SKIN LESION BORDER DETERMINATION IN DERMATOLOGIC CLINICAL IMAGES USING SPLINE CURVE TECHNIQUE

1. SKIN LESION BORDER DETERMINATION IN DERMATOLOGIC CLINICAL IMAGES USING SPLINE CURVE TECHNIQUE • A. Erol Fazlıoğlu

2. Problem? • A significant number of malignant melanomas(skin tumors), especially early melanomas curable by excision, are not diagnosed correctly in the clinical setting.

3. Findings-1 • The diagnostic sensitivity reported for unaided dermatologist observers ranges from a low of about 66% to about 81%. • Diagnostic accuracy for non-dermatologists is believed to be lower.

4. Findings-2 • The relatively low diagnostic sensitivity of dermatologist and non-dermatologist detection of malignant melanoma demonstrates the uncertainty involved in skin lesion analysis. Paramount in the process of skin lesion analysis is the identification of features that can be consistently interpreted by dermatologists and nondermatologists in the recognition of abnormal skin lesions.

5. Status in the USA • For 2002, there are 53,600 new cases and 7400 deaths estimated from malignant melanoma in the United States. This is a 4% increase in invasive melanoma from 2001.

6. Our Approach, Cure? • Melanoma is easily cured if detected at an early stage.

7. Aim • In order to determine the skin lesion borders, spline curve function will be taken into consideration.

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10. Refreshing our memories on empirical approximation issues -1 • We sometimes know the value of a function f(x) at a set of points x1, x2, . . . , xN (say, with x1 < . . . < xN), but we don’t have an analytic expression for f(x) that lets us calculate its value at an arbitrary point. For example, the f(xi)’s might result from some physical measurement or from long numerical calculation that cannot be cast into a simple functional form. Often the xi’s are equally spaced, but not necessarily.

11. Refreshing our memories on empirical approximation issues -2 • The task now is to estimate f(x) for arbitrary x by, in some sense, drawing asmooth curve through the x i. If the desired x is in between thelargest and smallest of the xi’s, the problem is called interpolation; if x is outsidethat range, it is called extrapolation,

12. Refreshing our memories on empirical approximation issues -3 • There is an extensive mathematical literature devoted to theorems about whatsort of functions can be well approximated by which interpolating functions. Thesetheorems are, alas, almost completely useless in day-to-day work: If we knowenough about our function to apply a theorem of any power, we are usually not inthe pitiful state of having to interpolate on a table of its values!

13. Distinction between Interpolation & Function Approximation • Interpolation is related to, but distinct from, function approximation. That taskconsists of finding an approximate (but easily computable) function to use in placeof a more complicated one. In the case ofinterpolation, you are given the function fat points not of your own choosing. For the case of function approximation, you areallowed to compute the function f at any desired points for the purpose of developingyourapproximation.

14. Spline Functions • In situations where continuity of derivatives is a concern, one must usethe “stiffer” interpolation provided by a so-called spline function. • Cubic splinesare the most popular. They produce an interpolated function that is continuousthrough the second derivative. • Splines tend to be stabler than polynomials, with lesspossibility of wild oscillation between the tabulated points.

15. Polynomial orders

16. Choosing the best technique • Simple Spline(Rank 3), • Base Spline, • Natural Spline, • Bezier • De Casteljau's algorithm

17. Cubic Spline Interpolation • Given a tabulated function yi = y(xi), i = 1...N , focus attention on oneparticular interval, between xj and xj+1. Linear interpolation in that interval givesthe interpolation formula y = Ayj + Byj+1 These two equations are a special case of the general Lagrange interpolation.

18. The Goal of Cubic Spline Interpolation • Since it is (piecewise) linear, first equation has zero second derivative inthe interior of each interval, and an undefined, or infinite, second derivative at theabscissas xj . • The goal of cubic spline interpolation is to get aninterpolation formulathat is smooth in the first derivative, and continuous in the second derivative, bothwithin an interval and at its boundaries.

19. Cubic Spline Interpolation

20. Taking derivatives of y function with respect to x, using the definitions of A,B,C,D to compute dA/dx, dB/dx, dC/dx, and dD/dx. • The result, for the first derivative is And, for the second derivative.

21. Problem with Cubic Spline Interpolation • The only problem now is that we supposed the ’s to be known, when, actually,they are not. However, we have not yet required that the first derivative, computedfrom first derivation equation, be continuous across the boundary between two intervals. • Thekey idea of a cubic spline is to require this continuity and to use it to get equationsfor the second derivatives

22. Bringing to Light of Spline Function • The required equations are obtained by setting equation (first derivation) evaluated forx = xj in the interval (xj-1, xj) equal to the same equation evaluated for x = xj butin the interval (xj, xj+1). With some rearrangement, this gives (for j = 2, . . . , N-1)

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24. Thanks for your attention! • Any questions? • A. Erol Fazlıoğlu