Remote Sensing & GIS Applications 3(2+1)

Lesson 10 Digital Image

10.2 Image Rectification and Restoration

Remotely sensed images are taken from a great distance from the surface of the earth affected by various parameters such as atmosphere, solar illumination, earth rotation, sensor motion with its optical, mechanical, electrical components etc. which causes distortion in the imagery. The intent of image rectification and restoration is to correct these distortions arises in image acquisition process both geometrically and radiometrically. Obviously, the nature of such procedures varies considerably with factors such as the digital image acquisition type (digital camera, along-track scanner, across-track scanner), platform (airborne, satellite), atmosphere and total field of view.

10.3 Geometric Correction

Raw digital images acquired by earth observation systems usually contain geometric distortions so they do not reproduce the image of a grid on the surface faithfully. Geometric distortions are the errors in the position of a pixel relative to other pixels in the scene and with respect to their absolute position in a particular projection system.

The geometric distortions are normally two types: (i) systematic or predictable, and (ii) random or unpredictable.

Due to rotation of earth and platform’s forward motion the scan lines are not perpendicular to the direction of ground track, which makes the scanned area skewed and scale variation due to earth’s curvature found in both push-broom and whisk-broom scanner.  But in case of whisk-broom scanner, there are additional two errors founds (a) scale variation along the scan direction due to its scanning process, (b) error due to nonlinearity in scan mirror velocity.

Random distortions and residual unknown systematic distortions are difficult to account mathematically. These are corrected using a procedure called rectification.

Rectification is process of correcting an image so that it can be represented on a plain surface. It includes the process known as rubber sheeting, which involves stretching and warping image to georegister using control points shown in the image and known control points on the ground called Ground Control Point (GCP). Rectification is also called georeferencing.

The image analyst must be able to obtain both the distinct sets of co-ordinates: image co-ordinates and map co-ordinates (x,y) associated with GCPs. These values are used in a least squares regression analysis to determine geometric transformation coefficients that can be used to interrelate the geometrically correct (map) coordinates and the distorted- image coordinates. The GCPs are chosen such that these can be easily identified on the image and whose geographic co-ordinates can be obtained from ground (using GPS) or from existing geo-referenced map or image. The two common geometric correction procedures generally used are: image-to-map rectification and image-to-image registration, which uses an existing georeferenced map and image correspondingly. Fig 10.2 shows rectified map which can be used to rectify an unrectified image.

Fig. 10.2. Image to map rectification.

Two basic operations are performed for geometric correction: (1) Spatial interpolation and (2) Intensity interpolation. Spatial interpolation accounts for the pixel position correction and intensity interpolation accounts for the DN value assignation from input image to its corresponding output pixel position.

(1)   Spatial interpolation

The first process of geometric correction involves the pixel position correction using rubber sheeting procedure. The input and output image co-ordinate systems can be related as:

x= f₁ (X, Y) and  y = f₂(X, Y)                               (10.1)

where,

      (x, y) = distorted input image coordinates (column, row)

      (X, Y) = correct image coordinates

      f₁, f₂ = transformation functions

The mapping function could be polynomial. A first degree polynomial can model six kinds of distortions: translation and scale changes in x and y, skew and rotation.

The 1^st degree polynomial (i.e. linear equation) can be mathematically expressed as

x = a₀ + a₁ X + a₂ Y                                     (10.2)

y = b₀ + b₁ X + b₂ Y                                    (10.3)

The co-efficients a_i and b_i are evaluated using GCPs.

(2)    Intensity interpolation

Intensity interpolation involves the extraction of DN value from input image to locate it in the output image to its appropriate location. Fig 10.3 used as an example, for an output pixel (3, 5) grey shaded, the corresponding input pixel is (2.4, 4.8). This means the output image may not have one-to-one match with the input image. Therefore an interpolation process is applied which is known as resampling. This process accounts the position of output pixel to its corresponding position of input pixel and the DN values of input pixel and their surroundings.

Fig. 10.3. a) Distorted image; b) Corrected image pixels matrix.

There are generally three kinds of resampling process-

a)   Nearest-neighbour interpolation

Let for an output pixel (3, 5), in Fig 10.3 the input pixel location is (2.4, 4.8). In nearest-neighbor interpolation, the output pixel (3, 5) will be filled up with DN value nearest to the input pixel location (2.4, 4.8) in the present case is (2, 5). In this process the radiometric values or DN values do not alter and it is computationally efficient. But features in the output matrix may be offset spatially by + 0.5 pixel, thus a ‘patchy’ appearance arise at the edges and in linear features.

b)  Bilinear interpolation

Bilinear interpolation finds brightness values in two orthogonal direction of the desired input location (2.4, 4.8) in the input image. Distance-weighted average of DN values of four nearest pixels ((1, 4), (1, 5), (2, 4), (2, 5)) is computed in the input distorted image. Closer to the input location will have more weight; DN value of highest weighted pixel will be assigned to the output pixel.

where, R_k are the surrounding four DN values, D²_k are the distance squared from the desired input location (2.4, 3.8).

