of image qualities. Note that aberrations are intrinsic to the mechanism of the image formed through refraction and reflection of light, even if the lenses were perfect in terms of the geometrical shapes, surface qualities, and centering on the optical axis, the image will still be the subject to aberrations, which become significant as the aperture and the field depart more and more from infinitesimal values. The reason that aberrations exist is that the simple paraxial model is not an accurate and flawless model of the real circumstances in the formation of an image through an optical system.
Optical aberrations could be divided into monochromatic aberrations, which are also known as Seidel Aberrations, named after Philipp Ludwig von Seidel, a German mathematician whose calculation method first described them in 1857; and chromatic aberrations. There are five Seidel aberrations. Three of them – Spherical Aberration, Coma, and Astigmatism – cause basic deterioration of the image qualities, making it blurred. The remaining two – Petzval Field Curvature and Distortion – alter the geometries of images.
Chromatic Aberration is a kind of aberration arising from a chromatic light source entering the lenses.
The phenomenon of chromatic dispersion is the cause of chromatic aberration. As shown in figure 1 below, when white light passes through a prism, it can be decomposed into different colors. The reason for this decomposition is that the wavelengths of various colors are different, and the refractive index of light with different wavelengths is different. This means that short-wavelength light has a large refractive index and long-wavelength light has a small refractive index.
Figure 1. Chromatic Dispersion in a triangular prism. A ray of a white light source is separated into polychromatic lights.
Lenses also produce the same phenomenon as prisms (See figure 2). Since the red light has a small refractive index and the blue light has a large refractive index, after passing through the lens, the focus of the red light is at the rear, and the focus of the blue light is at the front (of the ideal focal point). This deviation from the predicted focal point is called chromatic aberration. This means that a beam of white light emitted from a point light source and reaching the film or image sensor cannot be imaged as a point, but as a colored spot composed of different colors.
There are two types of chromatic aberration: Axial Chromatic Aberration and Magnification Chromatic Aberration. Axial chromatic aberration refers to the chromatic aberration phenomenon resulting from the focus position on the optical axis due to different wavelengths; magnification chromatic aberration refers to the lateral position change of the image color on the image plane perpendicular to the optical axis resulting from the difference in wavelength around the image plane. This is an off-axis aberration that increases with the angular field of view (Click on the link to learn about What is Field of View)
Figure 2. Chromatic Aberrations in an optical lens.
Axial chromatic aberration is related to the focal distance of imaging, causing the separation of colors or flares; while magnification chromatic aberration is related to the magnitude of the imaging plane, causing color staggering around the screen, forming diffuse color fringes, this is known as the fringing phenomenon. Chromatic aberration affects the color reproduction of images on the color film, and also reduces the resolution of images made on black and white film.
The common approach to correct Axial Chromatic Aberrations is using an Achromatic Doublet Lens constructed by lenses of different refractive indices/dispersion indices so that their chromatic aberrations cancel each other out, the lens group often consists of a positive crown lens and a negative flint lens. A converging crown lens has a low refractive index and little dispersion, while a diverging flint lens has a high refractive index and greater dispersion.
The correction of Magnification Chromatic Aberration is difficult compared to the correction of axial chromatic aberration, and its deterioration effect on image qualities will increase with the increase of focal length, and will not decrease with the reduction of aperture. An effective measure to correct the chromatic aberration of magnification is to use lenses made from abnormal/ultra-low dispersion optical glass.
Field Curvature, also known as “Curvature of Field” or “Petzval Field Curvature”, is a common optical problem. It is the phenomenon that an object plane perpendicular to the principal optical axis can not form a flat image field, but instead, the image field conjectured to be planar is inward bent into a curved, bowl-like shape. The consequence of field curvature is a flat object fractionally appearing sharp in a certain part(s) of the frame, instead of appearing sharp across the entire film frame. The reason is: due to the curved nature of optical elements, there are actually two image planes, the main image plane is the focal plane of the transverse tangent line, and the inferior image plane is the focal plane of the radial line, the optical lenses projecting the image in a curved manner, rather than flat.
