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Chapter 4 Radiometric Processing of Remote Sensing Data Remote Sensing Ayman F. Habib 1 Overview • Radiometric calibration – Sensor calibration • DN/grey values → At-sensor radiance. – Atmospheric correction • At-sensor radiance → Surface radiance. – Solar and topographic correction • Surface radiance → Surface reflectance. • Radiometric image processing: – Spatial/image domain. • Noise removal. • Point and edge detection. – Frequency domain. Remote Sensing Ayman F. Habib 2 Radiometric Calibration Remote Sensing Ayman F. Habib 3 Recorded Digital Numbers (DN) • Recorded grey values (digital numbers) at any pixel location is not a record of the true ground- leaving radiance. • The signal is attenuated due to atmospheric absorption. • The directional properties of the signal are altered by refraction and scattering. • Incoming radiance is augmented by scattered path radiance. – Introduces haze and reduces the image contrast. Remote Sensing Ayman F. Habib 4 Recorded Digital Numbers (DN) Sensor S1 S2 Q P Remote Sensing Ayman F. Habib 5 Recorded Digital Numbers (DN) • All of the signal appears to originate from point “P” on the ground. However: – The scattering at S1 redirects some of the radiation into the sensor’s field of view. – Some of the energy from point “Q” is scattered at S2 into the sensor’s field of view. – The cumulative effect of the scattering at S1 and S2 is known as “the atmospheric path radiance”. – The exitance from “P” and “Q” is attenuated by the atmosphere. Remote Sensing Ayman F. Habib 6 Recorded Digital Numbers (DN) • Moreover, recorded energy by the sensor depends on: – The solar elevation angle, the solar azimuth, the slope of the ground, and the sensor look angle. • The effect of these factors is described by the bidirectional reflectance function. – The Earth-Sun distance. • There are some techniques that can be used to estimate the atmospheric and viewing angle effects: Radiometric Calibration. Calibration Remote Sensing Ayman F. Habib 7 Atmospheric Correction: Why? (I) • Compute the ratio between two bands of a multi- spectral image (e.g., Normalized Difference Vegetation Index – NDVI). – The scattering effect is inversely proportional to the wavelength. – The involved channels will be unequally affected by scattering. – The computed ratio will be a biased estimate of the true ratio. Remote Sensing Ayman F. Habib 8 Digital Numbers Digital Numbers reduced by absorption increased by scattering 60 Digital Numbers 50 40 30 20 10 0.5 0.6 0.7 0.8 1.1 Wavelength (µm) • Combined effects of scattering and absorption on Digital Numbers (DN) recorded by remote sensing systems. Remote Sensing Ayman F. Habib 9 Atmospheric Correction: Why? (II) • One might use the recorded digital numbers (grey values) to infer some properties of the surface: – Surface normal. – Surface roughness. – Surface reflectance properties. • Reliable recovery of such properties is only possible if the atmospheric component of the signal is estimated and removed. Remote Sensing Ayman F. Habib 10 Object Recognition / Atmospheric Effects • Expected high intensity yellow color response: RGB: 255, 255, 0. Recorded Intensities R: 255 G: 249 B: 29 Remote Sensing Ayman F. Habib 11 Atmospheric Correction: Why? (III) • Comparison of remote sensing data captured at different times. – Environmental monitoring. – Resource management. – Evaluation of disaster effects. – Change detection applications. • Reliable comparison is only possible after removing the atmospheric effect. Remote Sensing Ayman F. Habib 12 Change Detection Analysis Calgary, 1956 Calgary, 1999 Remote Sensing Ayman F. Habib 13 Change Detection Analysis Remote Sensing