THÈS EE - CESBIO - Université Toulouse III - Paul Sabatier

Chapitre 3. Article 1- p(B), the a priori probability of class B,- d, a measurement of the distance between the retained classification threshold r t and thethreshold r 0 (optimal threshold when the two classes are equiprobable).1) Equal a priori probabilitiesWhen p(A)=p(B), the retained classification threshold r t is optimal when equal to r 0 , thereforeparameter d=r t /r 0 equals 1, and the expression of the probability of error reduces to:PE = hL ∆r(11)( )Figure 2 presents the probability of error PE as a function of ∆r, for different values of L,when the classification threshold is r 0 .The figure can be used to assess the number of looks required to achieve a given accuracy inthe classification when r A and r B are known.Figure 2 can also be used to assess the accuracy that can be expected when the systemparameters (number of looks) and class characteristics (r A and r B ) are known. Unsurprisingly,the error decreases when L and ∆r increase.5045PE (%)403530252015105L=1L=2L=4L=8L=16L=32L=64L=12800 1 2 3 4 5 6 7 8∆r (dB)Figure 2. Probability of error (in %) of the ratio method as a function of the change inintensity ratio ∆r (in decibels) between the two classes, for a number of looks L varyingbetween 1 and 128.2) General caseWe can derive from (4) that the optimal threshold is found for a specific value of parameter d:12L⎛1−p(B)⎞∆r⎜ ⎟ −1p(B)d = dopt= roptr0=⎝ ⎠(12)1⎛1−p(B)⎞ 2L∆r− ⎜ ⎟⎝ p(B)⎠Thus, for the general case, the optimal classification threshold depends on r 0 , L, ∆r and p(B).However, when the method is to be used in several different scenes, p(B) is not known inmost cases. It is then suggested to use r 0 as a threshold in the classification scheme. Theadditional classification error compared to the optimal case needs to be assessed, in order tomake sure that such a practice is acceptable.Figure 3 shows the additional error due to the use of r 0 rather than r opt , as a function of the77