Tutorial: Using Convolutional Neural Networks to Detect Object Keypoints
Horizontal’Transformations’( )’ KeyPoints’ summary page.pdf · Microsoft Word -...
Transcript of Horizontal’Transformations’( )’ KeyPoints’ summary page.pdf · Microsoft Word -...
Parent Functions 𝒇(𝒙) and the Transformed Function 𝒈(𝒙)
𝑓(𝑥) = 𝑥! à 𝑔(𝑥) = 𝑎[𝑘(𝑥 − 𝑑)]! + 𝑐
𝑓(𝑥) = √𝑥 à 𝑔(𝑥) = 𝑎!𝑘(𝑥 − 𝑑) + 𝑐
𝑓(𝑥) = !
! à 𝑔(𝑥) = !
!(!!!)+ 𝑐
𝑓 𝒙 = 𝒙 0 0 1 1 4 2 9 3
𝑓 𝒙 =𝟏𝒙
-‐2 -‐0.5 -‐1 -‐1 -‐0.5 -‐2 0 undef. 0.5 2 1 1
Transformations of Functions
𝒈(𝒙) = 𝒂𝒇[𝒌(𝒙 − 𝒅)] + 𝒄
Inverse Algebraically 1) change 𝑓(𝑥) to 𝑦 2) swap the 𝑥 and the 𝑦 3) re-‐arrange equation to isolate 𝑦 4) change 𝑦 to 𝑓!!(𝑥)
Key Points
Inverse: 𝒇!𝟏(𝒙)
Example: 𝑓(𝑥) = 3𝑥 + 5 𝑦 = 3𝑥 + 5 𝑥 = 3𝑦 + 5 𝑥 − 5 = 3𝑦 𝑥 − 53
= 𝑦
𝑓!!(𝑥) =𝑥 − 53
Inverse Graphically -‐ To draw the inverse of a function all you need to do is swap the 𝑥 and 𝑦 coordinates of each point.
Horizontal Transformations (𝒙)
STRETCH/COMPRESS/REFLECT (𝒌 value) Horizontal Stretches (multiply x-‐values by !
! )
-‐ stretch if −1 < 𝑘 < 1 by a factor of !!
Horizontal Compression (multiply x-‐values by !
! OR divide x-‐values by 𝑘)
-‐ compression if 𝑘 > 1 or 𝑘 < −1 b.a.f.o !!
Horizontal Reflection across the y-‐axis if 𝑘 < 0 (change sign of all 𝑥 values)
SHIFT LEFT/RIGHT (𝒅 value) Shift Right if 𝑑 > 0 (add 𝑑 value to x-‐coordinates) Shift Left if 𝑑 < 0 (add 𝑑 value to x-‐coordinates)
Vertical Transformations (𝒚)
SHIFT UP/DOWN (𝒄 value) Shift UP if 𝑐 > 0 (add 𝑐 value to y-‐coordinates) Shift DOWN if 𝑐 < 0 (add 𝑐 value to y-‐coordinates)
STRETCH/COMPRESS/REFLECT (𝒂 value) Vertical Stretches (multiply y-‐values by 𝑎) -‐ stretch if 𝑎 > 1 or 𝑎 < −1 by a factor of 𝑎 Vertical Compression (multiply y-‐values by 𝑎) -‐ compression if −1 < 𝑎 < 1 by a factor of 𝑎 Vertical Reflection across the x-‐axis if 𝑎 < 0 (change sign of all 𝑦 values)