1 W. Philpot, Cornell University The Digital Image

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1 1 The Digital Image DEFINITION: A grayscale image is a single-valued function of 2 variables: ff(xx 1, xx 2 ). Notes: A gray scale image is a single-valued function of two spatial variables, ff(xx 11, xx 22 ). A standard color (RGB) image is a 3-valued function of two spatial variables, ff(xx 1, xx 2 ). A multispectral or hyperspectral image is an n-valued function of two spatial variables where n = # spectral bands. The limit of two spatial variables is a matter of convention and convenience (think CAT scan, MRI, or ground penetrating radar). soil temperature time Spatial features: resolution (sampling, pixel size and spacing) image size (ground coverage) Spectral features number of spectral bands location of spectral bands band width & shape Trade-offs: data volume: day1 day 2 day 3 day 4 gray value ==> quantization sampling ==> average over 2 hrs 40 min Digitization: The process of representing a continuous function by a finite set of discrete observations. Sampling: digitization in the x,y (spatial) domain. Quantization: digitization in the function domain (intensity/radiometric). DEFINITION: A digital image is an image that has been sampled and quantized. Sampling & quantization ==> information loss The sampling and quantization procedures can be designed to minimize information loss. Radiometric features quantization (radiometric precision) dynamic range image size (bytes) = (width/x-spacing) * (length / y-spacing) * # bytes/pixel * # spectral bands Sensitivity: narrow spectral bands less energy per band smaller pixels less energy per pixel

2 2 Data volume image size = wwwwwwwwh llllllllllh/(pppppppppp ssssssssssssss) 2 # bbbbbbbbbb/pppppppppp # ssssssssssssssss bbbbbbbbbb Swath (square) Pixel size Pixels / Bytes / Spectral Data Volume (km) (m) band pixel Bands (MB) 1,000 1,000 1.E , E , E , E , E ,000 Sampling & quantization information loss The sampling and quantization procedures can be designed to minimize information loss. # of samples vs. resolution and positional accuracy # of gray levels vs. intensity resolution and radiometric precision # bits # gray vals. n 2 n range ==> bit, binary, bi-tonal image (fax, screen, half-tone) ==> photographs, printed images ==> 1 byte = 8 bits (digital camera) ,536 -/+32,768 ==> 2 bytes = 16 bits word = the fixed-sized piece of data handled as a unit = 1, 2, 4, 8, or even 16 bytes dynamic range: difference between the maximum and minimum signals upper limit ==> saturation lower limit ==> dark current sensitivity (quantization interval): smallest interval ( v) between 2 consecutive values. false contouring: apparent boundaries in an image resulting from inadequate quantization. probability density function (pdf): The probability that a pixel, randomly selected from an image will have a gray value, k.

3 3 AVIRISng 480 bands (1 mrad, 14-bit) VIIRS 375 m m, 12-bit WV-3 (1.24 m VNIR, 3.7 m, SWIR, 11-bit) ASTER (15+30 m, 8-bit) OLI (30 m, 12-bit) ETM+ (30 m, 8-bit) MSS (80 m, 7-bit) AVHRR (1000 m, 10-bit) wavelength (microns) Reflectance AVIRIS reflectance Salton Sea Water Vegetation wavelength (nm) ITC's database of satellites and sensors: ETM+ (8-bit) IKONOS (11-bit) AVHRR (11-bit)

4 4 Bright areas are saturated (roofs), dark areas are black (building shadow) and subtle variations are washed out (parking spaces). Source: False Contouring: apparent boundaries in an image resulting from inadequate quantization Temperature distribution map (Image quantized to 256 gray values) Greater bit depth allows for either increased sensitivity (IKONOS) or greater dynamic range (AVHRR) Image quantized to 4 gray values

5 5 Plane angles: examples s = 2h tan (θ/2) h = altitude s = perpendicular distance (at range h) θ = the angle subtended by s at range h If θ < 5 small angle approximation sin θ ~ tan θ ~ θ in radians s ~ h θ GIFOV altitude (h) IFOV (θ) length (s) 100 m 1 mrad 10 cm s 0.1 mrad 1 cm (visual res. limit) 0.01 mrad 1 mm 1 km 1 mrad 1 m 10 km 1 mrad 10 m 800 km 1 mrad 800 m (AVHRR) 0.1 mrad 80 m (MSS; 1972) 0.01 mrad 8 m (SPOT; 1986) mrad (1 µrad) 0.8 m (Quickbird; 2001) Solid Angles A solid angle is defined as an area on the surface of a sphere, divided by the square of the radius of the sphere, ω = A/r 2 ~ A'/r 2 for small angles, the area on the surface of the sphere is approximated by the area of the intersecting planar surface. ω = solid angle = A /r2 (steradians) ω 1/2 = plane angle = s x / r (IFOV) GIFOV: the projection of the instrument IFOV on to the ground The GIFOV is dependent on the altitude and look angle. Sometime referred to as the GRC or Ground Resolution Cell θ h Typical Assumptions 1. The GIFOV is a pure pixel 2. The reflectance is typically assumed to be Lambertian (i.e., the reflected radiance is independent of the illumination direction and the viewing direction.) Reality 1. The GIFOV is a mixed pixel. 2. The reflectance is very dependent on the illumination and viewing geometry

