Den här bilden är suddig, men hennes leende är så fint att jag ändå vill spara den.

Because bilden is the definite form of bild.

• en bild = a picture
• bilden = the picture

In Swedish, when you say this/that with a noun, you usually use:

• den här + definite noun for common-gender words
• det här + definite noun for neuter words

So:

• den här bilden = this picture
• not den här bild

This is a very common pattern in Swedish.

Den här means this.

It is used with en-words (common gender):

• den här bilen = this car
• den här bilden = this picture

Compare:

• det här huset = this house (for a t-word / neuter noun)

So in the sentence, Den här bilden means This picture.

Swedish uses sin/sitt/sina when the possessor is the subject of the same clause.

Here, the clause is:

• hennes leende är så fint = her smile is so lovely

The subject is leende (smile), not hon (she). Since the possessor is not a reflexive subject in this clause, Swedish uses hennes = her.

Compare:

• Hon tvättar sitt hår. = She is washing her own hair.
• Jag gillar hennes leende. = I like her smile.

So hennes is correct here.

Because leende is a neuter noun (ett leende), and adjectives match the gender of the noun.

• en fin bild = a beautiful picture
• ett fint leende = a beautiful smile

In the sentence:

• hennes leende är så fint = her smile is so lovely

So the -t on fint is there because leende is an ett-word.

Yes, here leende is a noun.

It comes from the verb le = to smile, but ett leende means a smile.

So:

• le = to smile
• ett leende = a smile

This is similar to English forms like a smile from to smile, although the Swedish word is built differently.

In the sentence:

• hennes leende = her smile

Suddig means blurry, fuzzy, or out of focus.

Here:

• bilden är suddig = the picture is blurry

It is an adjective. You may also see forms like:

• en suddig bild = a blurry picture
• ett suddigt foto = a blurry photo
• suddiga bilder = blurry pictures

So the form changes depending on the noun, just like many Swedish adjectives do.

Så ... att means so ... that.

In the sentence:

• så fint att jag ändå vill spara den
• literally: so lovely that I still want to save it

This structure is very common in Swedish:

• Hon är så trött att hon somnar direkt. = She is so tired that she falls asleep immediately.

• Det var så kallt att vi gick hem. = It was so cold that we went home.

So here, the smile is so lovely that the speaker still wants to keep the picture.

Here ändå means still, anyway, or all the same.

The idea is:

• the picture is blurry
• but despite that, the speaker wants to save it

So:

• jag ändå vill spara den = I still want to save it / I want to save it anyway

It adds a sense of contrast: something negative is true, but the speaker’s decision remains the same.

Den refers back to bilden (the picture).

Since bild is an en-word, the object pronoun is den:

• en bild → den
• ett foto → det

So:

• jag vill spara den = I want to save it
• and it here means the picture

If the noun were neuter, you would use det instead.

Because in Swedish subordinate clauses, words like inte, alltid, ofta, ändå, and similar sentence adverbs usually come before the finite verb.

Main clause:

• Jag vill ändå spara den.

Subordinate clause after att:

• ... att jag ändå vill spara den

This is a very important Swedish word order rule.

Compare:

• Main clause: Hon kommer inte idag.
• Subordinate clause: ... att hon inte kommer idag

So att jag ändå vill spara den is the normal and correct order.

Because men means but, and it connects two main clauses:

• Den här bilden är suddig
• men hennes leende är så fint att jag ändå vill spara den

Using a comma before men is very normal in Swedish writing, especially when the clauses are fairly complete and clear on their own.

Even if punctuation rules can vary a little by style, this comma is natural and helpful here.

Yes:

• Den här = this
• bilden = picture / the picture
• är = is
• suddig = blurry
• men = but
• hennes = her
• leende = smile
• är = is
• så = so
• fint = lovely / beautiful
• att = that
• jag = I
• ändå = still / anyway
• vill = want
• spara = save / keep
• den = it

So the full idea is:

This picture is blurry, but her smile is so lovely that I still want to save it.

Yes. Spara can mean several related things, including:

• save a file or picture
• keep something instead of deleting or throwing it away
• save up money, in other contexts

In this sentence, since it is about a picture, spara den could naturally mean:

• save it if talking about a digital image
• keep it if talking more generally

So the exact English choice depends on context, but the Swedish word spara works well for both ideas here.