Ayman F. Habib 14 Change Detection / Radiometric Differences Calgary 56 Calgary 99 Remote Sensing Ayman F. Habib 15 Radiometric Calibration Procedure • Sensor calibration: – DN/grey values → At-sensor radiance. • Atmospheric correction: – At-sensor radiance → Surface radiance. • Solar and topographic correction: – Surface radiance → Surface reflectance. Remote Sensing Ayman F. Habib 16 Sensor Calibration Digital Numbers → At-Sensor Radiance Remote Sensing Ayman F. Habib 17 DN → At-Sensor Radiance • Objective: Convert digital numbers (DN) to absolute radiance values. • The conversion process accounts for the exact form of the analog to digital (A/D) response function of the sensor under consideration. • Detectors are designed to have a linear response to incident spectral radiance. • The sensor’s response to incoming radiance is monitored using onboard calibration lamps. Remote Sensing Ayman F. Habib 18 DN → At-Sensor Radiance Remote Sensing Ayman F. Habib 19 DN → At-Sensor Radiance 255 DN = Digital Number Slope = channel Gain (G) 0 Lmin L = Spectral Radiance Lmax Remote Sensing Ayman F. Habib 20 DN → At-Sensor Radiance DNmax DN DNmin Lmin L Lmax DN = DNmin + (DNmax - DNmin)/(Lmax - Lmin) * (L – Lmin) DN = DNmax/(Lmax – Lmin) * L - DNmax/(Lmax – Lmin) * Lmin DN = Channel Gain * L + Channel Offset Remote Sensing Ayman F. Habib 21 DN → At-Sensor Radiance Lmax L=Spectral radiance Lmax − Lmin Slope = 255 Lmin 0 DN = Digital number 255 Remote Sensing Ayman F. Habib 22 Lmin and Lmax for LANDSAT MSS Spectral Bands Satellite 4 5 6 7 Lmin Lmax Lmin Lmax Lmin Lmax Lmin Lmax LANDSAT 1 0.0 24.8 0.0 20.0 0.0 17.6 0.0 40.0 LANDSAT 2 1.0 21.0 0.7 15.6 0.7 14.0 1.4 41.5 (6/22 to 7/16/75) LANDSAT 2 0.8 26.3 0.6 17.6 0.6 15.2 1.1 39.1 (> 7/16/75) LANDSAT 3 0.4 22.0 0.3 17.5 0.3 14.5 0.3 44.1 (3/5 to 6/1/78) LANDSAT 3 0.4 25.9 0.3 17.9 0.3 14.9 0.3 38.3 (> 6/1/78) LANDSAT 4 0.2 23.0 0.4 18.0 0.4 13.0 1.0 40.0 • L = (Lmax – Lmin) / DNmax * DN + Lmin • L is expressed in milli-Watts cm-2 sr-1 µm-1 (spectral radiance). Remote Sensing Ayman F. Habib 23 Sensor Radiance Equation Remote Sensing Ayman F. Habib 24 Paths of Radiance Received by the Sensor Remote sensing Total radiance system E0 at the sensor L s Solar irradiance 90° Lp Lt Lp Tθ 0 2 Tθ v Diffuse sky 4 irradiance Ed 1,3,5 1 Atmosphere θv 3 θ0 Li 5 Reflectance from Reflectance from neighboring area study area Remote Sensing Ayman F. Habib 25 Paths of Radiance Received by the Sensor • Path 1: – Electromagnetic energy from the sun that was attenuated very little before illuminating the target within the IFOV. • Path 2: – Electromagnetic energy that might never reach the Earth surface (due to atmospheric scattering). – This energy might be scattered into the IFOV. • Path 3: – Energy from the sun that has undergone some Rayleigh, Mie, and/or non-selective scattering before illuminating the target within the IFOV. Remote Sensing Ayman F. Habib 26 Paths of Radiance Received by the Sensor • Path 4: – Reflected radiation from nearby terrain elements into the sensor IFOV. • Path 5: – Reflected radiation from nearby terrain elements into the atmosphere and then scattered onto the target within the sensor IFOV. • The radiance received by the sensor is the cumulative summation of the radiation carried by paths 1-5. Remote Sensing Ayman F. Habib 27 Target Irradiance • For a given spectral interval (λ1 : λ2), the solar irradiance reaching the Earth’s surface can be represented by: λ2 E g = ∫ ( E0λ Tθ 0 cos θ 0 + Ed λ ) dλ (Wm − 2 ) λ1 E0λ Tθ 0 cos θ 0 ≡ Path 1 Spectral Radiance Ed λ ≡ Diffuse Sky Spectral Radiance ( Paths 3 and 5) Remote Sensing Ayman F. Habib 28 Atmospheric Transmittance • Atmospheric transmittance (Tθ) accounts for the