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

• A. Erol Fazlıoğlu

Problem?Problem?

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

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• 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.

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• 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.

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.

Our Approach, Cure?

• Melanoma is easily cured if detected at an early stage.

Aim

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

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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.

Refreshing our memories on empirical approximation issues -2

• The task now is to estimate f(x) for arbitrary x by, in some sense, drawing a smooth curve through the x i. If the desired x is in between the largest and smallest of the xi’s, the problem is called interpolation; if x is outside that range, it is called extrapolation,

Refreshing our memories on empirical approximation issues -3

• There is an extensive mathematical literature devoted to theorems about what sort of functions can be well approximated by which interpolating functions. These theorems are, alas, almost completely useless in day-to-day work: If we know enough about our function to apply a theorem of any power, we are usually not in the pitiful state of having to interpolate on a table of its values!

Distinction between Interpolation & Function Approximation

• Interpolation is related to, but distinct from, function approximation. That task consists of finding an approximate (but easily computable) function to use in place of a more complicated one. In the case of interpolation, you are given the function f at points not of your own choosing. For the case of function approximation, you are allowed to compute the function f at any desired points for the purpose of developing your approximation.

Spline Functions

• In situations where continuity of derivatives is a concern, one must use the “stiffer” interpolation provided by a so-called spline function.

• Cubic splines are the most popular. They produce an interpolated function that is continuous through the second derivative.

• Splines tend to be stabler than polynomials, with less possibility of wild oscillation between the tabulated points.

Polynomial orders

Choosing the best technique

• Simple Spline(Rank 3),

• Base Spline,

• Natural Spline,

• Bezier

• De Casteljau's algorithm

Cubic Spline Interpolation

• Given a tabulated function yi = y(xi), i = 1...N , focus attention on one particular interval, between xj and xj+1. Linear interpolation in that interval gives the interpolation formula

y = Ayj + Byj+1

These two equations are a special case of the general Lagrange interpolation.

The Goal of Cubic Spline Interpolation

• Since it is (piecewise) linear, first equation has zero second derivative in the interior of each interval, and an undefined, or infinite, second derivative at the abscissas xj .

• The goal of cubic spline interpolation is to get an interpolation formula that is smooth in the first derivative, and continuous in the second derivative, both within an interval and at its boundaries.

Cubic Spline Interpolation

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.

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, computed from first derivation equation, be continuous across the boundary between two intervals.

• The key idea of a cubic spline is to require this continuity and to use it to get equations for the second derivatives

Bringing to Light of Spline Function

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

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Thanks for your attention!Thanks for your attention!

• Any questions?

• A. Erol Fazlıoğlu