This technique alters the DN value and generates smoother appearance in the resampled image.

c)     Cubic convolution interpolation

An improved restoration of the image is obtained in cubic convolution interpolation process. Similar to bilinear interpolation this technique is uses 16 neighbor pixel in the input image. It provides smoother appearance and sharpens edge and also changes the DN values but requires more computational time than previous processes.

Any interpolation is a low pass filtering operation and it produces a loss of higher frequency information depending on the method used.

10.4 Radiometric Correction and Noise Removal

Radiance value recorded for each pixel represents the reflectance/emittance property of the surface features. But the recorded value does not coincide with the actual reflectance/ emittance of the objects due to some factors like: sun’s azimuth and elevation, viewing geometry, atmospheric conditions, sensor characteristics, etc. To obtain the actual reflectance/ emittance of surface features, radiometric distortions must be compensated. The main purpose of radiometric correction is to reduce the influence of errors. An error due to sensor’s characteristic is also referred as noise.

Radiometric correction is classified into the following three types:

1)   Sun angle and topographic correction

2)   Atmospheric correction

3)   Detector response calibration

a)       De-striping

b)      Removal of missing scan line

c)       Random noise removal

d)      Vignetting removal

1) Sun angle and topographic correction: The sun elevation correction accounts for the seasonal position of the sun relative to the earth (Fig. 10.4). The correction is usually applied by dividing each pixel value in a scene by the sine of the solar elevation angle or the cosine of zenith angle (which is simple 90⁰ minus the solar elevation angle).

DN_correction = DN / sinα                                           (10.5)   

DN_correction = DN / cosθ                                          (10.6)

α = solar elevation angle, θ = zenith angle (90^o- elevation angle)

Fig. 10.4. Effects of seasonal changes on solar elevation angle.

(Source: landsathandbook.gsfc.nasa.gov/data_properties/prog_sect6_3.html; and modified )

The earth- sun distance correction is applied to normalize for the seasonal changes in the distance between the earth and the sun. The earth- sun distance is usually expressed in astronomical units. (An astronomical unit is equivalent to the mean distance the earth and the sun, approximately 149.6×10⁶km). The irradiance from the sun decreases as the square of the earth-sun distance (Lillesand and Kiefer, 2003).

2) Atmospheric correction: During transmission of EM wave from sun to objects and from objects to sensor through atmosphere, various kinds of absorption and scattering takes place. Objects receive not only the direct solar illumination, but also the scattered radiation from atmosphere and neighbouring objects and similarly the sensor receives reflection or radiation from target, its neighbouring objects, and scattered radiation from atmosphere (which is called path radiance or haze).

Therefore the sensor receives a composite reflection/ radiation which can be expressed as-

where,

 L_a = apparent spectral radiance

          measured by sensor

 ρ = reflectance of objects

 E = irradiance on object

 T = transmission of atmosphere

 L_p = path radiance / haze

 Fig. 10.5. Effect of solar illumination in atmosphere

For multispectral and hyperspectral data, the approaches of atmospheric correction is different.

The errors due to solar elevation, sun-earth distance and atmospheric effects can be reduced in two procedures: (i) absolute correction and (ii) relative correction.

Absolute correction

The absolute correction procedure of Landsat data is stated below:

The radiance recorded by a sensor is converted in DN values in a range 0 – 255 (in 8 bit data). Therefore to obtain reflectance value of terrain objects it is required to convert DN values back to its equivalent the radiance value, and radiance value to reflectance value.

 where,

 L_λ = Spectral radiance at the sensor’s aperture

  LMAX _λ = Spectral at-sensor radiance that is scaled to Q_{cal min}.

  LMIN _λ = Spectral at-sensor radiance that is scaled to Q_{cal max}.

  Q_cal= Quantized calibrated pixel value (DN).

  Q_{cal max}= Maximum quantized calibrated pixel value corresponding to LMIN _λ.

  Q_{cal min}= Maximum quantized calibrated pixel value corresponding to LMAX _λ.

where,

  ρ_λ = TOA reflectance

  L_λ = Spectral radiance at the sensor’s aperture

  d= Earth-Sun distance

  ESUN_λ= Mean exoatmospheric solar irradiance

  θ_s= Solar zenith angle

The values of L_λ, LMAX_λ, LMIN_λ, Q_cal, Q_{cal max}, Q_{cal min},  θ_s are obtained from the sensor parameters and satellite position.

