Pattern Recognition - Chapter 4: Discriminant function with Gaussian distribution
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Equation (1)
![](https://static.wixstatic.com/media/64c52b_f035eef659fe41c49a3ee56bd1f44e2f~mv2.png/v1/fill/w_511,h_90,al_c,q_85,enc_auto/64c52b_f035eef659fe41c49a3ee56bd1f44e2f~mv2.png)
![](https://static.wixstatic.com/media/64c52b_1f520e4667f24b429e9c3400fd6a2a3a~mv2.png/v1/fill/w_491,h_91,al_c,q_85,enc_auto/64c52b_1f520e4667f24b429e9c3400fd6a2a3a~mv2.png)
![](https://static.wixstatic.com/media/64c52b_03020cbd209d434e8033a61c7486893f~mv2.png/v1/fill/w_374,h_43,al_c,q_85,enc_auto/64c52b_03020cbd209d434e8033a61c7486893f~mv2.png)
1. Independent features with the same (common) covariance
![](https://static.wixstatic.com/media/64c52b_d86dce075cd049ea9b3b0ef3faa75a8c~mv2.png/v1/fill/w_68,h_36,al_c,q_85,enc_auto/64c52b_d86dce075cd049ea9b3b0ef3faa75a8c~mv2.png)
![](https://static.wixstatic.com/media/64c52b_1c0d9c7c28a341dca08e1e2e93d2a063~mv2.png/v1/fill/w_112,h_30,al_c,q_85,enc_auto/64c52b_1c0d9c7c28a341dca08e1e2e93d2a063~mv2.png)
![](https://static.wixstatic.com/media/64c52b_97de676c2a274bfbbcd114b09089e9bd~mv2.png/v1/fill/w_87,h_49,al_c,q_85,enc_auto/64c52b_97de676c2a274bfbbcd114b09089e9bd~mv2.png)
![](https://static.wixstatic.com/media/64c52b_13d18fe3ee9340a7956df5a1f9af8b17~mv2.png/v1/fill/w_402,h_54,al_c,q_85,enc_auto/64c52b_13d18fe3ee9340a7956df5a1f9af8b17~mv2.png)
![](https://static.wixstatic.com/media/64c52b_65f5aba816c8474c8157caa3d45c4933~mv2.png/v1/fill/w_257,h_55,al_c,q_85,enc_auto/64c52b_65f5aba816c8474c8157caa3d45c4933~mv2.png)
![](https://static.wixstatic.com/media/64c52b_84694b4fc66740e983f7cf3e49252300~mv2.png/v1/fill/w_320,h_70,al_c,q_85,enc_auto/64c52b_84694b4fc66740e983f7cf3e49252300~mv2.png)
![](https://static.wixstatic.com/media/64c52b_c7fb52c6a5ae436fbe7f5d3eeccd35b5~mv2.png/v1/fill/w_306,h_285,al_c,q_85,enc_auto/64c52b_c7fb52c6a5ae436fbe7f5d3eeccd35b5~mv2.png)
(*Source: Tso B. and Mather P. Classification Methods for Remotely Sensed Data)
![](https://static.wixstatic.com/media/64c52b_6d6898df4b4d4af390f89ea4dd72845d~mv2.png/v1/fill/w_268,h_43,al_c,q_85,enc_auto/64c52b_6d6898df4b4d4af390f89ea4dd72845d~mv2.png)
![](https://static.wixstatic.com/media/64c52b_e0546d6092d7468a9bbbe9e7fc752637~mv2.png/v1/fill/w_217,h_106,al_c,q_85,enc_auto/64c52b_e0546d6092d7468a9bbbe9e7fc752637~mv2.png)
![](https://static.wixstatic.com/media/64c52b_ff33de8073604b26ad1c0deb5ac4f2cb~mv2.png/v1/fill/w_640,h_213,al_c,q_85,enc_auto/64c52b_ff33de8073604b26ad1c0deb5ac4f2cb~mv2.png)
(*Source: Richard O.Duda et al. Pattern Recognition)
2. Dependent features with the same (common) covariance
![](https://static.wixstatic.com/media/64c52b_b8c49fc7f6d94c2aba306052dc3cd424~mv2.png/v1/fill/w_59,h_32,al_c,q_85,enc_auto/64c52b_b8c49fc7f6d94c2aba306052dc3cd424~mv2.png)
![](https://static.wixstatic.com/media/64c52b_9f420f001eb5437eae13ba920be54877~mv2.png/v1/fill/w_425,h_48,al_c,q_85,enc_auto/64c52b_9f420f001eb5437eae13ba920be54877~mv2.png)
![](https://static.wixstatic.com/media/64c52b_b29f2a0ebc8a4d2595d1d0373dd5a758~mv2.png/v1/fill/w_319,h_41,al_c,q_85,enc_auto/64c52b_b29f2a0ebc8a4d2595d1d0373dd5a758~mv2.png)
![](https://static.wixstatic.com/media/64c52b_20daf235be8b429a95ad5d867d62ea4c~mv2.png/v1/fill/w_471,h_50,al_c,q_85,enc_auto/64c52b_20daf235be8b429a95ad5d867d62ea4c~mv2.png)
![](https://static.wixstatic.com/media/64c52b_7f362c3e6e694ac8aed21a29bd7e392e~mv2.png/v1/fill/w_108,h_46,al_c,q_85,enc_auto/64c52b_7f362c3e6e694ac8aed21a29bd7e392e~mv2.png)