atmospheric scattering and absorption. • In the absence of the atmosphere, the transmittance of solar radiant energy to the ground would be 100%. • The atmospheric transmittance (Tθ) may be computed as: – Tθ = e-τ/cosθ. – Where τ is the atmospheric optical thickness, – and θ is the solar/sensor zenith angle. Remote Sensing Ayman F. Habib 29 Atmospheric Optical Thickness (τ) • The optical thickness of the atmosphere at a certain wavelength, τ(λ), equals the sum of all the attenuating coefficients which are made up primarily of: – Rayleigh scattering τR. – Mie scattering τM. – Selective atmospheric absorption τA. τ (λ ) = τ R + τ M + τ A Where : τ A = τ H O + τ O + τ O + τ CO 2 2 3 2 Remote Sensing Ayman F. Habib 30 Radiance Received by the Sensor • A portion of the solar irradiance onto the target is reflected into the sensor’s IFOV. • Assuming a diffuse (Lambertian) target, the radiance received by the sensor: λ2 1 LT = π ∫ R Tθ λ v ( E0λ Tθ 0 cos θ 0 + Ed λ ) dλ (W m − 2 sr −1 ) 1 – The division by (π) is needed for the transformation from irradiance (Wm-2) to radiance (Wm-2sr-1). – R represents the target reflectance. – Tθv is the atmospheric transmittance from the target to the sensor. Remote Sensing Ayman F. Habib 31 Radiance Received by the Sensor • The radiance recorded by the sensor, LS, does not equal LT. – There is some additional radiance from other paths (2&4). LS = LT + LP (Wm −2 sr −1 ) • LP is known as the atmospheric path radiance. • The first objective of atmospheric correction is to minimize or remove the contribution of the atmospheric path radiance (LP). Remote Sensing Ayman F. Habib 32 Atmospheric Correction At-sensor radiance → Surface radiance Atmospheric Path Radiance Remote Sensing Ayman F. Habib 33 Atmospheric Path Radiance • LP represents the scattered path radiance, which introduces haze in the imagery and reduces image contrast. • LP is wavelength dependent. – Shorter wavelengths normally manifest greater scattering effect. • Haze compensation aims at minimizing the influence of path radiance effects. • Haze compensation of multi-spectral data can be achieved by observing recorded radiance from targets with zero reflectance. – Deep clear water bodies. Remote Sensing Ayman F. Habib 34 Atmospheric Path Radiance • Any signal observed over such an area represents the path radiance. • Observed signal can be subtracted from all pixels in that band. • Haze compensation is usually applied uniformly throughout the whole scene. – This may or may not be valid. • When extreme viewing angles are involved, one might need to compensate for varying atmospheric path lengths within the scene. Remote Sensing Ayman F. Habib 35 10,000 10,000 Frequency Frequency Original Data 5,000 5,000 Band1 Band 2 0 125 255 0 125 255 10,000 Brightness Values 10,000 Brightness Values Frequency Frequency 5,000 5,000 Band 4 Band 3 10,000 0 125 255 10,000 0 125 255 Band Minimum Maximum Brightness Values Brightness Values 1 51 242 Frequency Frequency 2 17 115 5,000 5,000 3 14 131 Band 5 4 4 105 Band 7 5 0 193 7 0 128 0 125 255 0 125 255 Brightness Values Brightness Values Remote Sensing Ayman F. Habib 36 10,000 10,000 Frequency Frequency Adjusted Data 5,000 5,000 Band1 Band 2 10,000 0 125 255 10,000 0 125 255 Brightness Values Brightness Values Frequency Frequency 5,000 5,000 Band 4 Band 3 0 125 255 0 125 255 10,000 Brightness Values 10,000 Brightness Values Band Minimum Maximum Frequency Frequency 1 0 191 2 0 98 5,000 5,000 3 0 117 Band 5 Band 7 4 0 101 5 0 193 0 125 255 0 125 255 7 0 128 Brightness Values Brightness Values Remote Sensing Ayman F. Habib 37 Surface Radiance → Target Reflection Remote Sensing Ayman F. Habib 38 Target Reflection λ2 1 LT = π ∫ R Tθ λ v ( E0λ Tθ 