6 6 A simple mechanical scanner geometry Sampling interval & res. element sampling interval (sx, sy) - the separation of adjacent samples (equidistant sampling). resolution cell size (rx, ry) (also resolution element size) - the area covered, or solid angle subtended by a single sample. Instantaneous Field of View (IFOV) The field of view of a single detecting element when all motion is stopped. The term IFOV is frequently used as a synonym for resolution cell size. Although not proper, it is rarely misleading because the actual difference is usually quite small.

7 7 The standard definition of spatial resolution is the measure of the separability of line pairs (Rayleigh Criterion). Sampling interval & res. element picture element (pixel, pel) - a single sample pure pixel: a pixel which lies entirely within a single target class. mixed pixel: a pixel which lies partly in two or more target classes. mixed pixel pure pixel Rule of thumb: To insure at least one pure pixel, the largest sampling interval should be to 1/3 the smallest dimension of the target. ss XX = ss yy dd/3

8 8 Resolution criteria: pure pixel vs. sampling pure pixel criterion: s d 1/3 d1 d1 sampling criterion: s d 2/2 d3 d3 d2 d1 d2 d1 Important definitions picture element (pixel, pel) a single sample sampling interval (sx, sy) - the separation of adjacent samples (equidistant sampling). resolution cell size (r x, r y ) - resolution cell size (rx, ry) (also resolution element size) - the area covered by a single sample. Instantaneous Field of View (IFOV) - the field of view of a single detecting element when all motion is stopped. Ground Instantaneous Field of View (GIFOV) - the IFOV projected on the ground (specific to a particular altitude, and viewing angle.) pure pixel: a pixel which lies entirely within a single target class. mixed pixel: a pixel which lies partly in two or more target classes. Digital samples: pixel display size display cell (dx, dy) - a single pixel on a particular display device. For an image to appear undistorted (without prior geometric correction) the display cell size should be proportional to the sampling intervals, i.e.: dx/dy = sx/sy <== aspect ratio

9 9 Sampling and resolution Wavelength (l) - the distance between consecutive maxima of a repeating waveform. Spatial frequency (k = 1/ l) - the number of times per unit distance that a spatial feature repeats. Sampling: 1-D example Sampling the sine wave (assume that r ==> 0) One dimensional examples amplitude phase frequency 1. Oversampling (s < λ/2) as s ==> 0 excellent excellent excellent as s ==> λ/2 x good good 2. Critical sampling (s = λ/2) x x good 3. Undersampling (s > λ/2) x x x Critical Sampling interval, (sc) The lowest sampling interval capable of correctly characterizing a specific spatial frequency: k = 1/ λ. sc = λ /2 Nyquist frequency, (kn) The highest spatial frequency which can be represented when sampling at an interval, s. kn = 1/2s Aliasing - The appearance of artifacts in an image as a result of undersampling. There is a direct relationship between the original wavelength, λ, the sampling interval, s, and the aliased wavelength, λ': If λ < 2s the original frequency is: k = = + λ 2s s and the aliased frequency is: k' = = λ' 2s s

10 10 Aliasing: 2-dimensional example kx = spatial frequency in the x-direction ky = spatial frequency in the y-direction k = frequency in two-dimensional system: 2 2 x y k = k + k direction of two-dimensional spatial frequency: 1 ky ϕ= tan kx (a) Original Image (SPOT pan, 10 m GIFOV) (b) Resampled to 20 m GIFOV (c) Resampled to 30 m GIFOV (d) Resampled to 40 m GIFOV Note that the image, when observed at a constant scale, does not appear blurry if the sample size remains constant. Increasing the sample size (resolution cell size) tends to damp out the aliased features.

11 11 Aliasing in two-dimensions summary Example: sampling interval: sx sy Nyquist frequency: Nx Ny actual spatial freq.: kx = Nx + Dkx ky = Ny + Dky aliased spatial freq.: kx' = Nx Dkx ky' = Ny Dky 2 ( ) ( ) 2 k' = aliased frequency = k ' + k ' 1 k y ' ϕ= ' aliased direction = tan k x ' Aliasing: the effect of resolution cell sizeresolution cell size (r), sampling interval (s) r ==> 0 - The sampled value will approach the local function value. - susceptible to random noise. 0 < r < s - has the effect of a low-pass filter. Frequencies higher than 1/2r will be damped out. r = s r > s x - will damp out (but not remove) frequencies higher than Nyquist frequency ==> no significant loss of information for frequencies lower than the Nyquist frequency. - damps out aliased frequencies - damps out highest real frequencies. y