Fuzzy Medical Image Segmentation Presentation-I for Pattern Recognition Class Mohammed Jirari

Fuzzy Logic Fuzzy Logic Definition: A branch of logic that uses degrees of membership in sets rather than a strict true/false membership.

Fuzzy Logic A tool to represent imprecise, ambiguous, and vague information Its power is the ability to perform meaningful and reasonable operations Fuzzy logic is not logic that is fuzzy -- it is a logic of fuzziness. It extends conventional Boolean logic to recognize partial truths and uncertainties.

Linguistic Variables Fuzzy logic quantifies and reasons about vague or fuzzy terms that appear in our natural language Fuzzy Terms are referred to as linguistic variables Definition: Linguistic Variable Term used in our natural language to describe some concept that usually has vague or fuzzy values Examples: Linguistic Variable Typical Values Temperature hot, cold Height short, medium, tall Speed slow, creeping, fast

Example of a Fuzzy Set The graph shows how one might assign fuzzy values to various temperatures based on 68 degrees = room temp. Climate for a given temperature is defined as: 60d = {1 c, 0 w, 0 h} 68d = {0.5 c, 1 w, 0.5 h} 70d = { 0.15 c, 0.15 w, 0.85 h} Sum of fuzzy values not always 1 -- often it is more than 1 1 The fuzzy set is constructed by taking the domain of raw data and assigning corresponding fuzzy values to each member of the fuzzy set. The fuzzy values for “cool” is depicted in blue. “Warm” is in green. “Hot” is in red. We will use 68 degrees Fahrenheit , standard room temperature, as our optimal temperature. Fuzzy values are measured on a 0.0 to 1.0 scale, not unlike probability. However, fuzzy values are NOT probability. Let’s take a straightforward example, 60 degrees. As you can see, 60 degrees is very strongly considered “cool” and is assigned a fuzzy value of 1. The others have fuzzy value of 0. At this temperature, the fuzzy set value is little different from the Boolean value of “X is cool”. But look at 68 degrees. Since it is the optimal room temperature, we can understand that 68 degrees is “warm” with a fuzzy value of 1. However, it is also judged “cool” with fuzzy value 0.5 and “hot” with fuzzy value 0.5. This is because some people will find room temperature to be too cold, while others find it too hot. Again, all three are simulanteously true. Note that the fuzzy values add up to 2.0, not 1.0. This is what makes them different from probability. 68 degrees is not 50% cool, it has a fuzzy value of coolness of 0.5. There is no limit on the sum of the fuzzy values. The interpretation of 70 degrees should be straightforward to you at this point. 60 d 68 d 76 d Cold Warm Hot

Example of a Fuzzy Set: Asymmetric Version Fuzzy sets are rarely symmetric. This might be considered by some to be a more accurate description of a room climate: 60d = {1 c, 0 n, 0 w, 0 h} 68d = {0.5 c, 1 n, 0.8 w, 0.5 h} 70d = { 0.15 c, 0.7n, 0.95 w, 0.85 h} Could also be represented as: WARM = (0/60, .8/68, .95/70) 1 The depiction of the fuzzy set on the previous slide might lead you to believe that fuzzy sets often have a symmetry to them. Such is not the case. If anything, fuzzy sets are rarely symmetric. To demonstrate this, let us extend our temperature example to interpret “warm” as being “slightly too hot”, and therefore add a fourth member called “nice”. In this case, “cool” and “hot” might remain symmetric with each other, but you can see that “nice” and “warm” are not. You are probably asking yourself, “How are these fuzzy values determined?” The knowledge engineer is responsible for helping extract this from the expert. Often, it is the result of a series of test cases -- providing a number of temperatures to the expert and asking, “How would you interpret this?” The knowledge engineer then tries to fit a curve to the answers such that the expert generally agrees with the results. It is not easy, especially if it is a decision that multiple experts must agree upon. 60 d 68 d 76 d Cold Nice Warm Hot

Short Medium Tall 1 Membership Value 0.5 4 5 6 7 Height in Feet An individual at 5’5 feet would be said to be a member of “medium” persons with a membership value of 1, and at the same time, a member of “short” and “tall” persons with a value of 0.25. Fuzzy Rule: IF The person’s height is tall THEN The person’s weight is heavy A fuzzy rule maps fuzzy sets to fuzzy sets

Fuzzy Sets Fuzzy sets are used to provide a more reasonable interpretation of linguistic variables A fuzzy set assigns membership values between 0 and 1 that reflects more naturally a member’s association with the set A fuzzy set is an extension of the traditional set theory That generalizes the membership concept by using the Membership function that returns a value between 0 and 1 that represents the degree of membership an object x has to set A.

Employing Fuzzy Rules Conventional expert system - when a condition becomes true, the rule fires. Fuzzy expert system - if the condition is true to any degree, the rule fires. Example rules: If the room is hot, circulate the air a lot If the room is cool, leave the air alone If the room is cool and moist, circulate the air slightly

Fuzzy Expert System Process Fuzzification -- convert data to fuzzy sets Inference -- fire the fuzzy rules Composition -- combine all the fuzzy conclusions to a single conclusion Different fuzzy rules might conclude that the air needs different circulation levels Defuzzification -- convert the final fuzzy conclusion back to raw data

Fuzzy Logic vs. Probability Theory Probability = likelihood that a future event will occur probability event is in a set Fuzzy Logic = measures ambiguity of event that has already occurred degree of membership in a set

Weaknesses Limitations of Fuzzy Logic: Increases complexity of the expert system For large systems, fuzzy logic might be horribly inefficient -- combining with conventional logic is often difficult Validation and verification can be complex

Image Interpretation The process of labeling image data, typically in the form of image regions or features, with respect to domain knowledge Centers on the problem of how extracted image features are bound to domain knowledge All image interpretation methods rely to some extent on image segmentation and feature extraction

Image Segmentation Boundary-driven methods extract features such as edges, lines, corners or curves that are typically derived via filtering models which model or regularize differential operators in various ways Region-based methods typically involve clustering, region growing, or statistical models Methods can be combined into a hierarchical feature extraction/segmentation model which partitions images into regions as a function of how these partitions can minimize the statistical variations within feature regions

Seed Segmentation 1-Compute the histogram 2-Smooth the histogram by averaging to remove small peaks 3-Identify candidate peaks and valleys 4-Detect good peaks by peakiness test 5-Segment the image using thresholds 6-Apply connected component algorithm

What next? Use fuzzy logic to do segmentation Use fuzzy region growing to do segmentation Compare the results of the two methods Compare results with other non-fuzzy methods