Chapter 3: Image Processing PDF

Chapter 3. Image processing Contents 3.1 Pixel Processing 3.2 LTI Systems (Linear Time Invariant) and Convolution 3.3 Linear Image Filters 3.4 Non-Linear Image Filters  Chapter 3. Image processing 3.1 Pixel Processing We can transform the brightness value of a pixel based on...

Chapter 3. Image processing Contents 3.1 Pixel Processing 3.2 LTI Systems (Linear Time Invariant) and Convolution 3.3 Linear Image Filters 3.4 Non-Linear Image Filters  Chapter 3. Image processing 3.1 Pixel Processing We can transform the brightness value of a pixel based on the value itself, independently of its location or the values of other pixels in the image. It is basically a mapping of one brightness value to another brightness value or one color to another color. F(x,y) is the value, g(x,y) is the result of the transformation T. 𝑔 𝑥, 𝑦 = 𝑇(𝑓 𝑥, 𝑦 )  Chapter 3. Image processing 3.1 Pixel Processing The following figures illustrate an example of such transformations.  Chapter 3. Image processing 3.1 Pixel Processing  Chapter 3. Image processing 3.2 LTI Systems (Linear Time Invariant) and Convolution Systèmes invariant aux décalage linéaires We will present this concept using one-dimensional signals before extending to multiple dimensions. Figure below shows an LTI system with input f(x) and output g(x).  Chapter 3. Image processing 3.2 LTI Systems (Linear Time Invariant) and Convolution Systèmes invariant aux décalage linéaires Linear: The output is scaled by the same amount as the input. Its response to a weighted sum of inputs is equal to the weighted sum of its responses to each of the inputs. Time invariant: The output is delayed the same as the input. If delaying the input to the system delays the output by the same amount of time.  Chapter 3. Image processing 3.2 LTI Systems (Linear Time Invariant) and Convolution The unit impulse response for DT systems The simplest input signal to a system is a single unit impulse. H[n] = System output (response) to a unit impulse input [n]. The importance of the unit impulse response h[n] is that for LTI Systems (Linear Time Invariant) we can use superposition to construct the output y[n] for any input x[n] just by scaling and shifting h[n]. Thus, the unit impulse response completely characterize the system.  Chapter 3. Image processing 3.2 LTI Systems (Linear Time Invariant) and Convolution DT systems response for shifted/scaled unit impulse  Chapter 3. Image processing 3.2 LTI Systems (Linear Time Invariant) and Convolution DT systems response for shifted/scaled unit impulse. We suppose that we have some sequences x[n] with the value x[k] at n =k. Output is now scaled by the value of x[k].  Chapter 3. Image processing 3.2 LTI Systems (Linear Time Invariant) and Convolution Example: Given the following impulse response h[n] for the system, find the output y[n] for the input x[n] shown. Hint: Treat each member of x[n] individually as scaled/shifted impulses. Determine the output for each member of x[n] separately as scaled/shifted versions of h[n]. Add all results to get y[n]. Given:  Chapter 3. Image processing 3.2 LTI Systems (Linear Time Invariant) and Convolution Solution: The procedure is called convolution or convolution sum.  Chapter 3. Image processing 3.2 LTI Systems (Linear Time Invariant) and Convolution Solution: The procedure is called convolution or convolution sum.  Chapter 3. Image processing 3.2 LTI Systems (Linear Time Invariant) and Convolution Review DT Sampling (shifting) property ∞ ∞ 𝑥 𝑘 = ෍ 𝑥 𝑛 𝛿[𝑘 − 𝑛] 𝑥 𝑛 = ෍ 𝑥 𝑘 𝛿[𝑛 − 𝑘] 𝑛=−∞ 𝑘=−∞ Interpretations: 1. The -function samples the n-th term of the sequence x[k] 2. The sequence x[n] is represented as a weighted sum of shifted unit impules  Chapter 3. Image processing 3.2 LTI Systems (Linear Time Invariant) and Convolution Systèmes invariant aux décalage linéaires DT systems response for shifted/scaled unit impulse  Chapter 3. Image processing 3.2 LTI Systems (Linear Time Invariant) and Convolution Systèmes invariant aux décalage linéaires DT system output for general input sequence Next sum the previous result over all values of k. Since system is linear, the output is also summed (superposition principle).  Chapter 3. Image processing 3.2 LTI Systems (Linear Time Invariant) and Convolution Systèmes invariant aux décalage linéaires DT system output for general input sequence Next sum the previous result over all values of k. Since system is linear, the output is also summed (superposition principle). ∞ 𝑥 𝑛 = ෍ 𝑥 𝑘 𝛿[𝑛 − 𝑘] 𝑛=−∞  Chapter 3. Image processing 3.2 LTI Systems (Linear Time Invariant) and Convolution Systèmes invariant aux décalage linéaires The system’s output y[n] for any input sequence x[n] just by knowing the impulse response h[n]. y 𝑛 = σ∞ 𝑘=−∞ 𝑥 𝑘 ℎ[𝑛 − 𝑘] y 𝑛 = 𝑥 𝑛 ∗ ℎ[𝑛]  Chapter 3. Image processing 3.2 LTI Systems (Linear Time Invariant) and Convolution Systèmes invariant aux décalage linéaires Given x[n] and h[n], find the output y[n]. 1. Break up the input x[n] into individual shifted/scaled impulses 2. Determine the output y[n] for each of these impulses by shifting/scaling h[n] accordingly. 3. Add all the results together to get y[n].  Chapter 3. Image processing 3.2 LTI Systems (Linear Time Invariant) and Convolution Systèmes invariant aux décalage linéaires  Chapter 3. Image processing 3.2 LTI Systems (Linear Time Invariant) and Convolution Systèmes invariant aux décalage linéaires Given x[n] and impulse response h[n], find the output y[n]. y 𝑛 = σ∞ 𝑘=−∞ 𝑥 𝑘 ℎ[𝑛 − 𝑘] Step1: Rename x[n] -> x[k] Step2: Change h[n] into h[-k]  Chapter 3. Image processing 3.2 LTI Systems (Linear Time Invariant) and Convolution Systèmes invariant aux décalage linéaires Step3: Shift h[-k] by n, where n