0 cos θ 0 + Ed λ ) dλ (W m − 2 sr −1 ) 1 • For small bandwidth, the above equation can be simplified as: 1 LT = R Tθ v ( E0∆λ Tθ 0 cos θ 0 ∆λ + Ed ) (W m − 2 sr −1 ) π • Having estimates of: – Solar irradiance at the top of the atmosphere (E0), – Diffuse sky scattering (Ed), – Solar zenith angle (θ0), – Sensor zenith angle (θv), and – Atmospheric optical thickness (τ); • One can derive an estimate of the target reflection (R). Remote Sensing Ayman F. Habib 39 Example • Solar Zenith angle (θ0) = 38°. • Sensor Zenith angle (θv) = 0°. • Atmospheric optical thickness (τ) = 0.15. • Tθ0 = e-0.15/cos38° = 0.827. • Tθv = e-0.15/cos0° = 0.861. • Solar irradiance at the top of the atmosphere (E0) = 256 Wm-2. • Diffuse sky scattering (Ed) = 0 Wm-2. • LT = 1/π R * 256 * 0.827 * 0.861 * cos(38°) = 45.72 R Remote Sensing Ayman F. Habib 40 Sun Angle Normalization Sun-Earth Distance Normalization Remote Sensing Ayman F. Habib 41 Ta ng en Satellite tp lan i th e n Ze Zenith angle Summer Sun Equator Sun Fall Spring Sun Winter Remote Sensing Ayman F. Habib 42 Sun Angle Normalization • Sun elevation correction accounts for the seasonal changes in the position of the Sun relative to the Earth. • Through Sun angle normalization, image data acquired under different solar illumination angles are normalized. – This can be done by calculating the digital numbers DN assuming the Sun was at the zenith. • The correction is applied by dividing the DN by the cosine of Sun’s zenith angle. Remote Sensing Ayman F. Habib 43 Sun Angle Normalization E0 E θ E0 = E / cos(θ) Remote Sensing Ayman F. Habib 44 Sun-Earth Distance Normalization • The Earth-Sun distance normalization is applied to compensate for the seasonal changes in the distance between the Earth and the Sun. • The Earth-Sun distance is expressed in astronomical units. – An astronomical unit is equivalent to the mean distance between the Earth and the Sun ≈ 149.6 x 106 km. • The irradiance from the Sun decreases as the square of the Earth-Sun distance increases. E0 (Solar radiance at mean Earth − Sun Distance) = E * d 2 Remote Sensing Ayman F. Habib 45 Combined Normalization • E0 = E * d2 / cos(θ0). • Where: – E = Solar irradiance. – E0 = Normalized solar irradiance at the mean Earth- Sun distance while the Sun is at the zenith. – θ0 = Sun’s angle from the zenith. – d = Earth-Sun distance in astronomical units. • Information regarding the solar elevation angle and Earth-Sun distance for a given scene is normally part of the ancillary data supplied with the purchased imagery. Remote Sensing Ayman F. Habib 46 Radiometric Image Processing Spatial Domain Remote Sensing Ayman F. Habib 47 Radiometric Image Processing: Overview • Noise Removal: – Median filter. – Neighborhood averaging filter. • Convolution. • Primitive extraction. – Interest points. – Edge detection. • Image processing in the frequency domain. – Fourier transform. Remote Sensing Ayman F. Habib 48 Noise Removal • Noise is present in acquired imagery due to: – Poor-sampling by the sensor (data recording process). – Atmospheric conditions. • Noise removal precedes any subsequent image processing activity (e.g., classification and positioning). • Principle: Apply spatial domain smoothing techniques in local neighborhoods within the acquired image. Remote Sensing Ayman F. Habib 49 Noise Removal: Median Filter • Sort the intensity values, within a local neighborhood, in an ascending or descending order. • Choose the median as the new centre value. • Characteristics: – Removes pixels in the neighborhood that are significantly different (i.e., due to noise) from the rest. – It preserves image sharpness/details. Remote Sensing Ayman F. Habib 50 Noise Removal: Median Filter 93 100 98 105 25 95 → 107 102 100 25 93 95 98 100 100 102 105 107→ 93 100 98 105 100 95 107 102 100 Remote Sensing Ayman F. Habib 51 Noise Removal: Median Filter Before After Remote Sensing Ayman F. Habib 52 Noise Removal: Neighborhood Averaging • Process of averaging all pixels in the neighborhood. 