Preprocessing - Filtering: noise suppression Flashcards
What types of noise are in cameras?
• Photon noise:
Variation in the number of photons falling on a pixel per time interval T
Schwankung der Anzahl der Photonen, die pro Zeitintervall T auf ein Pixel fallen
• Thermal noise:
Heat can free electrons and generate a response when there is none
Wärme kann Elektronen freisetzen und eine Reaktion erzeugen, wenn keine vorhanden ist
• Electronic noise
• Camera response may be non-linear over the number of photons falling on a surface (camera gamma)
Die Reaktion der Kamera kann nichtlinear über die Anzahl der auf eine Oberfläche fallenden Photonen sein (Kamera-Gamma)
• Saturation: each pixel can generate only a limited amount of charge
jedes Pixel kann nur eine begrenzte Menge an Ladung erzeugen
• Blooming: saturated pixel can overflow to neighboring pixels
gesättigte Pixel können auf benachbarte Pixel überlaufen
• Burned pixels
How is noise detected?
Basic logic:
each imaging system records an image by counting photons
This type of noise can be modeled
• as an independent additive noise
• by a zero-mean Gaussian
Better model: Poisson distribution
• Individual photon detections: independent events that follow a
random temporal distribution
• Photon counting: classic Poisson process
• discrete probability distribution
• Parameter λt controls the mean and the variance:
var{N} = E{N} = λt
• Symmetry of the distribution depends on the mean value
What is the Salt and pepper noise?
= Data drop-out - Errors in the data transmission - burned/corrupted pixels
• maximum value (“snow in the image”)
• zero (“peppered”)
• isolated/localized noise: affects only individual pixels
What is linear transformation?
Realized by linear systems
A transformation T is linear if for any input functions 1f and 2f and for any constant scalar a ∈ R, we have:
T{a · 1f} = a · T{1f} and T{1f + 2f} = T{1f} + T{2f}
Consequences:
• Multiplication in the input corresponds to multiplication in the output.
• Filtering an additive image is equivalent to filtering each image separately and then adding the results.
What is the Shift-Invariant Transformation?
A linear system is shift-invariant if
T{δj−μ,k−ν} = gj−μ,k−ν
Without loss of generality, the position of the unit impulse can be set to
μ = ν = 0.
Then, the impulse response is
gjk =T{δjk}
Remark: The filter T does not change as we shift it in the image (as we move it from one position to the next).
If not, shift-variant & depends on μ and ν
What is the convolution?
Let the pattern [fjk ] be the input of a linear, shift-invariant system with impulse response [gjk ]. Then, the output [hjk ] can be computed using the discrete convolution of [fjk ] and [gjk ]
Algebraic properties: commutative, associative, distributive
In general, patterns defined only for finite interval
- Filtering often involves replacing value of a pixel in input image by weighted sum of its neighbors
- Represent the weights as a matrix/image, g
- g is usually called the kernel
- The operation for computing the weighted sum is called convolution
How can the Discrete Fourier Transform be used to compute the output of a linear system?
- Fast Fourier Transform for efficient computation
- Visualization of the effect of a linear system
• Sequence of samples [fjk ] completely defined by the Mx My values fjk
• Notionally, the sequence can be continued periodically by
f(j +μMx,k +νMy)=f(j,k)=fjk
• Finite sequence [fjk] interpreted as one period of a periodic sequence [fjk]
What is the circular convolution?
- Let [ ̃f jk] and [g ̃ jk ] be two periodic sequences with the common period length Mx , My .
- Use DFT to compute the periodic sequences [F ̃ μν] and [G ̃μν].
• Consequence: one period of the circular convolution equals the discrete convolution if Mx′ ≥Mx +mx −1 and My′ ≥My +my −1.
What does the Mean Filter?
smoothing via simple averaging, used for removing image noise
Properties
• Sensitive to outliers
• Typically blurs edges
• Often causes ringing effects
What does the Gaussian Filter?
Gaussian smoothing: smoothing via weighted averaging
• Pixels closest to the center pixel weighted more heavily
• The Gaussian function has exactly that profile.
• Gaussians better approximate the behavior of a defocused lens.
Isotropic Gaussian filter: circularly symmetric
• Small σ: almost no effect (weights at neighboring points are negligible)
• Large σ: blurring (neighbors have almost the same weight as the central pixel)
• Commonly used σ values:
Let w be the size of the kernel. Then σ = w/5.
For example for a 3 × 3 kernel: σ = 3/5 = 0.6
Properties
• Works very well for images affected by Gaussian noise
• Not very effective in removing salt & pepper noise:
• small σ values do not remove the salt & pepper noise
• large σ values blur the image too much
What is the ringing effect?
Loss of (precision in) high frequency components
What is the Median Filter?
Median filter: non-linear smoothing
• Replaces a pixel value with the median of all pixels in the neighborhood
• Relatively slow because it involves sorting
• Cannot be implemented via convolution
Properties
• Preserves high spatial frequency details
• Works well if less than 50% of the neighboring pixels are corrupted by noise
• Not very effective in removing Gaussian noise