Table 10.1 Solar spectral irradiances for bands of ETM+ sensor

Band	watts/(meter squared * µm)
1	1969.000
2	1840.000
3	1551.000
4	1044.000
5	225.700
7	82.07
8	1368.000

(Source: landsathandbook.gsfc.nasa.gov/data_prod/prog_sect11_3.html)

Table 10.2 Earth-Sun distance in astronomical units

Julian Day	Distance	Julian Day	Distance	Julian Day	Distance	Julian Day	Distance	Julian Day	Distance
1	0.9832	74	0.9945	152	1.0140	227	1.0128	305	0.9925
15	0.9836	91	0.9993	166	1.0158	242	1.0092	319	0.9892
32	0.9853	106	1.0033	182	1.0167	258	1.0057	335	0.9860
46	0.9878	121	1.0076	196	1.0165	274	1.0011	349	0.9843
60	0.9909	135	1.0109	213	1.0149	288	0.9972	365	0.9833

(Source:landsathandbook.gsfc.nasa.gov/data_prod/prog_sect11_3.html)

Noise Removal: Image noise is any unwanted disturbance in image data that is due to limitations in the sensing, signal digitization, or data recording process. Radiometric errors due to detector response calibration are also termed as noise.

3) Detector response calibration

Destriping: Striping or banding in imagery occurs due to error in sensor adjustment or inability in sensing the radiation by sensors. This error was common in Landsat MSS sensor, where six detectors were used for each band drifted over time, resulting in relatively higher or lower values along every sixth line in the image. One common method of striping error correction is done by comparing the histogram between a set of data. A histogram is evaluated from scan lines 1, 7, 13, and so on; for a second lines 2, 8, 14, and so on; and so forth. These histograms are then compared in terms of their mean and standard deviations of six detectors to identify the problem detector(s) for the problem lines to resemble those for the normal data lines. Fig. 10.6(a) shows a stripped image and Fig. 10.6(b) is the output result after de-striping algorithm is applied.

Fig. 10.6. Destriping: (a) striped image, (b) destriped image.

(Source: blamannen.wordpress.com/2011/07/12/destripe-landsat-7-etm-some-thoughts/)

Line drop: Line drop correction is similar to strip error. Line drop error is found when a sensor completely fails to scan. This can be minimized by normalizing with respect to their neighboring features or image values i.e. the data for missing lines is replaced by the average image value of neighbors. Fig. 10.7 shows the line drop correction.

Fig. 10.7. Line drop correction: (a) original image (b) restored image.

Random noise removal: Random noise is not systematic. The noise in a pixel does not depend on the spatial location of the pixel in the imagery. It causes images to have a “salt and pepper” or “snowy” appearance. Moving window of 3 x 3 or 5 x 5 pixels is used for removing these errors.

Vignetting removal: Vignetting is found when the central portion image is more brighten than the periphery or corners. This error is found in images obtained from optical sensor, which uses lens. Images taken using wide-angle lenses often vignette due to refraction property of light. Proper calibration between the irradiance and sensor output signal is generally used for this correction. Fig. 10.8. shows the vignetting effect in left image and the right image is the output after vignetting removal is applied. 

Fig. 10.8. Left image have vignetting effect in contrast to the uniform image.

(Source: www.rbc.ufrj.br/_pdf_56_2004/56_1_06.pdf)

Keywords: Digital image, Rectification, Geometric correction, Radiometric correction, Noise.

References

• blamannen.wordpress.com/2011/07/12/destripe-landsat-7-etm-some-thoughts/

• geoinformatics.sut.ac.th/sut/vichagan/DIP/Chapter_07.pdf

• landsathandbook.gsfc.nasa.gov/data_properties/prog_sect6_3.html

• landsathandbook.gsfc.nasa.gov/data_prod/prog_sect11_3.html

• Lillesand, T. M., Kiefer, R. W., 2002, Remote sensing and image interpretation, Fourth Edition, pp. 471-488.

• www.rbc.ufrj.br/_pdf_56_2004/56_1_06.pdf

• www.wamis.org/agm/pubs/agm8/Paper-5.pdf

Suggested Reading

• Bhatta, B., 2008, Remote sensing and GIS, Oxford University Press, New Delhi, pp. 311-322.

• Joseph, G., 2005.Fundamentals of Remote Sensing, Second Edition, Universities Press (India) Pvt. Ltd., pp. 320-321.

• nature.berkeley.edu/~penggong/textbook/chapter5/html/sect52.htm

• www.e-education.psu.edu/natureofgeoinfo/book/export/html/1608