1 1 1 93 100 98 1 ∗ 1 1 1 * 105 25 95 = 92 9 1 1 1 107 102 100 Remote Sensing Ayman F. Habib 53 Convolution • Convolution is a simple mathematical operation which is fundamental to many image processing operators. • Convolution provides a way of multiplying together two arrays of numbers, generally of different sizes, but of the same dimensionality, to produce a third array of numbers of the same dimensionality. • This is used in image processing to implement operators whose output pixel values are simple linear combinations of certain input pixel values. Remote Sensing Ayman F. Habib 54 Convolution • In an image processing context, one of the input arrays is two-dimensional gray-level image. • The second array is usually much smaller, two- dimensional, and known as the kernel or convolution mask. • The convolution is performed by sliding the kernel over the image, generally starting at the top left corner. – The kernel is moved through all the positions where the kernel fits entirely within the boundaries of the image. Remote Sensing Ayman F. Habib 55 Convolution • Each kernel position corresponds to a single output pixel: – The value at that pixel is calculated by multiplying together the kernel value and the underlying image pixel value for each of the cells in the kernel, and then adding all these numbers together. • If the image has M rows and N columns, and the kernel has m rows and n columns, then the size of the output image will have: – (M-m+1) rows and (N-n+1) columns. Remote Sensing Ayman F. Habib 56 Convolution: Numerical Example 5 10 12 3 4 7 6 7 9 8 6 11 13 1 4 10 2 4 6 8 9 7 8 10 8 7 5 1 2 3 Mask Coefficients Original Values 10 9 12 8 10 10 11 3 3 4 10X7 W1 W2 W3 Z1 Z2 Z3 Z4 Z6 W4 W5 W6 Original Value Z5 Average Filter W7 W8 W9 Z7 Z8 Z9 1/9 1/9 1/9 Original Image 1/9 1/9 1/9 1/9 1/9 1/9 8 7 6 6 5 9 5 R = w1 z1 + w2 z 2 + L + w9 z9 9 9 9 8X5 = ∑ wi zi i =1 After Processing Remote Sensing Ayman F. Habib 57 Remote Sensing Ayman F. Habib 58 Smoothing Using Gaussian Mask x2 + y 2 − 1 2σ2 G ( x, y ) = e 2πσ 2 For σ = 1.0 Remote Sensing Ayman F. Habib 59 Gaussian Smoothing Remote Sensing Ayman F. Habib 60 Primitive Extraction Point Primitives Edge Detection Remote Sensing Ayman F. Habib 61 Interest Operators for Point Primitives • For some applications, one might be interested in automatic extraction of distinct points from the input imagery. – For example, photogrammetric 3-D restitution of the object space from overlapping 2-D imagery. • Two common approaches for automatic point extraction are: – Moravec Interest operator. – Förstner Interest operator. Remote Sensing Ayman F. Habib 62 Moravec Interest Operator • Interest points (feature points) are image locations where the interest operator computes a high variance value. • In other words, certain pixel within the image corresponds to an interesting point if: – There is a significant difference between the grey-value at this pixels and its neighboring pixels. • This comparison is quantified using variance computation. Remote Sensing Ayman F. Habib 63 Moravec Operator (Variance Computation) I1 = ∑ [ f ( x, ( x, y ) ∈ S y ) − f ( x , y + 1)] 2 I2 = ∑ [ f ( x, ( x, y ) ∈ S y ) − f ( x + 1, y )] 2 I3 = ∑ [ f ( x, ( x, y) ∈ S y ) − f ( x + 1, y + 1)] 2 I4 = ∑ [ f ( x, ( x, y ) ∈ S y ) − f ( x + 1, y − 1)] 2 Where S represent all the pixels in the window Remote Sensing Ayman F. Habib 64 Moravec Operator • Edge pixels