of for 分別

This method can establish a good image of our company. ~ 對
He can cancel the participation of activity. ~ 錯

應該是:
He can cancel the participation in the activity.

首先看 noun 的 collocations 連語, 字的搭配, 可參考好的字典

image (n) of something (someone) 某事物 (某人) 的形象
participation (n) in something 參與某事
image of the company = company's image

例子:
The aim is to improve the public image of the company.
Participation in the program was voluntary.

The chairman's participation in fund-raising for poor children will improve the image of the company.

even 用 participate (verb) 都是用 in
participate in something

Prepositions:

OF (preposition)
"of" is used to indicate possession. It can also used to indicate other kinds of relationship between people or things.

IN (preposition)
You can use "in" as a preposition to say where someone/something is, or where someone/something goes.
in + place
in + abstract noun

FOR (preposition), 為
If something is for someone, they are intended to have it or benefit from it.

The new chairman will reveal his plans for the company.

This method can establish a good image of our company. - OK
He can cancel the participation IN AN activity


OF
The window of the house is white in colour
He jumps out of the window of his home
The tail of dog is injured.
He was standing in front of me when the firework started.


For
It may be harmful for health so do not eat it.
It is for Mary but not you.
We are not going out today for (= because) the weather forecast it will rain heavily.
The shirt is too big for you.
The table is too heavy for you to lift up.
The test is difficult for all of us.