Notes

Outline

Computer Vision, Lecture 3 Professor Hager http://www.cs.jhu.edu/~hager

Outline for Today Image noise Filtering by Convolution Properties of Convolution

IMAGE  NOISE

Image Noise An experiment: take several images of a static scene and look at the pixel values

Noise Models Noise is commonly modeled using the notion of “additive white noise.” Scalar example:  I(x) = I*(x) + n(x) Images: I(i,j,t) = I*(i,j,t) + n(i,j,t) Note that n(i,j,t) is independent of n(i’,j’,t’) unless i’=i,j’=j,t’=t. Typically we assume that n (noise) is independent of image location --- that is, it is i.i.d

Properties of Noise Processes Properties of temporal image noise:

Image Noise An experiment: take several images of a static scene and look at the pixel values

PROPERTIES OF TEMPORAL IMAGE  NOISE (i.e., successive images)

Other Types of Noise Impulsive noise randomly pick a pixel and randomly set ot a value saturated version is called salt and pepper noise Quantization effects Often called noise although it is not statistical Unanticipated image structures Also often called noise although it is a real repeatable signal.

Temporal vs. Spatial Noise It is common to assume that: spatial noise in an image is consistent with the temporal image noise the noise is independent and identically distributed Thus, we can think of the image itself as an additive noise process

How to reduce noise Averaging is a common way to reduce noise instead of temporal averaging, how about spatial For a pixel in image I at I,j

DISCRETE  CONVOLUTION

How to Reduce Noise For a pixel in image I at I,j Computing this for every pixel location is the convolution of the image I with the template (or kernel) consisting of a 3x3 array of 1’s. Note that is this O(n2m2) for an nxn image and mxm template. Note we have to normalize the template to 1 to make sure we don’t introduce any scaling into the image.

Convolution With a Box Filter

Some Convolution Facts  We often write I’ = B*I to represent the convolution of I by B.  B is referred to as the kernel of the convolution (or sometimes the “stencil” in the discrete case). Note convolution is Associative Commutative Linear We are using a discrete convolution; we will see this is not always consistent with an underlying continuous convolution that we may wish to implement Convolution is formally defined on unbounded images and kernels. padding schemes: pad with zeros (same size vs. full size) compute only legal values

Understanding Convolution Another way to think about convolution is in terms of how it changes the frequency distribution in the image. Recall the fourier representation of a function F(u) = s f(x) e-2p i u x dx recall that e-2p i u x = cos(2p u x) – i sin (2 p u x) Also we have f(x) = s F(u) e-2p i u x du F(u) = |F(u)| ei F(u) a decomposition into magnitude and phase |F(u)|^2 is the power spectrum Questions: what function takes many many many terms in the Fourier expansion?

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The Fourier Transform and Convolution If H and G are images, and F(.) represents Fourier transform, then Thus, one way of thinking about the properties of a convolution is by thinking of how it modifies the frequencies of the image to which it is applied. In particular, if we look at the power spectrum, then we see that convolving image H by G attenuates frequencies where G has low power, and amplifies those which have high power. This is referred to as the Convolution Theorem

The Properties of the Box Filter

Computer Vision, Lecture 4 Professor Hager http://www.cs.jhu.edu/~hager

Outline for Today Gaussian Filtering Edges and derivative filters

What a Box Filter Does

The Gaussian Filter: A Better Noise Reducer Ideally, we would like an averaging filter that removes (or at least attenuates) high frequencies beyond a given range It is not hard to show that the FT of a Gaussian is again a Gaussian.  Hence, it operates as a low pass filter. Note that in general, we truncate --- a good general rule is that the width (w) of the filter is at least such that w > 5 s.  Alternatively we can just stipulate that the width of the filter determines s (or vice-versa). Note that in the discrete domain, we truncate the Gaussian, thus we are still subject to ringing like the box filter.

Computational Issues: Separability Recall that convolution is commutative.  Suppose I use the templates gx = exp(-i2/2 s2) and By = gy = exp(-j2/2 s2) Then gx * (gy * I) = (gx * gy) * I but, it is not hard to show that the first convolution is simply the 2-D Gaussian that we defined previously! In general, this means that we can “separate” the 2-D Gaussian convolution into 2 1-D convolutions with great computational cost savings. A good exercise is to show that the box filter is also separable.

Computational Issues: Minimizing Operations Note that for a 256 gray level image, we can precompute all values of the convolution and avoiding multiplication. For the box filter, we can implement any size using 4n additions per pixel. Also note that, by the central limit theorem, repeated box filter averaging yields approximations to a Gaussian filter. Finally, note that a sequence of filtering operations can be collapsed into one by associativity. in general, this is not a win, but we’ll see examples where it is ...

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Yet Another View of Convolution Suppose we consider the convolution template as a “vector”: T = [T1,T2,T3 .... Tn] Likewise, consider a region of the image to which the convolution is applied as a vector I = [I1,I2,...In] Then the value of the convolution at a point is just the “dot product” v = T ¢ I Thus, we can also think of convolution as a kind of “pattern match” where regions of the image that are “similar” to T respond more strongly than those that are dissimlar (up to a scale factor)

What Else Can You Do With Convolution? Thus far, we’ve only considered convolution kernels that are smoothing filters. Consider the following kernel: [ -1, 1]

What Else Can You Do With Convolution? Thus far, we’ve only considered convolution kernels that are smoothing filters. Consider the following kernel: [ -1;1]

The Image Gradient Recall from calculus for a function of two variables  f (x,y) : The gradient: points in the direction of maximum increase. Its magnitude is proportional to the rate of increase. The total derivative in the direction n = n ¢ r f The kernel [-1,1] is a way of computing the x derivative The kernel [-1;1] is a way of computing the y derivative

Some Other Interesting Kernals

Some Other Interesting Kernals

Smoothing Plus Derivatives One problem with differences is that they by definition reduce the signal to noise ratio (can you show this?) Recall smoothing operators (the Gaussian!) reduce noise. Hence, an obvious way of getting clean images with derivatives is to combine derivative filtering and smoothing: e.g. G * Dx * I = Dx * G * I