have no variance along the edge direction. • The minimum value of the previous directional variances are taken as the interest value at the central pixel, (xc, yc): – I(xc, yc) =min (I1, I2, I3, I4). • We have interest points if the interest value at their locations are local maxima and exceed a predefined threshold. Remote Sensing Ayman F. Habib 65 Original Image Remote Sensing Ayman F. Habib 66 Variance Image Remote Sensing Ayman F. Habib 67 Moravec Interest Points Remote Sensing Ayman F. Habib 68 Edge Detection • Acquired imagery are rich with linear features. • The linear features are bounded by “edges”. • An edge represents a discontinuity in the two dimensional gray value distribution function. • Abrupt change in the gray level intensity within an area of the image space constitutes an edge. • Edge detection refers to the process that examines the digital image for discontinuities in the grey level function. Remote Sensing Ayman F. Habib 69 Edge Detection Original Image Detected Edges Remote Sensing Ayman F. Habib 70 Edge Detection Techniques • Edge detection techniques can be classified into two categories. – First derivative operators. • For example, SOBEL and CANNY Edge Detection. – Second derivative operators. • For example, LOG - Laplacian of Gaussian. Remote Sensing Ayman F. Habib 71 Edge Detection Intensity f(x) x Remote Sensing Ayman F. Habib 72 First Derivative • Edges are characterized by maximal values in the first derivative of the intensity profile. f´(x) Threshold x Remote Sensing Ayman F. Habib 73 Non Maximal Suppression • To have a single response per edge: – Broad ridges in the first derivative magnitude array must be thinned so that only the magnitudes at the points of greatest local change remain. • Non maximal suppression suppress all values across the direction of the line that are not peak values. Remote Sensing Ayman F. Habib 74 Second Derivative Operator • Edges are characterized by zero crossings in the second derivative of the intensity profiles. f´´(x) Edge Location x Remote Sensing Ayman F. Habib 75 Edge Detection: SOBEL Filter Gx Gy a b c -1 -2 -1 -1 0 1 d e f 0 0 0 -2 0 2 g h i 1 2 1 -1 0 1 • SOBEL Filter in the x-direction: – Gx = (g+2h+i) - (a+2b+c). • SOBEL Filter in the y-direction: – Gy = (c+2f+i) - (a+2d+g). Remote Sensing Ayman F. Habib 76 Remote Sensing Ayman F. Habib 77 Edge Detection: SOBEL Filter • Disadvantage: – Only the derivatives in x and y directions are considered. – The x-y direction is ignored. – This will result in losing some edge pixels. • As an improvement, we can consider an expanded neighborhood of “e” and use “Extended SOBEL Filter.” Remote Sensing Ayman F. Habib 78 Extended SOBEL Filter • SOBEL Filter in the xy-direction: – Gxy = 4/6[(j+2d+2h+u) - (p+2b+2f+o)] • SOBEL Filter in the yx-direction: – Gyx = 4/6[(m+2f+2h+s) - (r+2b+2d+l)] p q r j a b c m k d e f n l g h i o s t u Remote Sensing Ayman F. Habib 79 Original Image Remote Sensing Ayman F. Habib 80 CANNY Edge Detection Remote Sensing Ayman F. Habib 81 Edge Detection & Enhancement • Chavez Kernel: – Apply the Average Filter. – Subtract the Average Filter from the original pixel value to get the high frequency (HF) component: • HF = pixel value - average value – By adding HF back to the original pixel, a high frequency enhancement will be achieved: • New value = Pixel Value + HF. Remote Sensing Ayman F. Habib 82 Chavez Kernel 1 1 1 1 AVG = ∗ 1 1 1 9 1 1 1 0 0 0 1 1 1 − 1 − 1 − 1 0 1 0 − 1 ∗ 1 1 1 = 1 ∗ − 1 8 − 1 HF = 9 9 0 0 0 1 1 1 − 1 − 1 − 1 0 0 0 − 1 − 1 − 1 − 1 − 1 − 1 0 1 0 + 1 ∗ − 1 8 − 1 = 1 ∗ − 1 17 − 1 NewValue = 9 9 0 0 0 − 1 1 − 1 − 1 1 − 1 Remote Sensing Ayman F. Habib 83 Radiometric Image Processing Frequency Domain Remote Sensing