Therefore all optical lenses have, associated with it, a basic field curvature, which is a function of the index of refraction of the lens elements and their surface curvatures. When taking a picture using a lens with field curvature, when the focus of the lens is on the center of the picture, the center is clear and the surroundings are blurred. Vice versa, when the focus of the lens is on the surroundings, the center becomes blurred. The sharpest image can only be formed on a curved focal surface rather than a flat focal plane. If the geometries of the image plane do not conform to this curved focal surface, image blurring is inevitable. In this manner, it is impossible to obtain an image that is clear in the center and on all sides on a flat image plane. Therefore, in some special cameras, the film is placed in an arc position on purpose to reduce the image of field curvature. This also explains when taking a photo with a set of wide-angle lenses, the camera sensor arranges the subjects in an arc shape to improve the image qualities of the peripheral field of view because the field curvature of the wide-angle lens is larger than that of other lenses.
Figure 3. Field Curvature
Distortion is a variation of the magnification with the field angle, causing changes in the shapes of the image according to the actual object. Distortion has no impact on the image qualities, but solely affects the similarities of the image to the object.
Due to distortion, a straight line in the object plane becomes a curve on the image side, causing a “false” image. Distortion has nothing to do with the relative aperture but is associated with the field of view of the lens. Therefore, special attention should be paid to the influence of distortion when using wide-angle lenses.
Distortion could be regarded as a departure from the perfect model of pinhole projection. In pinhole projection, the magnification of an object is negatively proportional to its distance to the camera along the optical axis, so that a camera pointing directly at a flat surface reproduces that flat surface. Distortion can be thought of as non-uniform stretching of the image.
Distortion could be divided into two types: Barrel Distortion and Pincushion Distortion. Barrel distortion is the situation when the image is magnified more than the center area around the optical axis than the perimeter. Pincushion distortion is the reverse of barrel distortion, which happens when the magnification ratio around the edges of the imaging screen is greater than the center. Barrel distortion usually exists in wide angle and fisheye camera lenses, whilst pincushion barrel often takes place at long focal lengths.
Figure 4. Barrel Distortion and Pincushion Distortion.
A single lens has no distortion for all object distances, but distortion will certainly be introduced if a diaphragm is set before and after a thin lens. No distortion will be introduced if you put the stop on the lens. In a successful camera design, a diaphragm is often placed between two or more almost symmetrical groups of lenses, which can correct part of the distortion along with astigmatism. There are also several algorithms aimed at rectifying distortions, such as findChessboardCorners, calibrateCamera, initUndistortRectifyMap, remap, etc. The algorithms are used in conjunction. The correction process is to convert a special object point from the world coordinate system to the camera coordinate system and then project it to the imaging plane coordinate system, and finally, the data on the imaging plane is transformed into the graphic pixel coordinate system.
Astigmatism is said to be present when the object point is not on the optical axis of the optical system, and the beam it emits has an inclination angle with the optical axis. Astigmatism is different from Coma. It is an off-axis aberration that describes the imaging defect of infinitely narrow beams and is only related to the field of view. The magnitude of the projection of the distance between the convergent point of the meridian narrow beam and the convergent point of the sagittal narrow beam on the optical axis is the value of astigmatism.
As far as the entire narrow beam is concerned, a short line perpendicular to the meridian plane is obtained at the meridian focus, which is called the Meridian Focal Line; a short line perpendicular to the meridian focal line and located at the sagittal focus is obtained on the meridian plane, which is called the Sagittal Focal Line. At other positions, the beam cross-section is an elliptical diffuse spot; at the middle position of the two focal lines, the beam cross-section is a circular diffuse spot, and the beam with this structure is called an astigmatic beam, and this imaging defect is called astigmatism.
Due to the presence of astigmatism, the image quality of the off-axis field of view is significantly reduced. Even if the aperture is opened very small, very clear images cannot be obtained in the meridional and sagittal directions at the same time. The size of the astigmatism is only related to the angular field of view, not the size of the aperture. Therefore, astigmatism is more obvious in the wide-angle lens, and the subject should be placed in the center of the picture as much as possible when shooting.
Tags: Optical Basics: Chromatic Aberration, Field Curvature, Distortion, and Astigmatism