Radiometric, Geometric Correction and Image Enhancement

Radiometric and geometric image correction

Image correction removes distortions that the sensor, the atmosphere, the platform and the terrain introduce into raw data, so that brightness values and pixel positions represent the scene faithfully. Radiometric correction restores digital numbers (DN) to physically meaningful radiance or reflectance, while geometric correction assigns each pixel its correct ground coordinates. Both are usually applied before quantitative analysis.

Positioning precision and accuracy. The NAVSTAR GPS constellation nominally consists of at least 24 operational satellites in six orbital planes. Precision describes the repeatability of repeated measurements and is expressed by their standard deviation, whereas accuracy describes closeness to the true value, so a set of positions can be precise yet biased. Carrier-phase relative static post-processing gives centimetre-to-millimetre positions, which is better than code-based DGPS, and geodetic receivers that use carrier-phase observations are more accurate than code-only receivers.

Polynomial georeferencing. An image is georeferenced by fitting a polynomial that relates image coordinates (x,y) to map coordinates (X,Y). A polynomial of order n has (n+1)(n+2)/2 coefficients for each coordinate direction, so the minimum number of ground control points (GCPs) equals this count, and the points must not be collinear. Each GCP supplies one equation in x and one in y. A first-order transformation needs 3 points, a second-order transformation needs 6 points, and a third-order transformation needs 10 points.

De-striping. Striping is a systematic pattern of lines with abnormal brightness, caused by calibration differences between detectors in a line-scanning or push-broom sensor. De-striping equalizes the statistics, such as the mean and standard deviation, of each detector so that the stripes disappear. It is therefore a correction for sensor defects, not for atmospheric or geometric effects.

Atmospheric correction. Atmospheric scattering adds path radiance, and absorption attenuates the signal. Absolute methods such as radiative transfer models (for example 6S or MODTRAN) compute these effects from atmospheric conditions. Relative methods normalize one scene to another by histogram adjustment or by regression against pseudo-invariant features. Image-to-image registration, which matches the coordinates of one image to another, is a geometric correction and does not remove atmospheric effects.

Image-to-image registration. Registration aligns two images of the same area so that corresponding pixels refer to the same ground location. The images may be acquired on different dates, which is common in multi-temporal change detection, and they need a significant geographic overlap so that enough common features can be matched. They need not share the same spatial resolution, because resampling maps them onto a common grid.

Geometric errors in aerial imagery. External errors arise from random movement of the aircraft during exposure. Variation in flying height changes the image scale, and changes in attitude (roll, pitch and yaw) tilt the image plane and displace image points. Relief displacement is caused by terrain elevation rather than by aircraft motion, and focal length is a fixed calibration parameter of the camera.

Linear contrast enhancement

Linear contrast enhancement stretches the range of input digital numbers linearly onto a wider output range, such as 0 to 255 for 8-bit data, so that grey-level differences become visible. The transformation is a straight-line mapping applied to each pixel independently, so it is a point operation.

Min-max stretch. The minimum and maximum DN values of the image define the input range, and the output is DN_{out} = \frac{DN_{in}-DN_{min}}{DN_{max}-DN_{min}} \times (2^k - 1) for a k-bit image. An 8-bit image uses the output range 0 to 255, and a 6-bit image uses 0 to 63. For example, if an 8-bit image has DN values from 40 to 200, a pixel of value 100 becomes \frac{60}{160} \times 255 \approx 96. Results are normally rounded to the nearest integer.

Percentage and standard deviation stretches. The percentage stretch clips a fixed fraction of the histogram tails before stretching. The standard deviation stretch sets the limits at the mean plus or minus a multiple of the standard deviation. Both are linear, but they assume that the histogram is roughly Gaussian, so that the centre and spread describe the data well.

Piece-wise linear stretch. When the histogram is non-Gaussian, for example bimodal or multimodal, a single linear function wastes output range on sparsely populated grey levels. A piece-wise linear stretch divides the input range into segments, each with its own slope and intercept, so that densely populated parts of the histogram receive more output range.

Reading a piece-wise stretch. Each segment is defined by its input limits and output values, and the output minimum and maximum are set by the segment endpoints rather than automatically by the histogram limits. For a segment joining input a to output c and input b to output d, the output is c + (BV_{in} - a)\frac{d-c}{b-a}. For example, a segment from input 10 to 60 mapped onto 0 to 100 gives an input of 30 an output of \frac{20}{50} \times 100 = 40.

Linear stretch versus histogram equalization. Histogram equalization maps each grey level through the cumulative frequency distribution, so output levels are spread according to how often the input levels occur. The linear stretch ignores frequency. Equalization needs the histogram and its cumulative sum, so it is computationally more expensive than a single linear mapping. Both methods are point operations that depend only on the pixel's own value, not on a neighbourhood. If the input histogram is already uniform, its cumulative distribution is a straight line, and equalization gives the same result as a linear stretch over the full range.

Contrast enhancement types

Contrast enhancement techniques can be grouped by the form of their transfer function. Linear and piece-wise linear stretches use straight-line segments, whereas nonlinear stretches such as logarithmic, exponential, Gaussian and histogram equalization use curved transfer functions.

Logarithmic stretch. A logarithmic transform of the form DN_{new} \propto \log(1 + DN_{old}) produces a concave transfer curve that rises steeply at low input values and flattens at high values. It expands the contrast of dark pixels and compresses bright pixels, which suits images dominated by low brightness.

Contrast enhancement as a radiometric enhancement. Contrast enhancement changes the brightness values of pixels within a single image, so it is classed as a radiometric enhancement. This differs from spectral enhancement (band ratios, principal components), spatial enhancement (filtering, edge detection) and temporal enhancement (analysis of multi-date images).

Piece-wise linear stretch with saturation. A piece-wise contrast stretch may be defined with thresholds. Input values at or below a lower limit map to 0, values above an upper limit map to the maximum output, and the intermediate values follow a linear rule. The intermediate rule does not have to join the saturated ends continuously. For example, if inputs up to 40 map to 0, inputs above 160 map to 255, and the intermediate inputs follow BV_{out} = 3BV_{in} - 90, then an input of 100 gives an output of 210.