Ayman F. Habib 84 Fourier Series 1 0.5 (1/1)sin(2*pi*1*x) 0 -0.5 -1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1 0.5 0 -0.5 -1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Remote Sensing Ayman F. Habib 85 Fourier Series 1 0.5 (1/1)sin(2*pi*1*x) (1/3)sin(2*pi*3*x) 0 -0.5 -1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1 0.5 0 -0.5 -1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Remote Sensing Ayman F. Habib 86 Fourier Series 1 0.5 (1/1)sin(2*pi*1*x) (1/3)sin(2*pi*3*x) 0 (1/5)sin(2*pi*5*x) -0.5 -1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1 0.5 0 -0.5 -1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Remote Sensing Ayman F. Habib 87 Fourier Series 1 0.5 (1/1)sin(2*pi*1*x) (1/3)sin(2*pi*3*x) 0 (1/5)sin(2*pi*5*x) -0.5 (1/7)sin(2*pi*7*x) -1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1 0.5 0 -0.5 -1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Remote Sensing Ayman F. Habib 88 Fourier Series 1 0.5 (1/1)sin(2*pi*1*x) (1/3)sin(2*pi*3*x) 0 (1/5)sin(2*pi*5*x) -0.5 (1/7)sin(2*pi*7*x) (1/9)sin(2*pi*9*x) -1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1 0.5 0 -0.5 -1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Remote Sensing Ayman F. Habib 89 Fourier Series • F(x) = (1/1)sin(2*pi*1*x) + (1/3)sin(2*pi*3*x) + (1/5)sin(2*pi*5*x) + (1/7)sin(2*pi*7*x) + (1/9)sin(2*pi*9*x) + …. • We have generated a square wave by adding different sine waves. • These sine waves have different amplitudes and wavelengths/frequencies. Remote Sensing Ayman F. Habib 90 Fourier Series • The reverse is the Fourier series: Any signal can be decomposed into many sinusoidal waves. • Each sine wave is characterized by its amplitude and wavelength (or frequency). • The input signal is time/location dependant or is said to be in time/spatial domain. Remote Sensing Ayman F. Habib 91 Fourier Series • Fourier series is the process of transforming a signal from time/spatial domain to frequency domain. • In this example: Summation of sine waves f(x) with different amplitudes Fourier Series Square wave and wavelengths that make up the signal. Remote Sensing Ayman F. Habib 92 Fourier Series Amplitude Frequency 1 1 f(x) Fourier Series 1/3 3 1/5 5 Spatial domain 1/7 7 1/9 9 … … To infinity Frequency domain Remote Sensing Ayman F. Habib 93 Fourier Transform of Discrete Functions • Given a discrete function gl, where 0 ≤ l ≤ N - 1 Fourier Transform: Spatial → Frequency Domain N −1 2π kl −i Gk = ∑g l =0 l e N 0 ≤ k ≤ N −1 Inverse Fourier Transform: Frequency → Spatial Domain N −1 2 π kl i gl = ∑G k =0 k e N 0 ≤ l ≤ N −1 Remote Sensing Ayman F. Habib 94 Image: Spatial and Frequency Domains Low frequency components Remote Sensing Ayman F. Habib 95 Image: Spatial and Frequency Domains Remote Sensing Ayman F. Habib 96 Fourier Transform & Image Processing • Low frequency components correspond to the trend within the input image. • High frequency components correspond to the details within the input image. • Image smoothing can be established by maintaining low frequency components while eliminating high frequency components. • Image sharpening can be established by maintaining high frequency components while eliminating low frequency components. Remote Sensing Ayman F. Habib 97 Smoothing in the Frequency Domain Remote Sensing Ayman F. Habib 98 Sharpening in the Frequency Domain Remote Sensing Ayman F. Habib 99 Fourier Transform & Image Enhancement Spatial Domain Frequency Domain Spatial Domain Frequency Domain Remote Sensing Ayman F. Habib 100 Fourier Transform & Image Enhancement Spatial Domain Frequency Domain Remote Sensing Ayman F. Habib 101 Fourier Transform & Image Enhancement Frequency Domain Spatial Domain Remote Sensing Ayman F. Habib 102