![](https://static.wixstatic.com/media/64c52b_ec21749770954ffd84be8f94094a2a26~mv2.png/v1/fill/w_182,h_56,al_c,q_85,enc_auto/64c52b_ec21749770954ffd84be8f94094a2a26~mv2.png)
![](https://static.wixstatic.com/media/64c52b_57fd9686085d4c5bb83aa2b5c61720f6~mv2.png/v1/fill/w_172,h_39,al_c,q_85,enc_auto/64c52b_57fd9686085d4c5bb83aa2b5c61720f6~mv2.png)
![](https://static.wixstatic.com/media/64c52b_0f1b7fa192af4e9fb304e87741f2e97d~mv2.png/v1/fill/w_403,h_45,al_c,q_85,enc_auto/64c52b_0f1b7fa192af4e9fb304e87741f2e97d~mv2.png)
![](https://static.wixstatic.com/media/64c52b_992fdc203c5c4e24b7293bf98cf86ca4~mv2.png/v1/fill/w_310,h_35,al_c,q_85,enc_auto/64c52b_992fdc203c5c4e24b7293bf98cf86ca4~mv2.png)
![](https://static.wixstatic.com/media/64c52b_da9dbdce78ab4991a15165fd435d0804~mv2.png/v1/fill/w_236,h_75,al_c,q_85,enc_auto/64c52b_da9dbdce78ab4991a15165fd435d0804~mv2.png)
3. Dependent features with the scaled covariance
![](https://static.wixstatic.com/media/64c52b_291b70b272b94713bb14068665b86a5f~mv2.png/v1/fill/w_73,h_28,al_c,q_85,enc_auto/64c52b_291b70b272b94713bb14068665b86a5f~mv2.png)
![](https://static.wixstatic.com/media/64c52b_7e7a596fad67420f9438d1d0a06124a1~mv2.png/v1/fill/w_463,h_39,al_c,q_85,enc_auto/64c52b_7e7a596fad67420f9438d1d0a06124a1~mv2.png)
![](https://static.wixstatic.com/media/64c52b_d4a3de80fe734826bf2c3d2549a26a1c~mv2.png/v1/fill/w_103,h_59,al_c,q_85,enc_auto/64c52b_d4a3de80fe734826bf2c3d2549a26a1c~mv2.png)
![](https://static.wixstatic.com/media/64c52b_5298ed3a1e734d7a8ac000854531e4cc~mv2.png/v1/fill/w_386,h_50,al_c,q_85,enc_auto/64c52b_5298ed3a1e734d7a8ac000854531e4cc~mv2.png)
![](https://static.wixstatic.com/media/64c52b_332a5cc377534ff698148f3a6e1f61f7~mv2.png/v1/fill/w_468,h_49,al_c,q_85,enc_auto/64c52b_332a5cc377534ff698148f3a6e1f61f7~mv2.png)
![](https://static.wixstatic.com/media/64c52b_7f43c344c56d4d17a750f23f43a4c732~mv2.png/v1/fill/w_401,h_49,al_c,q_85,enc_auto/64c52b_7f43c344c56d4d17a750f23f43a4c732~mv2.png)
![](https://static.wixstatic.com/media/64c52b_ef800a6b59684d668bc91f5de242f157~mv2.png/v1/fill/w_320,h_101,al_c,q_85,enc_auto/64c52b_ef800a6b59684d668bc91f5de242f157~mv2.png)
4. Dependent features with the arbitrary covariance
![](https://static.wixstatic.com/media/64c52b_cedd20522df940caa3b92f7b82aee72b~mv2.png/v1/fill/w_432,h_47,al_c,q_85,enc_auto/64c52b_cedd20522df940caa3b92f7b82aee72b~mv2.png)
![](https://static.wixstatic.com/media/64c52b_db1006fee8704027bb4a63dc99646a5f~mv2.png/v1/fill/w_526,h_42,al_c,q_85,enc_auto/64c52b_db1006fee8704027bb4a63dc99646a5f~mv2.png)
![](https://static.wixstatic.com/media/64c52b_8b112fc0515f4bc5b41d736433984375~mv2.png/v1/fill/w_520,h_51,al_c,q_85,enc_auto/64c52b_8b112fc0515f4bc5b41d736433984375~mv2.png)
![](https://static.wixstatic.com/media/64c52b_b366bf7c086c44968d2dd843b3962ab9~mv2.png/v1/fill/w_432,h_117,al_c,q_85,enc_auto/64c52b_b366bf7c086c44968d2dd843b3962ab9~mv2.png)
![](https://static.wixstatic.com/media/64c52b_c81284fbe14249abb56824f4e12e9539~mv2.png/v1/fill/w_266,h_265,al_c,q_85,enc_auto/64c52b_c81284fbe14249abb56824f4e12e9539~mv2.png)
![](https://static.wixstatic.com/media/64c52b_e08199fc289641deb4bdd043e3d3bc51~mv2.png/v1/fill/w_216,h_162,al_c,q_85,enc_auto/64c52b_e08199fc289641deb4bdd043e3d3bc51~mv2.png)
![](https://static.wixstatic.com/media/64c52b_239d0df873fb4b5087eaad6608ba5585~mv2.png/v1/fill/w_529,h_260,al_c,q_85,enc_auto/64c52b_239d0df873fb4b5087eaad6608ba5585~mv2.png)
(*Source: Richard O.Duda et al. Pattern Recognition)