Unwanted golf ball on iMac Catalina log in screen

Open System Preferences. When you hover over the picture next to your name, it will say "Edit". Click it and you can select the pic you want. Or just drag the pic you want over it until you see the green plus symbol and drop it.

My post did not direct you to replace your "desktop picture."

In order to follow the instructions previously provided, you have to go to Menu>System Preferences...>Users & Groups>Current User>Password...and change to the picture of your choice.

Change a user’s picture on Mac

"You can change the picture that appears next to a user name in the login window.

1. On your Mac, choose Apple menu  > System Preferences, then click Users & Groups.
2. Click the lock icon  to unlock it.
3. Enter an administrator name and password. You can change your own picture without unlocking Users & Groups preferences.
4. Select the user account on the left, then click the picture on the right.
5. Instead of clicking the picture, you can drag an image file from the Finder  onto the picture.
6. If other users are logged in to this Mac now, you can’t select them.
7. Do one of the following:

• • Use a picture included with macOS: Click Defaults, then select a picture.
  • Take your picture using your Mac: Click Camera; when you’re ready, click Take Photo.
  • Use a picture from the Photos app: Click Photos. To see specific photos from a time, place, or album, click a disclosure triangle below photos, then select a group of photos. Select a picture, then click Next.

1. Adjust the picture.

• • Zoom in or out: Drag the slider left or right.
  • Move the picture: Drag the picture around within the circle."

According to: Change Users & Groups User preferences on Mac - Apple Support

"User picture

Click the picture to edit it, then select a new picture. 

• Use a picture included with macOS: Click Defaults, then select a picture.
• Take your picture using your Mac: Click Camera; when you’re ready, click Take Photo.
• Use a picture from the Photos app: Click Photos. To see specific photos from a time, place, or album, click a disclosure triangle below photos, then select a group of photos. Select a picture, then click Next.

Instead of clicking the picture, you can drag an image file from the Finder onto the picture."

There's also a golf ball in my emails where most people have a their initials. I have gmail.

FWIW neither the golf ball or any other picture will be seen by recipients of your emails unless they have that picture chosen for you in their own contacts.

Hopefully it worked for you. Of course I didn't have to go through all that.